Module 1:  Preliminary Assignment - Assignment 10

Major Quiz over Module 1 is assigned as part of Assignment 10

Orientation as instructed, including Transition to Physics II , which includes Timer, error analysis I and II, fitting a straight line, hypothesis testing time intervals, Fundamental Systems

Asst

Introductory Problem Set

lab/activity

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outline

Class Notes

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query

Fluids

·         Density, pressure, energy relationships, continuity and Bernoulli’s Equation.

·         Buoyancy

·         Viscosity and surface tension.

Thermal Energy and Thermodynamics

·         Specific heat, calorimetry, latent heat, energy conservation.

·         Kinetic theory of gases, ideal gas laws.

·         First and second laws of thermodynamics.

·         Analysis of multi-cycle heat engines.

Symbols used:

• `d or `delta:  D
• `lambda:      l
• `omega:      w
• `rho:           r
• `theta:         q
• `mu:             m
• `pi:               p
• `phi:             f
• `gamma:      g
• `sigma:        s
• `Sigma:       S

Note that symbols might not be correctly represented by your browser.  For this reason the Greek letters will be spelled out, with ` in front of the spelling.  You should substitute the appropriate symbol when making notes.

Suggestion:  If you want to memorize formulas, the formula at the beginning of each bulleted paragraph is probably worth the effort.   Those formulas that aren't worth the effort are enclosed in parentheses.

Fluids:

Relate {(t_i,x_i) | 0 <= i <= n} U {`dt_i, `dx_i | 1 <= i <= n} U {definition of average rate} U {average rate of change of position with respect to clock time on ith interval, vAve_i, average velocity on ith interval} U {t_mid_i, `dt_mid_i, approximate average rate of change of velocity between midpoints, approximate acceleration at t_i, a_Ave_i | 1 <= i <= n-1}  knowledge of Physics I and prerequisite mathematics

• P = F / A (definition of pressure). Use if you know pressure and force, or force and area, or area and pressure.

• `rho g h + .5 `rho v^2 + P is constant (Bernoulli's Eqn). If density `rho and gravitational acceleration g are constant, as with water near Earth's surface, we see that fluid velocity v, fluid altitude h and fluid pressure P vary in such a way that if one term goes up something else has to go down to compensate. Usually one of the three quantities will be constant so that one will go up and the other will go down. For water `rho = 1000 kg / m^3; for air `rho is about 1.4 kg/m^3.

• A1 * v1 = A2 * v2:  For a confined fluid, A1 * v1 = A2 * v2 (continuity equation for incompressible fluids—the amount flowing past one point is equal to the amount flowing past another). Ratio of velocities inverse to ratio of areas: v2 / v1 = A1 / A2. Remember area proportional to the square of the diameter or of the radius.

Thermal Energy Transfers and Materials:

• `dQ = m c `dT:  The specific heat c is typically given as number of Joules per kg, per Celsius degree. Thermal energy required to raise a sample is specific heat, multiplied by number of kg, multiplied by number of degrees temp change in Celsius. In equation form, `dQ = m c `dT.

• `dQ / `dt = k A ( `dT / `dx):  The rate of thermal energy transfer is proportional to temp gradient `dT / `dx and temp difference: `dQ / `dt = k A ( `dT / `dx). Note t is time and T is temp. `dx is typically thickness of container. `dQ / `dt is energy transfer in Joules / sec.

Kinetic Theory of Gases:

One particle mass m, vel. v in cylinder length L, moving parallel to axis of cylinder exerts average force F = `d(mv) / `dt = 2 m v / ( 2 L / v) = m v^2 / L on either end.

• If area of end is A then:

pressure = F / A = m v^2 / (L * A) = m v^2 / V, where V is volume.

Noting

• If a single mass is divided into large number of particles and they still move parallel to the axis, there is still the same momentum change in same time interval so the above expression for pressure is not changed.

• KE = 3/2 (nRT):  A large number of particles wouldn't keep moving in the same direction because of collisions. The energy would rapidly get divided equally among three independent directions in space. Only one direction affects a given end of the cylinder, so now pressure is 1/3 as great as in the previous model:

·         P = 2/3 KE / V or PV = 2/3 KE

·         PV = n R T so n R T = 2/3 KE so KE = 3/2 (nRT)

(n is the number of moles. To find KE per particle divide by the number of moles to get the KE of 1 mole, then by Avagadro's Number to get the KE of 1 particle.)

(note for General College and especially University Physics students:   the KE referred to here is the KE of a single particle at at the rms velocity)

Gas Laws:

• PV = n R T with R = 8.31 J / (particle Kelvin) so, for example:

V2 / V1 = T2 / T1:  If P and n are constant then V is proportional to T and you can use 'straight ratios' V2 / V1 = T2 / T1 to calculate desired temp or volume.

V2 / V1 = P2 / P1:  If V and n are constant then P is proportional to T and you can use 'straight ratios' V2 / V1 = P2 / P1 to calculate desired pressure or volume.

P2 / P1 = V1 / V2:  If T and n are constant then P is inversely proportional to V and you can use inverse ratios P2 / P1 = V1 / V2 (note that one of the ratios is 'upside down') to calculate desired temp or volume.

If you know three of the four quantities P, V, n and T you can find the fourth.

Thermodynamics:

• When a system converts thermal energy to mechanical work as it runs through a repeating cycle, energy `dQin is transferred to the system from the 'hot source', `dQout is transferred out to the 'cold sink' and the system does work `dW.
• `dQin = `dW + `dQout:   Energy conservation tells us that `dQin = `dW + `dQout.
• efficiency = `dW / `dQin:  The efficiency of such a system is

efficiency = work done by system / thermal energy transfered into system, or

efficiency = `dW / `dQin .

• If we know two of the three quantities `dQin, `dQout and `dW we can find the third and hence compute efficiency.

The maximum possible efficiency of any such system is (Th – Tc) / Th, where Th and Tc are the temperatures of the hot and cold sources.  This efficiency is theoretically achievable only by a Carnot Cycle, which consists of two adiabatic processes bounded by two isothermals.

Fluids:

Analyze position vs. clock time data to obtain information about velocity and acceleration of the behavior of a fluid flowing from a hole in the side of a cylindrical container.

Where x_i is the vertical position of the fluid surface relative to the outflow hole and t_i the clock time at which this position occurs:  Relate {(t_i,x_i) | 0 <= i <= n} U {`dt_i, `dx_i | 1 <= i <= n} U {definition of average rate} U {average rate of change of position with respect to clock time on ith interval, vAve_i, average velocity on ith interval} U {t_mid_i, `dt_mid_i, approximate average rate of change of velocity between midpoints, approximate acceleration at t_i, a_Ave_i | 1 <= i <= n-1}  knowledge of Physics I and prerequisite mathematics

Analyze the behavior of a fluid flowing from a hole in the side of a cylindrical container. 

• rho is fluid density (assumed constant), y depth with respect to fluid surface, P gauge or absolute pressure as appropriate, A_cs_cyl the cross-sectional area of the cylinder, v_exit the velocity of fluid exiting the cylinder, v_surface the velocity of the fluid surface as it descends, A_exit the cross-sectional area of the exit hole, P_fluid the pressure inside the fluid at the altitude of the outflow and P_atm the pressure of the atmosphere (gauge or absolute, but the same for both), `dP the pressure difference between atmosphere and inside the fluid at the level of the outflow hole, and for a presumably short time interval  we have `dy the change in the vertical position of the fluid surface, `dV the change in the volume of fluid in the cylinder, `dm the change in the mass of the fluid in the cylinder, `dPE the change in the gravitational potential energy of the fluid in the cyilinder, `dKE difference between the kinetic energy of the mass `dm at the surface of the fluid and the same mass as it exits the cylinder, `dL the length of a cylinder of volume `dV and cross-sectional area A_exit.

Relate {rho, y, P} for vertical fluid column, by applying vertical equilibrium of forces. 

Relate {y, A_cs_cyl, `dy, `dV, `dm, `dWeight, rho, `dPE, `dKE, A_exit, v_exit | A_cs_cyl >> A_exit }

Relate {A_exit, `dL, `dP, rho, `dW, `dKE, v_exit}

Relate {P_fluid, P_atm, rho, v } for liquid flowing in absence of dissipative forces from a hole in a large container using Newton’s 2d Law, and using work-ke theorem

Relate {y_surface, y_exit, v_exit} for liquid flowing in absence of dissipative forces from a hole in a large container using energy conservation

Relate {P_fluid, P_atm, rho, v_surface, v_exit, y_surface, y_exit} using Bernoulli’s equation

Relate {P_1, P_2, y_1, y_2, v_1, v_2, rho_1, rho_2, A_1_cs, A_2_cs) using Bernoulli’s equation and continuity.

Thermal Energy Transfers and Materials:

Analyze flow of thermal energy.  dQ / dt is rate of thermal energy flow, k is thermal conductivity, T_1 and T_2 are temperatures at two parallel surfaces, L is the perpendicular distance between the surfaces,  A_cs is the area of either of the surfaces:

• Relate {dQ/dt, k, L, A_cs, dT/dx, T_1, T_2} qa_2, class notes

Analyze the changes in dimension of an object (composed of a single material of uniform density) due to temperature changes.  L_0, A_0 and V_0 are the length, area and volume of or enclosed by the object at temperature T_0, alpha the coefficient of linear expansion, beta the coefficient of volume expansion, `dT the change in temperature, `dL, `dA and `dV the changes in length, area and volume:

• Relate {`dL, L_0, `dT, T_0, T_f, alpha, A_0, `dA, V_0, `dv, beta} qa_2, class notes

Analyze the thermal change of an object composed of a single material of uniform density.  m is the mass of the object, c its specific heat, `dQ the thermal energy change, `dT the change in temperature, T_0 and T_f the initial and final temperature.

• Relate {c, m, `dQ, `dT, T_0, T_f} qa_2, class notes

Analyze the behavior of an isolated system consisting of n of objects at a number of initial temperatures.  For the ith object, c_i is its specific heat, m_i its mass, `dQ_i the change in its thermal energy, T_0_i its initial temperature, l_f_i its latent heat of fusion, l_v_i its latent heat of vaporization and T_f the final temperature approached by the system.

• Relate {c_i, m_i, l_f_i, l_v_i, T_0_i, T_f} qa_2, class notes

Kinetic Theory of Gases:

For a mass m moving at speed v colliding elastically with the ends of a cylinder of uniform cross-sectional area A_cs, where velocity is axial and all collisions are elastic.  F_ave is the average force exerted on one of the ends, P_ave the average pressure on that end, p the magnitude of the momentum of the particle, `dp the change in momentum at collision, `dt the time between collisions.

• Relate {m, v, p, `dp, `dt, F_ave}
• Relate {m, v, p, `dp, `dt, F_ave, A_cs, P_ave}

For a large number N of masses m all moving at identical speed v (assume that the particles do not collide with one another) colliding elastically with the walls and ends of a cylinder of uniform cross-sectional area A_cs, where particle velocities are random in 3 dimensions and all collisions are elastic.  F_ave is the average force exerted on one of the ends, P_ave the average pressure on that end. 

• Relate {m, N, v, L, F_ave, A_cs, P_ave}  

For a large number N of masses m having rms velocity v_rms (assume that the particles collide with one another with the resulting distribution of velocities), colliding elastically with one another and with the walls and ends of a cylinder of uniform cross-sectional area A_cs, where particle velocities are random in 3 dimensions and all collisions are elastic.  F_ave is the average force exerted on one of the ends, P_ave the average pressure on that end.  T is the Kelvin temperature of the system, n the number of moles, V the volume of the cylinder, C_p and C_v the molar specific heats at constant pressure and constant volume respectively, KE_ave the average kinetic energy of a particle, U the total internal energy of the system, and the masses are either monatomic or diatomic in nature, as specified:

• Relate {T, m_particle, KE_ave, v_rms, N, n, U, C_p, C_v, V, P_ave, monatomic/diatomic}

3 dimensions:

• {m, N, v, L, Fave, A_cs, P_ave, KE_ave, V, T}
• {direction of `dT, direction of `dKE, direction of `dP, direction of `dV, direction of `dN}
• {`dT as percent of T_0, `dP as percent of P_0, `dV as percent of V_0, `dN as percent of N_0, `dT/T_0, `dV/V_0, `dT/T_0, `dN/N_0 | percent changes are small }

 

Gas Laws:

Relate {P, V, n, T}, where P is the absolute pressure, V the volume, n the number of moles and T the absolute temperature of an ideal gas.

Relate {P_0, T_0, V_0, N_0, P_f, T_f, V_f, N_f}, where P is the absolute pressure, V the volume, n the number of moles and T the absolute temperature of an ideal gas and the subscript _0 indicates the value of the quantitiy in its initial state, while the subscript _f indicates the value of the quantity in its final state.

Bottle with Single tube: open 

• {A_cs,`dL, V_bottle, `dV_tube, `dV_tube/V_bottle , `dV_tube/V_bottle (as %) }
• {P, `dy, squeeze}
• {P, `dy, squeeze, P_atm}
• {P, `dy, squeeze, P_atm,T, V, `dV}

Tube only:   

• Atm pressure:  {rho_water, y2 – y1, L_0, `dL, (V / V_0)_air_column, P_atm}  

2-tube: 

• {P, V, T, y2 – y1, L_0, L}  

Thermodynamics:

For the system consisting of

• a container initially containing volume V_0 of an ideal gas (monatomic or diatomic, as specified) at pressure P_0 and temperature T_0

• a reservoir of water

• a thin vertical tube with one end in water, the other end open to the atmosphere at vertical position y_1 as measured relative to the surface of the water in the bottle

• a reservoir at vertical position y_1

• a hot sink at temperature T_f and a cold sink at temperature T_0

where P_1, V_1, T_1 constitute the state of the gas when water in the vertical tube is at vertical position y_1, P_2, T_2, V_2 the state when the temperature has reached T_f, `dQ the thermal energy flowing into the system between temperatures T_0 and T_f, `dQ_12 and `dQ_01 the thermal energy flowing into the system between the states indicated by the subscripts, `dPE the change in the gravitational potential energy of the system, `dU the change in the internal energy of the gas, `dW the work done by the gas as it expands at constant pressure, eff the 'practical efficiency' of the process (defined by `dPE / `dQ),  R the gas constant, C_p and C_v the molar specific heats of the gas at constant pressure and constant volume (respectively):

• Relate {P_0, V_0, T_0, n, y_max, y_1, P_1, V_1, T_1, T_f, P_2, V_2, T_2, `dQ, `dQ_12, `dQ_01, `dPE, `dU, eff, `dW, P vs. V graph, R, C_p, C_v}

For the same system taken through the first two cycles then returned to its original pressure by an adiabatic or isothermal expansion, as selected, where `dQ_h is now the total energy flowing into the system from the hot sink, `dQ_c the energy flowing into the cold sink, e the thermodynamic efficiency of the process and e_max the maximum possible thermodynamic efficiency achievable with the given hot and cold sinks:

• Relate {P_0, V_0, P_f, T_f, adiabatic\isothermal, `dW, `dU, `dQ_h, `dQ_c, e, e_max, P vs. V graph}

Asst

Introductory Problem Set

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Class Notes

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Lab Materials Form

Disk Contents

Disk Contents Experiments and Problem Sets

Transition to Physics II part of intro asst

query_0

blurry video uncertainties

Objectives

0.01 Observe a sequence of position vs. clock time events and present the results in a table.

Technically:

Observe for a sequence ot (t, x) events: {(t_i,x_i) | 0 <= i <= n}  knowledge of Physics I and prerequisite mathematics

·         x_i and t_i are the observed or actual position and clock time of an object moving along a line, relative to a fixed point on the line and an arbitrary instant of time.

·         n is the number of clock-time-and-position observations.

0.02  For an observed series of position vs. clock time events relate position vs. clock time information to displacements and time intervals between consecutive events.

Technically:

Relate {(t_i,x_i) | 0 <= i <= n} U {`dt_i, `dx_i | 1 <= i <= n}  knowledge of Physics I and prerequisite mathematics

·         `dx_i = x_i – x_(i-1) is called the displacement on the ith interval

·         | `dx_i | is the distance moved by the object corresponding to the ith interval

·         `dt_i is the time increment, or time interval, or change in clock time, corresponding to the ith interval

0.03  For an observed series of position vs. clock time events relate position vs. clock time information, displacements and time intervals between consecutive events, the definition of average rate of change, and the average rate of change of position vs. clock time for each interval.

Technically:

Relate {(t_i,x_i) | 0 <= i <= n} U {`dt_i, `dx_i | 1 <= i <= n} U {definition of average rate} U {average rate of change of position with respect to clock time on ith interval, vAve_i, average velocity on ith interval}  knowledge of Physics I and prerequisite mathematics

·         Average rate of change of A with respect to B on an interval is defined to be the change in A on the interval divided by the change in B on that interval.

·         The average velocity on an interval is defined to be the average rate of change of position with respect to clock time on that interval.

·         vAve_i stands for the average velocity on the ith interval.

0.04  Relate position vs. clock time information to average velocity vs. midpoint clock time and acceleration at each clock time.

Technically:

Relate {(t_i,x_i) | 0 <= i <= n} U {`dt_i, `dx_i | 1 <= i <= n} U {definition of average rate} U {average rate of change of position with respect to clock time on ith interval, vAve_i, average velocity on ith interval} U {t_mid_i, `dt_mid_i, approximate average rate of change of velocity between midpoints, approximate acceleration at t_i, a_Ave_i | 1 <= i <= n-1}  knowledge of Physics I and prerequisite mathematics

0.01 – 0.04 are sequenced, each being a superset of the preceding, with quantities new to the sequence underlined at each step

0.05  Relate

• uncertainty in measured quantity

• value of measured quantity

• percent uncertainty in measured quantity

• uncertainty in value of a given power of measured quantity 

knowledge of Physics I and prerequisite mathematics

0.06  Relate

• uncertainties in two measured quantities

• values of measured quantities

• percent uncertainties in measured quantities

• percent uncertainty in product or quotient of measured quantities

• uncertainty in sum or difference of measured quantities

knowledge of Physics I and prerequisite mathematics

0.07  Relate

measurement using given instrument

estimated precision of instrument

estimated accuracy of measurement

justification of estimate

 knowledge of Physics I and prerequisite mathematics

0.08  Estimate uncertainties in measured quantities as appropriate to means of measurement (ph1 background (see also ch 1), intro asst, asst 0 query, asst 1 query)  knowledge of Physics I and prerequisite mathematics

Asst

Introductory Problem Set

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Class Notes

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query

1

Deterioration of Difference Quotients

Principles of Physics students do  Brief Flow Experiment

query_1

measuring a pencil, rubber band, estimate flow expt uncertainties, first- and second-diff uncertainties

Objectives

01.01.  Relate

• domino measurements with largest and smallest ruler

• order of domino volumes

• uncertainty in each measurement

• uncertainty in volume calculation

• uncertainty in order of volumes

knowledge of Physics I and prerequisite mathematics

01.02. Relate for n measurements of clock time t and position x for a uniformly accelerating object

• percent uncertainty in x_i measurements

• percent uncertainty in t_i measurements

• percent uncertainty in inferred v_Ave_i

• percent uncertainty in inferred a_Ave_i

• percent uncertainty in slope of v_Ave_i vs. t_mid

knowledge of Physics I and prerequisite mathematics

01.03.  Precision and accuracy of measurements of various objects using ruler copies at various levels of reduction.

01.04.  Accumulation of error with successive difference quotients.

The path from here to there: systems, diagrams, etc. sufficient to lay foundation for playing the game at your level of ability, preparation, available time and dedication

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Introductory Problem Set

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Set 5, Probs 10-13

General College Physics and University Physics students do Flow Experiment

Brief Bottlecap and Tube Setups 

Formatting Guidelines and Conventions

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Class Notes

#03

query_2

therm energy transfer, expansion (proportionalities)

univ calorimetry

Objectives are specified when the situation is first encountered, and will typically be revisited during subsequent assignments, during which mastery is expected to build.

Identify and explain in terms of a simple example the proportionalities among dQ/dt (rate of thermal energy transfer), L (distance between points on a flow line), A_cs (cross-sectional area of flow) and `dT/`dx (temperature gradient).

Technically: 

Relate {dQ/dt, k, L, A_cs, dT/dx, T_1, T_2}

qa_2, class notes

Identify and explain in terms of a simple example the proportionalities among L_0 (the original length of an object), `dL (the change in its length due to thermal expansion), A_0 (original surface area), `dA (change in surface area), V_0 (original volume), `dV (chagne in volume) and `dT (the change in its temperature).

Technically:

Relate {`dL, L_0, `dT, T_0, T_f, alpha, A_0, `dA, V_0, `dB, beta}

qa_2, class notes

Relate

• c (specific heat of an object)

• m (mass of the object)

• `dQ (change in thermal energy of the object)

• `dT (change in temperature of the object)

• T_0 (initial temperature of the object)

• T_f (final temperature of the object)

qa_2, class notes

Relate for an isolated collection of n objects, where the subscript i refers to the i_th object, 1 <= i <= n

• c_i (specific heat of the i_th object)

• m_i (mass of the i_th object)

• `dQ_i (change in thermal energy of the i_th object)

• `dT_i (change in temperature of the i_th object)

• T_0_i (initial temperature of the i_th object)

• T_f (final temperature of the system

qa_2, class notes

univ calorimetry

System 5a (cylinder and mass(es) (BB’s) at vel v, elastic)

For a single particle of mass m moving at speed v perpendicular to and colliding elastically with a piston of cross-sectional area A_cs, with time `dt between collisions, where p is the momentum, `dp the change in momentum during a collision, F_ave the average force and P_ave the average pressure on the piston:

• Relate {m, v, p, `dp, `dt, F_ave}
• Relate {m, v, p, `dp, `dt, F_ave, A_cs, P_ave}

For a single particle of mass m moving at speed v perpendicular to and colliding elastically with the ends of a cylinder of cross-sectional area A_cs and length L, with time `dt between collisions, where p is the momentum, `dp the change in momentum during a collision, F_ave the average force and P_ave the average pressure on the piston:

• Relate {m, v, L, p, dp, KE }  
• Relate {m, v, L, `dt }
• Relate {m, v, L, p, `dp, `dt, F_ave, KE, A_cs, P_ave}

For a large number N of particles each of mass m_particle moving at speed v in a random direction in space, colliding elastically and end of a cylinder of cross-sectional area A_cs and length L, where F_ave is the average force and P_ave the average pressure on the end, KE_ave the average kinetic energy and v_Ave the (rms) average velocity of the particles, n the number of moles, U the total rotational and translational KE of the particles, T the absolute temperature, n the number of moles, R the gas constant, k the Boltzmann constanct, N_A Avagodro's number and the gas monatomic or diatomic:

• Relate {m_particle, N, v, L, F_ave, A_cs, P_ave}  
• Relate {T, m_particle, KE_ave, v_rms, N, n, U, C_p, C_v, monatomic/diatomic, R, k, N_A}

Where in addition V is the volume, T_0, V_0, P_0, N_0 the initial temperature, volume, pressure and number of particles, `dT the change in absolute temperature, `dKE the change in KE, `dP the change in pressure, `dN the change in the number of particles

• Relate {m, N, v, L, Fave, A_cs, P_ave, KE_ave, V, T}
• Relate {direction of `dT, direction of `dKE, direction of `dP, direction of `dV, direction of `dN}
• Relate {`dT as percent of T_0, `dP as percent of P_0, `dV as percent of V_0, `dN as percent of N_0, `dT/T_0, `dV/V_0, `dT/T_0, `dN/N_0 | percent changes are small }

context

Asst

Introductory Problem Set

lab/activity

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outline

Class Notes

other

query

3

Temperature vs. Clock Time Measurements

Are Your Labs Posted in Readable Format?

text_03

query_3

calorimetry, temp scales, gas laws

univ radiation

System 1a: 

Relate, where T_1 and T_2 are the initial temperatures of equal amounts of gas and T_f the final temperature when mixed:

•  {T_1, T_2, T_f}

Same gas same pressure:

Relate, for two samples of gas in a divided container at equal pressure, where T, V, n, m are absolute temperature, volume, number of moles and mass,

• {T_1, V_1, T_2, V_2, n_1, n_2, m_1, m_2, T_f}

Relate, for two samples of the same gas in a divided container

• {V_1, V_2, T_1, T_2, P_1, P_2, T_f}

… conceptual (involves gas laws as well)

Asst

Introductory Problem Set

lab/activity

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4

Set 5, Probs 14-16

Brief Bottle Experiments

Experiments to View

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Class Notes

#04

query_4

gas laws, energy equiv of thermal energy,

univ rms vel,

Relate, for a confined gas

• {P, V, n, T}

Relate, for a confined gas

• {P_0, T_0, V_0, N_0, P_f, T_f, V_f, N_f}

Asst

Introductory Problem Set

lab/activity

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outline

Class Notes

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query

5

Set 5, Probs 1-3

Raising Water in a Vertical Tube

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Class Notes

#05

query_5

water plug, buoyant force, force exerted by pressure

System 4: 

Bottle with Single tube: open 

Relate for a capped-bottle system with an open tube, where A_cs is the cross-sectional area of the tube, V_bottle the volume of the bottle, `dV_tube the volume of water displaced into the tube, `dy the change in the water level in the tube.

• {A_cs,`dL, V_bottle, `dV_tube, `dV_tube/V_bottle , `dV_tube/V_bottle (as %), `dy }

Relate for a capped-bottle system with an open tube, where P is the pressure inside the bottle, `dy the change in the water level in the tube, squeeze is perceived force exerted when squeezing the bottle, P_atm, T the absolute temperature of the gas, V the initial volume of the bottle, `dV the change in volume ..

• {P, `dy, squeeze}
• {P, `dy, squeeze, P_atm}
• {P, `dy, squeeze, P_atm,T, V, `dV}

Tube only:   [vernacular]  .. two tubes, indep measures of pressure (pressure tube indicates ratio of absolute pressure)

• Relate Atm pressure:  {rho_water, y2 – y1, L_0, `dL, (V / V_0)_air_column, P_atm}  

2-tube: 

• Relate {P, V, T, y2 – y1, L_0, L}  

Thermo: 

• Relate {P_0, V_0, T_0, n, y_max, P_1, V_1, T_1, T_f, P_2, V_2, T_2, `dQ, `dPE, `dU, eff}
• Relate {P_0, V_0, P_f, T_f, adiabatic\isothermal, `dW, `dU, `dQ}

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Introductory Problem Set

lab/activity

text

outline

Class Notes

other

query

6

Set 5, Probs 4-6

Kinetic Model Experiment

text_06

Class Notes

#06

query_6

bernoulli, kinetic model expt, cont eqn

{P, V, n, N, T}

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Introductory Problem Set

lab/activity

text

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other

query

7

Set 5, Probs 7-9

Measuring Atmospheric Pressure, Part 1

text_07

Class Notes

#07

query_7

video drag forces; more fluids

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Introductory Problem Set

lab/activity

text

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other

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8

Set 5, Probs 16-20

Bottle Thermometer

text_08

Class Notes

#08,09

Calculus in Thermo

query_8

kinetic theory, thermo

System 4:

• Relate {n, R, mon/diam, C_p, C_v, `dT, `dQ}
• Relate {m, v, p, `dp, `dt, F_ave, A_cs, P_ave} 

previously done:

3 dim 

• {m, N, v, L, F_ave, A_cs, P_ave}
• {T, m_particle, KE_ave, v_rms, N, n, U, C_p, C_v, monatomic/diatomic}

 System 5a: 

• {m, v, L, p, dp, KE }
• {m, v, L, `dt }
• {m, v, L, p, `dp, `dt, F_ave, KE, A_cs, P_ave}
• 3d:  {m, N, v, L, Fave, A_cs, P_ave, KE_ave} 

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9

text_09

Class Notes

#10,11

query_9

Waves

·         Relationships among wave velocity, frequency, wavelength, density of medium, tension of medium, amplitude, energy and period.

·         Superposition of waves

·         Nature of reflection at boundaries of various types and the formation of standing waves between two points.

·         Determining wavelengths of the harmonics of a standing wave given boundary conditions.

·         Determining frequencies of harmonics from wavelengths and propagation velocity.

·         Energy in standing and traveling waves

Waves and Optics:

• v = f * `lambda: The frequency f tells us how many peak-to-peak cycles of a traveling wave past in a unit of time. The wavelength `lambda tells us the distance between consecutive peaks of the wave. Thus when we multiply the frequency by the wavelength we obtain the distance travel by the disturbance in a unit of time, which is the velocity at which the disturbance propagates.

 T = 1 / f:  The period T is the time required for a peak-to-peak cycle, and is equal to the reciprocal of the frequency: T = 1 / f.

`omega = 2 `pi f:   The angular frequency `omega is the velocity of the point moving around the reference circle which models the simple harmonic motion of a single point as the wave passes. Since there are 2 `pi radians in a circle, `omega = 2 `pi f.

• y = A sin( (2 `pi f) (t - x / v) ): If the disturbance at the left-hand end of a wave traveling to the right has the form y = A sin ( (2 `pi f) t), then if the wave has propagation velocity v the disturbance will require a time delay `dt = x / v to reach position x. It follows that the disturbance at position x will satisfy the equation y = A sin( (2 `pi f) (t - x / v) ).

• v = `sqrt ( T / `mu ): In a uniform string whose mass per unit length is `mu = m / L, under uniform tension T, application of the impulse-momentum theorem shows that a transverse disturbance will travel at velocity v = `sqrt( T / `mu ).

v = `sqrt( B / `rho):  In a liquid medium the velocity with which a disturbance is propagated is given by v = `sqrt( B / `rho), where B is the bulk modulus and `rho the density of the liquid.

v = `sqrt( Y / `rho):  In a solid medium the velocity with which a disturbance is propagated is given by v = `sqrt( Y / `rho), where Y is the Young's modulus and `rho the density of the solid.

v = `sqrt( `gamma * p / `rho):  In an ideal gas we have propagation velocity v = `sqrt( `gamma p / `rho), where `gamma is the ratio Cp / Cv of specific heat at constant pressure to specific heat at constant volume, p is the pressure of the gas and `rho its density. Since for an ideal gas p / `rho = RT / M, where R is the gas constant, T the temperature and M the molar mass, we can also write v = `sqrt( `gamma * R T / M).

• `lambda = L * (2 / n), n = 1, 2, 3, ...: For a standing wave in a linear medium, where the wave is constrained to have nodes separated by distance L, then the number of node-antinode distances spanning distance L is a multiple of 2. Since 4 node-antinode distances are required to span a wavelength, the possible wavelengths are 2 L (corresponding to 2 node-antinode spans in distance L), L (corresponding to 4 node-antinode spans in distance L), 2/3 * L (corresponding to 6 node-antinode spans in distance L), 1/2 * L (corresponding to 8 node-antinode spans in distance L), etc..

The corresponding frequencies are the f = v / `lambda values v / ( 2 L), v / (L), v / (2/3 * L), etc., making up the series 1/2 * v / L, 1 * v / L, 3/2 * v / L, 4 * v / L, summarized by the formula f = (n / 2) * v / L.

The same results apply for much the same reasons when antinodes occur with separation L.

• `lambda = L * (4 / (2n - 1) ), n = 1, 2, 3, ... For a standing wave in a linear medium, where the wave is constrained to have nodes separated from antinodes by distance L, then the number of node-antinode distances spanning distance L is 1 plus a multiple of 2--i.e., an odd number. Since 4 node-antinode distances are required to span a wavelength, the possible wavelengths are 4 L (corresponding to 1 node-antinode spans in distance L), 4/3 L (corresponding to 3 node-antinode spans in distance L), 4/5 * L (corresponding to 5 node-antinode spans in distance L), 4/7 * L (corresponding to 7 node-antinode spans in distance L), etc..

The corresponding frequencies are the f = v / `lambda values v / ( 4 L), v / (4/3 * L), v / (4/5 * L), etc., making up the series 1/4 * v / L, 3/4 * v / L, 5/4 * v / L, 7/4 * v / L, summarized by the formula f = (2n-1)/4 * v / L.

• f ' = f / (1 - vSource/vSound). When a source of sound emitting pulses with frequency f is moving toward an observer with velocity vSource then if the speed of sound is vSound, the frequency detected by the observer is f ' = f / (1 - vSource/vSound). This is an instance of the Doppler Effect.

• f ' = f (1 + vObserver / vSound ): If an observer is moving toward a source which is emitting a sound with frequency f, then if the speed of sound is vSound the frequency detected by the observer will be f ' = f (1 + vObserver / vSound ).   This is another instance of the Doppler Effect.

• path difference = a sin(`theta). If waves are emitted in phase from two points separated by distance a, then if the waves are detected at a distance much greater than a, and at a position P such that a line from either source to P makes an angle `theta with the perpendicular bisector of the line segment connecting the sources, the distances traveled by the two waves will differ by distance a sin(`theta).

If this path difference is equal to half of a wavelength the two waves will arrive at P exactly out of phase and the net disturbance at P will be 0. We call this destructive interference.

If the path difference is equal to a whole wavelength or to any multiple of a whole wavelength the two waves will arrive at P exactly in phase and the net disturbance will be double that of either of the individual disturbances. We call this constructive interference. The condition for constructive interference is that path difference = n * `lambda for n = 0, 1, 2, ....

If the path difference is equal to a whole wavelength plus a half wavelength we will again have destructive interference. The condition for destructive interference is that path difference = (n + 1/2) * `lambda for n = 0, 1, 2, ... .

• sin(`theta1) / sin(`theta2) = n2 / n1: Snell's Law states that if `theta1 and `theta2 are the angles of incidence and refraction for a beam of electromagnetic radiation which is directed from one material into another, then the sines of those angles are inversely proportional to the indices of refraction n1 and n2 of the two materials. The index of refraction for a material is the number n such that v = c / n, where v is the propagation velocity of the electromagnetic radiation in the material and c the propagation velocity of electromagnetic radiation in a vacuum. For most materials n varies with the wavelength of the electromagnetic radiation, which leads to the phenomenon known as dispersion.

• Critical Angle (angle of total internal reflection):  Total internal reflection occurs whenever the angle of incidence is greater than that for which the angle of refraction is 90 degrees. If `theta1 and `theta2 are the angles of incidence and refraction and n1 and n2 the corresponding indices of refraction, then sin(`theta2) = n2 / n1 * sin(`theta1) and if sin(`theta1) = n1 / n2, then sin(`theta2) = 1 and the angle of refraction will be 90 degrees. The value of `theta1 for which sin(`theta1) = n1 / n2 is called the critical angle.   For any angle of incidence greater that the critical angle we will have total internal reflection.

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Introductory Problem Set

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other

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10

Set 6, Probs 1-10 (phy 122 only 1-8)

Experiments to View

text_10

System 7: 

• 2 independent pendulums:  {f_1, f_2, n, f_beat}

Class Notes

#12

Practice Test

query_10

wavelength, freq, etc; per wave mth; all introset

Relate for a traveling sine wave

• lambda (wavelength)
• f (frequency)
• omega (angular frequency)
• T (period)
• A (amplitude)
• 'time delay' between two points
• equation of motion of point at end
• equation of motion of arbitrary point
• waveform at given instant t

Relate for a an oscillation driven by two independent oscillators

• f_1 (frequency of first oscillator)
• f_2 (frequency of second oscillator)
• f_beat (beat frequency)
• n (number of beats occurring during time interval `dt)
• `dt (time interval)

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Introductory Problem Set

lab/activity

text

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other

query

11

Set 6, Probs 11-13

Measuring Atmospheric Pressure, Part 2 

Experiments to View

text_11

... label all problems by set and system

... require summary of introset topics, reln to goal set

... list each set where it first occurs; maybe divide into 'now' and 'ultimately'

... define sets for A, B, C grades

Class Notes

#13

query_11

harmonics

univ eqn of wave

Relate for longitudinal or transverse standing waves in a given object of given length

• boundary conditions
• positions of nodes and antinodes for fundamental oscillation
• lambda_1 (wavelength of fundamental oscillation)
• positions of nodes and antinodes for first n harmonics (n <= 4)
• pattern of positions of nodes and antinodes for subsequent harmonics
• positions of nodes and antinodes for nth harmonic
• lambda_n (wavelength of nth harmonic)
• c (propagation velocity)
• f_n (frequency of nth harmonic)
• A_max (amplitude of oscillation at antinode)
• A(x) (amplitude at position x)
• y(x, t) (position relative to equilibrium of point at position x along the object, at clock time t)
• sketch of y(x) for given t
• graph of y(t) for given x

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12

Set

text_12

Superposition: {y_1(x), y_2(x), y(x) }

Also graphical

Class Notes

#13

query_12

tension, mass, superposition, harmonics, reflection,

Relate for traverse standing wave in string or chain under uniform tension

• T (tension)
• c (propagation velocity)
• m (total mass)
• L (length)

Relate

• waveform y_1(x), given mathematically or graphically
• waveform y_2(x), given mathematically or graphically
• superposed waveform y_1(x) + y_2(x), constructed graphically
• superposed waveform y_1(x) + y_2(x), represented mathematically
• equation y_1(t, x) of traveling or harmonic wave
• equation y_2(t, x)  of traveling or harmonic wave
• equation of superposed wave

Relate

• traveling sine wave y_1(t, x) traveling in positive direction, traveling sine wave y_2(t, x) traveling in opposite direction, both with common propagation speed c, frequency f and amplitude A
• evolution of waveform of superposed wave between two specified points
• boundary conditions
• number of harmonic
• matching selection of points
• evolution of cycle of given harmonic

Relate

• traveling wave incident on fixed or free boundary
• reflected wave

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Introductory Problem Set

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other

query

13

Set 6, Probs 14-18 (Phy 122 do only 17 and 18)

Initial Activities with Waves and Optics. 

text_13

• For a uniform strip of material supported at its ends oscillating in its fundamental mode:

{length, amplitude at center, frequency, period, equation of motion at center, estimated amplitude at specified distance from end, estimated equation of motion at specified point} … done as {f, lambda, harmonic n, max amplitude, x, x(t)}

... later add propagation velocity and more

• For a uniform strip of material supported at its ends, to achieve sustained oscillation (i.e., harmonic):

{length, possible separation distances of intermediate supports}

• ... later add prop vel etc. and get freq etc.

• For uniform strip supported at middle:

{frequency, amplitude at endpoints, period, equation of motion at endpoint, wavelength}

Class Notes

#14

query_13

harmonics, energy in wave etc.x

Relate for a wave traveling through a medium

• amplitude
• frequency
• mass per unit length
• energy per unit length
• propagation velocity
• power

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Introductory Problem Set

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other

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14

Set 6, Probs 19-21

text_14

Doppler

{c, v_o, v_s, f, f ‘ }

Decibels

{I, Ampl_pressure, area}

{dB, I, I_0}

Energy

{density, A, f, omega, c, k … etc }

Class Notes

#15

query_14

doppler, decibel, beats, harmonics

Relate (Doppler effect) for a wave created by a source moving with velocity v_s and observed by an observer moving with velocity v_o  through a medium in which propagation velocity of the wave is c

• c
• v_o
• v_s
• f
• f ‘

Relate for a sound wave

• pressure amplitude
• intensity
• area
• power of source
• threshold intensity
• decibel level

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15

Set

Experiments to View

text_15

Class Notes

#16

query_15

al rod, dB, crab nebula, fetal heart

Relate for an aluminum rod which is supported at its middle and free at both ends

• length
• frequency of fundamental harmonic of longitudinal wave
• wavelength
• propagation velocity of sound in aluminum
• small change in position of support
• change in beat frequency as proportion of original beat frequency

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16

Set

Experiments to View

text_16

System 11a:

• Intersecting paths (will be principle rays): {path_1, path_2, x_int_1, x_int_2, y_int_1, y_int_2, int_12} … paths defined by connect-the-dot points; sketch then analyze by eqns and also by similar triangles

• {(x_0, 0), (0, y_1), (0, y_1+`dy), path difference d_13 – d_23 }

• Same U { (0, y_1 + j `dy) | 1 <= j <= n } U {pd_j, pattern}

query_16

more doppler etc

Relate for two broken-line paths, each defined by three points in the coordinate plane

• coordinates of the three defining points for each path
• x and y intercepts of the paths
• point(s) of intersection of the paths
• triangles formed by paths with one another and with the coordinate axes
• sets of similar triangles and relative dimensions

and apply to image formation by thin lenses.

Relate for a thin converging or diverging lens

• image distance
• object distance
• focal distance
• magnification
• nature of image
• diagrams of principle rays

Relate for a given point (x_0, 0) on the x axis of the coordinate plane, two points (0, y_1) and (0, y_2) on the y axis and a wavelength

• length of path 1 from (x_0, 0) to (0, y_1)
• length of path 2 from (x_0, 0) to (0, y_2)
• difference in path lengths
• difference in path lengths as a multiple of wavelength

For points (0, -a/2) and (0, a/2) on the coordinate plane and a wavelength lambda < a

• using a sketch depict the path difference corresponding to a distance observer, along paths making angle theta with the x axis
• calculate the path difference
• calculate the path difference as a multiple of lambda
• determine angles for which the path difference is an integer multiple of lambda
• determine angles for which the path difference is an integer multiple of lambda plus a half-wavelength
• apply to situations involving the two-slit phenomenon

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17

Set 6, Probs 22-24

Experiments to View

text_17

{theta_i, theta_r, n_a, n_b}

Single lens\mirror

{f, o, i, m, r\i}

Two lenses

{f_1, f_2, y_o, y_1, y_2, y_i, o_1, o_2, i_1, i_2}

Class Notes

#17, 18

query_17

refraction, lenses, prisms,

Relate for two lenses

• focal lengths
• position of object with respect to first lens
• position of image formed by first lens
• position of second lens
• position of image formed by second lens
• characteristics of each image

Relate for a plane wave incident from one medium at a plane interface with a second:

• indices of refraction
• angle of incidence
• angle of refraction
• speed of light in each medium

prism ...

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lab/activity

text

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other

query

18

Resolving Images

Experiment 26:  Circular Lens and Circular Mirror   (do the experiment but don't write it up--just respond to the query program).  Note the summary document Index of Refraction from Focal Point of Liquid-Filled Cylinder

text_18

Class Notes

#19

query_18

refraction, lenses, images etc.

1.  Sketch and analyze the path of light incident on an reflected or refracted by a thin lens, a mirror or a prism.

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Introductory Problem Set

lab/activity

text

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other

query

19

focusing_light_thru_cylindrical_lens

text_19

Path difference

{a, theta, L, path diff}

Class Notes

#20

query_19

interference

1.  Sketch and analyze the path of light incident on a thick cylindrical interface.

Relate

Relate

Relate

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Introductory Problem Set

lab/activity

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other

query

20

Image Formation; Combining Lenses

text_20

query_20

interference

1.  Analyze image formation by a combination of thin lenses.

2.  Analyze interference patterns for thin films, single slits, double slits and diffraction gratings.

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Introductory Problem Set

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query

21

Interference

text_21

polarization:

{phi, Imax, I}

{phase difference of sources, linearity, circularity or eccentricity of elliptical polarization (all three reduce to eccentricity of elliptical polarization)}

Class Notes

#21

query_21

polarization, interference

1.  Explain how radio waves could be polarized when passing through a series of vertical or horizontal metal posts, and whether the resulting waves would be polarized vertically or horizontally.

2.  Calculate Brewster's angle for light incident from a medium with index of refraction n_1 on a smooth flat surface with index of refraction n_2.

Relate

Relate

Relate

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Introductory Problem Set

lab/activity

text

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other

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22

text_22

Partial Summary of Intro Problem Set

query_22

query 22 same as query 20

There are no new objectives for Assignment 22.

Module 3, Assignments 23 - 33

Electrostatics

·         Coulomb’s Law, electric fields, superposition of fields.

·         Gaussian surfaces

·         Energy, potential difference, potential functions.

Electrical Circuits

·         Ohm’s Law, Series and Parallel Circuits, Kirchoff’s Laws, Energy Relationships

Magnetism

·         Magnetic fields.

·         Interaction between magnetic fields and moving electric charges.

·         Magnetic fields as the result of moving charges.

·         Magnetic flux, EMF resulting from changing magnetic flux.

• Coulomb's Law and its Consequences:

F = k * q1 * q2 / r^2, where k = 9 * 10^9 N m^2 / C^2, q1 and q2 are two point charges, r is the distance separating the charges and F is the force between the two charges. The force F is exerted on each charge by the other.  If q1 and q2 are of like sign the force will be one of attraction along the line between the two charges; otherwise the force will be one of repulsion in the direction of the line defined by the two charges. This law is known as Coulomb's Law.

• E = k q / r^2: The electric field E at a point P, due to a point charge q at another point Q, is the force experienced per Coulomb of test charge when the test charge is introduced at the point P. This field is directed along the line defined by P and Q, and is directed away from P if q is positive and toward P is q is negative.

• F = q * E: The force exerted on a charge q in electric field E is equal to the product q * E. If q is positive the force will be exerted in the direction of the field; if q is negative the force will be exerted in the direction opposite that of the field.

• Electrostatic Flux = 4 `pi k Q: The total electrostatic flux through a closed surface is 4 `pi k Q, where Q is the total charge enclosed by the surface.

E = k Q / r^2:  If the charge distribution is spherically symmetric, then the electric field due to that distribution will be spherically symmetric about the same central point. The field will be directed radially outward from the center and will hence be perpendicular to the surface of any concentric sphere. Thus for any such sphere we have uniform electric field E = 4 `pi k Q / ( 4 `pi r^2) = k Q / r^2, where Q is the total charge enclosed by the sphere. Note that this agrees with Coulomb's Law in the case of a point charge q.

E = 2 k `lambda / r:  If a charge distribution is cylindrically symmetric, then the electric field due to that distribution will be cylindrically symmetric about the central axis of the distribution. The field will be directed radially outward from the central axis and will hence be perpendicular to the curved surface of any coaxial cylinder. Thus if `lambda is the amount of charge per unit length enclosed by the cylinder, a cylinder of radius r and length L will enclose charge Q = `lambda * L. Since the curved surface of the cylinder will have area A = 2 `pi r * L, the field at the surface of the cylinder will be E = 4 `pi k Q / r^2 = 4 `pi k ( `lambda * L) / (2 `pi r * L) = 2 k `lambda / r.

E = 2 `pi k `sigma:  If charge is uniformly distributed over a large plane area, then symmetry arguments show that near that plane but not close to its edges the electric field due to the distribution is very nearly perpendicular to the plane. It follows that if the charge per unit area is `sigma, then a rectangular box with its central axis oriented perpendicular to the plane, with the box intersecting the plane and having cross-sectional area A perpendicular to the plane, will contain charge Q = `sigma * A. The electrostatic flux will exit this box through its two ends and not through its sides, so the flux will be E * ( 2 A). We therefore have E ( 2 A) = 4 `pi k ( `sigma * A) and E = 4 `pi k ( `sigma * A ) / ( 2 A) = 2 `pi k `sigma. Note that as long as we remain near the plane, in the sense already specified, the electric field remains constant.

Work, Energy, Potential Difference, Power:

• `dW(ON) = q * E * `ds: The work done ON a charge q which moves through a displacement `ds in the direction of a uniform electric field E is `dW = F * `ds = (q * E) * `ds.  The work done BY the charge against the field is the negative of this quantity.

• V = E * `ds. The potential difference V between two points is the work required per unit charge to move charge from the first point to the second. In a uniform electric field E the work to move a charge q through displacement `ds in the direction of the field is `dW(ON) = q * E * `ds, so the potential difference is V = `dW(ON) / q = q * E * `ds / q = E * `ds. The unit of this potential difference is the Joule / Coulomb, which is the Volt.

• F = q * `dV / `ds. If `dV is the potential difference corresponding to a displacement `ds in the direction of a constant electric field, then the potential gradient of the field is `dV / `ds. In this case the force on a charge q is the product of the charge and the potential gradient.

• `dKE = -`dW(BY). The change in the kinetic energy of a charge particle, in the absence of dissipative forces, is equal and opposite to the work done by the particle against an electrostatic field. Electrostatic fields are conservative, and the forces on charged subatomic particles within such fields usually exceed any dissipative forces to the extent that the dissipative forces are negligible. In this case energy conservation tells us that the kinetic energy change of such a particle will be equal to the work done on the system by the electrostatic field, or to the negative of the work done by the system against the electrostatic field.

• P = I * V: The power required to move charge at the rate I (standard unit Coulombs / sec or amps) through a potential difference V (standard unit Joules / Coulomb or volts) is the product I * V of the rate and potential difference (when we multiplied Coulombs/second by Joules/Coulomb we get Joules/second, or watts).

Conduction by Charge Carriers:

• (Rate of charge passage = N / L * vDrift).  If there are N charge carriers uniformly distributed over a length L, then the number of charge carriers per unit length is N / L. If those charge carriers, which typically have small mass and which therefore have large thermal velocity, have an average net drift velocity vDrift along the length, then the number of charges passing a given point per unit of time is given by N / L * vDrift.

• Rate of charge passage = charge density * A * vDrift.  The number of charges per unit of length in a uniform conducting wire depends on the volume density of charge carriers in the wire and on the cross-sectional area A of the wire. We have N / L = charge density * A, so that the number of charges passing a given point per unit of time is charge density * A * vDrift.

• The average drift velocity of the charge carriers in a substance is proportional to the strength of the electric field, which is equal to the potential gradient. For a uniform conducting wire with potential difference `dV between its ends, the strength of this electric field or potential gradient is `dV / L.

• (Rate of Charge Passage proportional to A * `dV / L).  The drift velocity is affected by various factors that vary with temperature and from one substance to another. For given substance at a given temperature the drift velocity is proportional to `dV / L, so that the number of charges passing a given point per unit time is proportional to charge density * A * `dV / L. Since charge density also depends on the substance and temperature, we can say that the number of charges passing a given point per unit of time for a given substance and temperature is proportional to A * `dV / L.

• (`dV / R proportional to A * `dV / L).  The number of charges passing a given point per unit of time is the current, designated I. Thus for given substance at a given temperature I is proportional to A * `dV / L. Ohm's Law tells us that I = `dV / R, where R is the resistance of the conductor to the flow of current (see below; for the present model `dV is the voltage V across the conductor). Thus `dV / R is proportional to A * `dV / L, so 1 / R is proportional to A / L and R is proportional to L / A.

• R = `rho * L / A.  We call the proportionality constant between R and L / A the resistivity of the substance, and we use the symbol `rho for the resistivity. Thus R = `rho * L / A. The resistance to the flow of current is proportional to the length of the conductor (greater L implies lower potential gradient, or electrical field strength, `dV / L, which results in lower drift velocity and therefore lower current) and inversely proportional to its cross-sectional area (greater A implies more charge carriers and hence more current flow for a give drift velocity).

Ohm's Law and Circuits:

• I = V / R. Ohm's Law expresses the fact that the current I in a conductor is proportional to the voltage across the conductor and inversely proportional to the quantity called resistance. Using V for the voltage and R for the resistance this is expressed as I = V / R.

• P = V^2 / R. If a voltage V is applied across a resistance R the resulting current is I = V / R. A current I passing through a potential difference V requires power P = I * V so P = (V / R) * V = V^2 / R.

• R = R1 + R2, series resistances. If a current I passes through two resistances R1 and R2 without branching, we say that the two resistances are in series. In this case voltages V1 = I * R1 and V2 = I * R2 are required across the resistances. Since the resistances, hence the voltages, are in series the total voltage across the combination is V1 + V2 = I * R1 + I * R2, or V = I * (R1 + R2). Thus I = V / ( R1 + R2) and we can say that the resistance of the series combination is R1 + R2.

• 1 / R = 1 / R1 + 1 / R2, parallel resistances: If a voltage V is maintained across a parallel combination of two resistances R1 and R2, then the entire voltage is experienced by both resistances. Hence current I1 = V / R1 flows through the first resistance and I2 = V / R2 flows through the second, giving a total current I = I1 + I 2 = V / R1 + V / R2. So I = V ( 1 / R1 + 1 / R2). Since I = V / R, we have 1 / R = 1 / R1 + 1 / R2.

Capacitors and Capacitance:

• C = Q / V. A capacitor, when a potential difference is maintained between its terminals, stores charge in proportion to that potential difference. Capacitance is defined as C = stored charge / voltage = Q / V.

• C = A / (4 `pi k Q) (parallel-plate capacitor). A parallel-plate capacitor consists of two flat plates each with area A, with a uniform separation d. If charge Q is taken from one initially uncharged plate and placed on the other, then the plates will have charges Q and -Q and charge densities Q / A and -Q / A. The resulting electric fields will therefore be E1 = 2 `pi k Q / A, directed away from the first plate, and E2 = 2 `pi k Q / A directed toward the second plate. Provided d is small compared to the dimensions of the plates, between the plates the field will therefore be 4 `pi k Q / A, directed from the first plate toward the second, and outside the plates field will be zero. The voltage between the plates will be V = E * d = 4 `pi k Q / A * d, and the capacitance, defined as Q / V, will be C = Q / (4 `pi k Q / A * d) = A / (4 `pi k * d).

• C = C1 + C2 (capacitors in parallel). A parallel combination of two capacitors with capacitance C1 and C2 will maintain the same potential difference across both. The total charge is therefore the sum of the two charges that would result from that potential difference, and as a result capacitance is the sum of the two capacitances.

• 1/C = 1/C1 + 1/C1 (capacitors in series). A series combination of two capacitors with capacitance C1 and C2 will result in equal charges Q on both capacitors. This results in series voltages V1 = Q / C1 and V2 = Q / C2 so that total voltage is V = Q ( 1 / C1 + 1 / C2) and we have V / Q = 1 / C1 + 1 / C2. Since the capacitance of the combination is C = Q / V, we see than 1 / C = 1 / C1 + 1 / C2.

Magnetism:

• B = k ' I * `dL / r^2: Magnetic fields are caused by electrical currents. A current I flowing thru a short straight segment `dL (think for example of a short segment of a straight conductor) will contribute k ' I `dL / r^2 to the magnetic field B at a point P, lying at distance r, provided that a line from the segment to P is perpendicular to the segment. k ' = 10^-7 Tesla / (amp meter). The direction of the field contribution at P is perpendicular to both `dL and to the line from the segment to P, according to the right-hand rule.

• B = k ' I * `dL / r^2 * sin(`theta): A current I flowing thru a short straight segment `dL will contribute k ' I `dL / r^2 sin(`theta) to the magnetic field B at a point P, lying at distance r, if the line from the segment to P makes angle `theta with the segment. If `theta is a right angle then this rule is identical to the rule of the preceding paragraph. k ' = 10^-7 Tesla / (amp meter). The direction of the field contribution at P is perpendicular to both `dL and to the line from the segment to P, according to the right-hand rule.

• B = 2 k ' * `pi I / a (field at the center of a single circular loop of radius a). If we add up all the k ' I `dL / r^2 contributions from a single loop of wire of radius a, adding to obtain the field at the center of the loop, we find that if the loop is in a horizontal plane and the current goes around the loop in the counterclockwise direction the magnetic field contributed by each segment is directed vertically upward. Thus all the individual k ' I `dL / r^2 contributions are in the same direction and their magnitudes add to the total magnitude of the magnetic field. Since the total `dL is equal to the circumference 2 `pi a of the loop and r = a we can replace `dL by 2 `pi a and r by a to get B = k ' I ( 2 `pi a ) / a^2 = 2 k ' * I / a.

• `phi = F * A * cos(`theta): The flux, designated `phi, of any constant field F which penetrate a planar surface whose area is A is equal to the product of the component of F perpendicular to the surface, and the area of the surface. Here `theta is the angle of the field F with a line perpendicular to the surface. The component of F perpendicular to the surface is therefore F * cos(`theta), and the flux is the product of this perpendicular component and the area: flux = F * cos(`theta) * A = F * A * cos(`theta).

• Vave = `d`phi / `dt (ave voltage = ave rate of change of electrostatic flux): When the electrostatic flux `phi = E * A cos(`theta) due to an electrostatic field E through a loop of area A is changed by amount `d`phi in a time interval of duration `dt, a potential difference with average value Vave = `d`phi / `dt is created around the loop.

• | Vave | = 4 * A * E * f (ave magnitude of a voltage produced when a loop of area A is rotated at frequency f in the presence of an electric field E): When a loop of area A is rotated with frequency f about an axis perpendicular to a constant electric field E, then the time required for the loop to rotate from perpendicular to parallel with the field is 1/4 of the period of rotation. The period of rotation is the reciprocal T = 1 / f of the frequency. So in `dt = 1/ 4 T = 1 / (4 f ) we have a change in flux from `phi = A * E to `phi = 0, a change of magnitude | `d `phi | = A * E. This implies average voltage vAve = `d`phi / `dt = A * E / (1 / (4 f) ) = 4 * A * E * f.

• F = q v B (force on a moving charge in the presence of the magnetic field, velocity and field perpendicular): When a charge q moves with velocity v perpendicular to a magnetic field B, it experiences a force F = q v B directed perpendicular to both the velocity v and the magnetic field B, with the direction determined by the right-hand rule.

• m v^2 / r = q v B (force on a moving charge in the presence of a perpendicular magnetic field is equal to the centripetal force on the charge): When a charge q moves with velocity v perpendicular to a uniform magnetic field B, the force being perpendicular to the velocity constitutes a centripetal force m v^2 / r. The particle will move in a circular orbit of radius r, where r is the solution to the equation m v^2 / r = q v B.

• v = | E / B | (velocity of a charge to particle moving in a straight line through 'crossed' electric and magnetic field): When a particle with charge q passes through uniform electric and magnetic fields which make right angles with one another, and the velocity of the particle is perpendicular to both of these fields, then if the particle travels in a straight line this means that net force on the particle is zero. This can only happen if the electrostatic force q E is equal and opposite to the magnetic force q v B, so that | q E | = | q v B |. Solving for v, the magnitude of the velocity, we easily obtain v = | E / B |.

• F = I L B sin(`theta) (force on a straight current segment in the presence of a uniform magnetic field): When a current I flows in a straight conductor of length L, in the presence of a magnetic field B making angle `theta with the conductor, then the field exerts a total force F = I L B sin(`theta), with the direction of the force perpendicular to both B and I, the direction determined by the right-hand rule.

23

Set 1, Probs 1-5

{q1, q2, k, epsilon_0, F_12, r}

{q1, q2, k, epsilon_0, F_12, PE}

{q1, q2, k, epsilon_0, r, F_12, PE}

{Q, q1, q2, k, epsilon_0, r, F_12, PE}

{Q, q1, q2, (x_1, y_1), (x_2, y_2), (x, y), r_1, r2, F_1, F_2, F, PE}

{q_i, (x_i, y_i), F_i | i = 1, ..., n} U {Q, (x, y), F, PE}

For sphere chg at center:  {Q, phi, a, E, q, r, PE(r),  V(r)}

... recommend notebook with answers to series of specified questions, headings etc. organized as appropriate by goal sets

{q_i, (x_i, y_i), F_i, E_i | i = 1, ..., n} U {Q, (x, y), F, E, PE, V}

{F, E, `ds, q, `dW, `dPE_elec, V}

Class Notes

#26

Physics I Initial Problem Sets:  Vectors

Overview of Electrostatic Forces, Fields and Energy

query_23

tape, vectors, coul law

Relate each of the following sets:

{q1, q2, k, epsilon_0, F_12, r}

{q1, q2, k, epsilon_0, F_12, PE}

q_1 and q_2 are charges, (x_1, y_1) and (x_2, y_2) their positions in the plane, r the distance between them, F the magnitude of the force of their interaction, k = 1 / (4 pi epsilon_0), epsilon_0 the permittivity of free space.

{q1, q2, k, epsilon_0, r, F_12, F_21, PE}

r is the vector from charge 1 to charge 2, F_12 the vector force exerted on charge 1 by charge 2, F_21, the force exerted by charge 1 on charge 2, PE the electrostatic potential energy of the two charges relative to infinite separation (quantities related to PE incorporated in asst 25)

{Q, q1, q2, (x_1, y_1), (x_2, y_2), (x, y), r_1, r_2, F_1, F_2, F, F / Q, PE, PE / Q}

(x, y) is a position different from the positions of q_1 and q_2, Q a hypothetical charge located at (x, y), r_1 and r_2 the vectors from q_1 and q_2, respectively, to (x, y), F_1 and F_2 the forces exerted on Q by q_1 and q_2, F (quantities related to PE incorporated in asst 25)

{q_i, (x_i, y_i), F_i | i = 1, ..., n} U {Q, (x, y), F, E_i, E, PE_system}

q_i is the ith of n charges, (x_i, y_i) its position, F_i the force of the ith charge and F the net force exerted by the n charges on the charge Q at (x, y), PE_system the total electrostatic potential energy of the n charges relative to infinite separation, E_i the contribution of the ith charge to the electric field at (x, y), E the electric field at (x, y) due to the n charges.

Asst

Introductory Problem Set

lab/activity

text

outline

Class Notes

other

query

24

Set 1, Probs 6-9

text_24

Class Notes

#27

query_24

electric field

Relate

Relate

Relate

Asst

Introductory Problem Set

lab/activity

text

outline

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other

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25

Set 1, Probs 10-23

Class Notes

#28

query_25

voltage, potential difference

Relate for a sphere with a single point charge at its center, where phi is the electrostatic flux through the sphere, Q the charge, a the radius of the sphere, E the magnitude of the electric field at a point on the surface of the sphere, q a charge at distance r from the center, PE(r) the electrostatic potential energy of the system, V(r) the electrostatic potential at the position of q due to the charge Q.

{Q, phi, A, E, q, r, PE(r),  V(r)}

Relate

{F, E, `ds, q, `dW_on, `dPE_elec, `dV}

where F is the force exerted on charge q at a certain position, E the field at that position, `ds a displacement of q small enough that the field E does not change significantly, `dPE_elec the change in the electostatic potential energy due to the displacement of q, `dW_on the done on q by the field during the displacement, `dV the change in electrostatic potential between the initial and final positions of q

Relate

Relate

Relate

Asst

Introductory Problem Set

lab/activity

text

outline

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other

query

26

Set 2, Probs 1-5

Current Flow and Energy

text_26

networks ...

For long cylinder, axially symmetric charge density:  {Q_enclosed, lambda, L, phi_E, a, E_radial}

For plane distribution const charge density, close to plane:  {`dA, sigma, `dQ, phi_E, N, E}

For parallel-plate capacitor:  {A, d, E, V, Q, C}

Class Notes

#22

query_26

current model, field

univ flux, electrostatics,

Relate for current I in a short segment of length `dL in a charge carrier in the presence of magnetic field B, where `dF is the force exerted by the field on the segment:

{I, B, `dL, `dF}

Relate for a circuit element with resistance R, where I is the current through the element and V the potential difference across the element:

{I, R, V}

Relate

{P, I, V, R}

for a circuit element with resistance R, where I is the current through the element and V the potential difference across the element, and P is the power required to maintain the current.

Relate for long cylinder carrying an axially symmetric charge distribution of density lambda, where phi_E is the electrostatic flux of the distribution, a its radius and E the magnitude of the electric field at distance r from the cylinder (where the distance from the point to the nearest end of the cylinder is much greater than r): 

{Q_enclosed, lambda, L, phi_E, a, E, r}

Relate for a plane distribution of constant charge density, at a point much closer to plane than to any of its edges: 

{`dA, sigma, `dQ, phi_E, N, E}

where sigma is the charge density, N a vector normal to the surface, `dA is the area of a cross section in a plane perpendicular to N of an appropriately positioned rectangular 'box' all of whose points are much closer to the plane than to any of its edges, `dQ the charge enclosed in the box, phi_E the electrostatic flux through an end of the box and E the electric field at the point.

Relate for a parallel-plate capacitor in air, where A is the area of each plate, d the separation, E the electric field between the plates, V the potential difference between the plates, C the capacitance and Q the magnitude of the charge on each plate:

• {A, d, E, V, Q, C}

Relate for two conducting elements of the same material, each of constant cross-section both subject to the same potential difference V between its ends:

{A_cs_1, A_cs_2, I_1, I_2, L_1, L_2}

where A_cs indicates cross-sectional area, I current, L length of the conductor in the direction normal to the cross-section.

Asst

Introductory Problem Set

lab/activity

text

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other

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27

Set 3, Probs 1-6

Capacitors and Current, Voltage and Energy

text_27

Class Notes

#30, 31

query_27

potential

Relate

{C, Q, I, R, V_s, V_c, `dt, `dQ, `dV_c},

for a series circuit consisting of a source, a capacitor and a resistance element, with V_s is the voltage of the source, C the capacitance, Q the charge on the capacitor, V_c the voltage of the capacitor, I the current, R the resistance, `dt a short time interval, `dQ the change in the charge on the capacitor, `dV_c the change in the voltage across the capacitor.

Relate

{V, R_1, R_2, `dV_1, `dV_2, I_1, I_2}

for a series or a parallel circuit with two resistance elements, where V is the voltage of the course, R_1 and R_2 the two resistances, `dV_1 and `dV_2 the respective voltage drops, I_1 and I_2 the currents through the respective resistors.

Relate

Relate

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Introductory Problem Set

lab/activity

text

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other

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28

Set 4, Probs 1-7

Batteries, Circuits and Measurement of Voltage and Current.

text_28

Class Notes

#22

query_28

capacitance, energy density

Relate for short current segment `dL, position r relative to `dL, the magnetic field contribution `dB at that position; theta is the angle of `dL with respect to r

{`dL, `dB, I, r, theta}

Relate {I, orientation, a, B}  for a circular current loop of radius a in which a current I flows with a given orientation, B the magnetic field at the center of the loop.

Relate for a plane region of area A, electric field of magnitude E whose angle with respect to the specified normal of the plane is theta,  phi_E the flux of the electric field through the region

{E, theta, A, phi_E}

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Introductory Problem Set

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29

Set 4, Probs 8-13

Charging and Discharging a Capacitor.

text_29

Class Notes

#23

query_29

generating voltage, current, resistance, AC etc

1.  Solve problems related to voltage induced by changing magnetic flux.

2.  Explain self-induction.

Relate for a plane region of area A, magnetic field of magnitude B whose angle with respect to the specified normal of the plane is theta,  phi_B the flux of the magnetic field through the region

{B, theta, A, phi_B}

Relate for a plane region of area A, magnetic field of magnitude B whose angle with respect to the specified normal of the plane changes from theta_1 to theta_2 it time interval `dt,  phi_B_1 the flux of the magnetic field through the region at the beginning of the interval and phi_B_2 the same quantity at the end, `dPhi_B the change in flux, ave_Rate the average rate of change of flux with respect to clock time, V_ave the approximate average induced voltage:

{B, theta, A, phi_B_1, phi_B_2, `dPhi_B, ave_Rate, V_ave}

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Introductory Problem Set

lab/activity

text

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other

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30

Set 4, Probs 14-18

The RC Circuit

text_30

{I, `dB, `dL, r}

Class Notes

#24,25

query_30

sources of mag field

Relate for charge q moving with velocity v in electric field E and magnetic field B, where F_E and F_B are the forces exerted by the respective fields

{q, E, B, v, F_E, F_B}

Relate for segment `dL carrying current I, in the presence of magnetic field B, where `dF is the force exerted on the segment,

{`dL, B, I, `dF }

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lab/activity

text

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31

Set

text_31

Class Notes

#31, 32

query_31

induction

Continue mastery of previous objectives.

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lab/activity

text

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other

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32

Set

query_32

Continue mastery of previous objectives.

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other

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33

Set

query_33

Continue mastery of previous objectives.

Module 4, Assignments 34 - 40

Modern Physics

·         Time dilation, length contraction, mass-energy.

·         Photoelectric effect, quantization of energy.

·         Uncertainty principle.

·         Atomic spectra, quantization of angular momentum, atomic structure.

·         Modes of nuclear decay, energy conservation.

Quantization of Energy in Electromagnetic Radiation:

• c = `lambda * f: Light, and generally all electromagnetic radiation, is characterized by a uniform speed c of approximately 3 * 10^8 m/s in a vacuum, and by a spectrum of wavelengths `lambda, which can be measured using diffraction gratings and photographic plates. Frequencies are easily determined once wavelengths are measured.

• E = h * f: The photoelectric effect demonstrates that electromagnetic radiation with frequency f transfers energy in discrete amounts equal to the product of Planck's constant h = 6.63 * 10^-34 J s and the frequency of the radiation. In this sense electromagnetic radiation behaves as a stream of photons, particles which carry energy but which have no mass.

• p = E / c: A photon with energy E will have momentum p = E / c.

Wave Properties of Particles:

• `lambda = h / p: A beam of massive particles (i.e., particles with mass) each with momentum p will exhibit interference effects identical to those of a wave having wavelength `lambda = h / p, where h is Planck's constant.  This wavelength `lambda is called the deBroglie wavelength.

• `dx * `dp = h (uncertainty principle): The uncertainty in x, the position of an object, and p, the momentum of the object, are denoted by `dx and `dp. It is not possible to measure both position and momentum to an arbitrarily high degree of precision. The more precisely one is measured the less precisely is possible to determine the other. The product of the uncertainties is approximately equal to Planck's constant. i.e., `dx * `dp = h.

Quantization of Orbital Energies in Atoms:

• m v^2 / r = k q^2 / r^2 (condition relating velocity and radius of 'orbit' of electron around proton). If an electron 'orbits' a proton at a distance r, then the velocity must be such that the Coulomb attraction k q^2 / r^2 (q = charge of electron) between the proton and electron constitutes the centripetal force m v^2 / r required to maintain the circular path. This relationship can easily be solved for v in terms of r, obtaining r = k q^2 / (m v^2).

• m * v * r = n ( h / (2 `pi) ), n = 1, 2, 3, ... . (quantization condition for angular momentum): Angular momentum can occur only in whole-number multiples of h / ( 2 `pi ). Combining this quantization condition on the angular momentum m * v * r with m v^2 / r = k q^2 / r^2 we conclude that for an electron orbiting a proton, only certain orbital radii and their associated velocities and energies can occur. The resulting model explains to a high degree of precision the spectrum observed as a result of energy transitions within a hydrogen atom.

• r = n^2 h^2 / ( 4 `pi^2 k q^2 m) = n^2 * .509 *10^-10 m approx., n = 1, 2, 3, ... . (possible radii of electron orbit around proton) Combining m v r = n ( h / (2 `pi) ), the quantization condition on angular momentum, with m v^2 / r = k q^2 / r^2, which states that Coulomb Force = centripetal force, we obtain r = n^2 h^2 / ( 4 `pi^2 k q^2 m). Using h = 6.63 * 10^-34 J s, k = 9 * 10^9 N m^2 / C^2, q = 1.60 * 10^-19 Coulombs and m = 9.11 * 10^-31 kg (mass of electron) we find that h^2 / ( 4 `pi^2 k q^2 m) = .509 * 10^-10 m, approx., so we can write r = n^2 * .509 Angstroms (approx).  An Angstrom is 10^-10 meter.

• 2 `pi r = n * `lambda   (orbital radius contains an integer number of particle waves):  For an orbital radius satisfying the quantization condition, the circumference of the orbital is a whole number of deBroglie wavelengths, suggesting that the electron in forms allowed orbits by creating a standing wave in each.

Conversion between Mass and Energy and Nuclear Decay Modes:

• E = m c^2: As predicted by Special Relativity and as confirmed by experiment, mass can be converted to energy and vice versa. The energy equivalent to mass m is E = m c^2.

• Alpha decay consists of the emission of a helium nucleus (2 protons and 2 neutrons all bound together) from the nucleus of a larger atom (i.e., one containing in its nucleus more than 2 protons and 2 neutrons). This process reduces the number of protons and neutrons in the larger nucleus each by 2. The mass of the resulting nucleus plus the mass of the alpha particle is less than the mass of the original nucleus. The resulting mass defect appears in the form of kinetic energy as the alpha particle and the remains of the original nucleus fly apart in opposite directions. In the usual case, where the alpha particle is much less massive then the remains of the original nucleus, almost all of this kinetic energy resides in the alpha particle.

• Beta decay consists of the emission of an electron from the nucleus of an atom. This process results when a neutron decays into a proton and an electron (note that the total charge is conserved--the original neutron has zero charge, and a proton and an electron have a net charge of zero). This process thus increases the number of protons in the nucleus by 1 and reduces the number of neutrons by 1. The mass of the resulting nucleus plus the mass of the beta particle (the electron) is less than the mass of the original nucleus. The resulting mass defect appears in the form of kinetic energy as the electron and the new nucleus fly apart in opposite directions. Since the electron is very much less massive then the new nucleus, virtually all of this kinetic energy resides in the electron.

• Gamma decay is analogous to a decrease in the orbital energy of an electron in an atom, but Gamma decay takes place within the nucleus and involves nuclear particles. The energies within the nucleus are much higher than the energies of atomic electrons, and the energies of the Gamma photons are hence much greater than the energies emitted by atomic electrons. The nucleus after a Gamma decay is less massive than before, and the mass defect is carry off by the energy of the photon.