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Class #1 - Class #12
Class #13-Class #21
Class #22-Class #33
Class #34-Class #41
Class Notes are posted here without video links, due to the large size of video files. Class notes including video links are distributed on CD's.Class Notes
Class Notes are posted here without video links, due to the large size of video files. Class notes including video links are distributed on CD's.Class #1-Class #12: Fluids and Thermodynamics
#01: Experiments conducted; no notes (demos on CD)
#02: Experiments conducted; no notes (demos on CD)
#03: Analyzing Experiments 2 and 4
#04: Pressure and Volume Relationships for an Ideal Gas
#05: Potential and Kinetic Energy Changes in a Fluid
#06: Fluids; Bernoulli's Equation
#07: Bernoulli's Equation
#08: Terminal Velocity results; Bottle Engine intro (thermodynamics)
#09: Bottle Engine: Work and Energy, Efficiency
#10: Bottle Engine: States 1, 2 and 3
#11: Kinetic Theory
#12: Thermodynamics; Introduction to Waves
Class #13-Class #21: Waves and Optics
#13: Introduction to Waves
#14: Harmonics in a String
#15: Doppler Shift
#16: Harmonics of a String; Music
#17: Snell's Law; Ray Tracing; Image Formation
#18: Focal Points, Image Formation
#20: Focal Distance, Object Distance, Image Distance, Magnification
#21: Interference
Class #22-Class #33: Fluids and Thermodynamics
#22: Current Flow and Resistance
#23: Series Resistance
#24: Magnetic Field from Pendulum Behavior
#25: Capacitors and Meters; Flux Change and Voltage
#26: Coulomb's Law
#27: Coulomb Forces, Energy, Potential Difference
#28: Electric Field
#29: More Electric Fields; Gauss' Law
#30: Field Lines, Equipotential Lines, Capacitance
#31: Capacitors
#32: Some DC Circuits
#33: Partial Review of E & M
Class #34-Class #41: Fluids and Thermodynamics
#34: The Bohr Model of the Hydrogen Atom
#35: Energy Levels of the Bohr Atom; Revisiting Induced Charge
#36: Alignment of Magnetic Dipoles
#37: Atomic Structure; Dual Nature of Light
#38: Time Dilation; Revisiting Source of Magnetic Field
Class Notes Further Notated
Analyzing Experiment 4: 'Bottle Thermometer' consisting of empty bottle articulating with U-tube containing alcohol 'plug', 'top' end closed; pressure inversely proportional to length of air column with small correction for difference in alcohol levels.
Estimating absolute zero: Extrapolating pressure vs. temperature data to horizontal axis.
Nonlinearity of many temperature measuring devices: The properties of substances used to measure temperature do not generally change linearly with temperature, but usually do change linearly over restricted temperature ranges.
Symbolizing thermal energy exchanges in Experiments 2-3: Specific heat is the rate at which the thermal energy of a substance changes, per unit of temperature, per unit of mass. Two objects placed in a closed system exchange thermal energy until their temperatures become equal (provided the mechancial energy of the system remains constant).
The ideal gas law is summarized by P V = n R T. If two of the quantities P, V, n and T remain constant the other two vary in a way determined by this law. The values of P, V and T constitute the state of the system.
When a quantity of water flows from hole in a container which lies lower than the water surface, mass at the level of the water is replaced by mass at the level of the hole, reducing the gravitational PE of the system. If dissipative forces are absent the KE of the exiting water is equal to the PE loss, thus determining the velocity of the exiting water.
Bernoulli's Equation, relating pressure, velocity, density and altitude at two different points of a continuous fluid, is an expression of energy conservation within that fluid.
Bernoulli's Equation is used to analyze pressure relationships between points within a fluid which flows through a narrowing tube.
We analyze data from an experiment in which a marble is accelerated to terminal velocity using an Atwood-type apparatus. Using curve fitting with power functions we obtain results for a marble of diameter 2 cm.
Vertically displacing water through a thin tube by means of heating the gas (at known initial temperature) in a closed in a water bath, we observe the height of the fluid and find the pressure necessary to support the fluid. We compare the temperature necessary to achieve that pressure with the temperature of the water bath.
We heat a gas to first raise a thin water column to a given height, then continue heating to displace the water to a reservoir at that height, thereby increasing the gravitational PE of the system.
The thermal energy required to move between states of the system (first at constant volume, then at constant pressure) can be compared with the PE change.
We observe PE change vs. the height to which the column is raise and at which the water is collected, conjecturing the height at which the ratio of the PE change to the thermal energy required is maximized.
The analysis of the bottle engine is represented on a graph of pressure vs. volume.
When the system is returned to its original pressure in preparation for another cycle, the graph, and the nature of energy exchanges, depend on the manner is which this is accomplished. We consider isothermal (slow) and adiabatic (quick) alternatives.
The pressure, volume, temperature and amount of gas in a system are related in a way that can be largely understood in terms of particles and elastic collisions. The predictions of this model agree very well with the behavior of gases.
The model correlates the average translational KE per particle with the temperature of the gas, and through molar mass and Avagadro's number to the rms velocities of the particles.
The law of equipartition of energy relates the total internal energy of an ideal monatomic or diatomic gas to its temperature.
The net work done during a complete thermodynamic cycle is equal to the area bounded by the corresponding graph of pressure vs. volume.
The efficiency of a thermodynamic cycle is equal to the ratio of the work done to the thermal energy input during the cycle. This efficiency has an upper limit that depends on the maximum and minimum temperatures at which the cycle operates.
Propagation velocity of a pulse in a string is measured by synchronizing a series of pulses with oscillations of a pendulum. This is repeated for a series of string tensions.
The frequency of the fundamental mode of oscillation is measured for a variety of string tensions.
Results are compared with the theoretical result for propagation velocity as a function of tension and mass per unit length.
Harmonics occur when a whole number of cycles 'fit' the length of the string.
Pulsing a string with increasing frequency, harmonics were observed to emerge at or near multiples of the fundamental frequency.
The formation of harmonics can be modeled by two waves with equal wavelength, moving through a section of string at equal speeds but in opposite directions.
If a source approaches or recedes from an observer, or vice versa, with the relative speed much less than propagation velocity, the observed frequency differs from the frequency of the source by approximate proportion v_relative / v_propagation. v_relative is positive when source and observer are approaching, negative when they are moving further apart.
If the condition that the relative speed is much less than propagation velocity, the approximation no longer valid. In this case the result can be reasoned out in a variety of ways, and it turns out that both the motion of the source and the observer, and not simply the relative motion, are required to determine the frequency shift.
For a standing wave in a homogeneous linear oscillator the possible sequences of nodes and antinodes are determined by the boundary conditions (mainly, by whether each end is a node or an antinode).
The sequence of possible frequencies depends on the end conditions. The natural frequency ratios are well approximated by powers of 2^(1/12), which doubles frequency in 12 'steps', i.e., a sequence of 12 frequencies with common ratio 2^(1/12).
Oscillating objects typically oscillate with a combination of harmonics. Possible waveforms are obtained by superposition of different multiples of the possible frequencies. The relative strengths of the various harmonic frequencies determine the tone of the oscillator.
Snell's Law governs the refraction of light at the interface between two light-transmitting media, specifying how the angles of incident and refracted light (as measured with the normal direction, i.e., the direction perpendicular to the surface) are related to the indices of refraction of the two materials.
Two parallel rays entering a circular disk in which the index of refraction is greater than that of the medium surrounding it are refracted in such a way that the lines of the refracted rays meet. Parallel rays which would, if undeflected, pass close to the center of the disk are refracted along lines that meet close to a single point of focus, or 'focal point'. For most materials the focal point lies outside the circle so that the rays would be refracted once more before reaching the focal point, and would not actually converge at that point.
(Two parallel rays entering a circular disk in which the index of refraction is less than that of the medium surrounding it are refracted in such a way that the refracted rays, continuing along the refracted paths, diverge and therefore never meet. If the lines of the refracted rays are extended back in the opposite direction, they do meet 'before' reaching the circular disk. Parallel rays which would, if undeflected, pass close to the center of the disk are refracted along lines which when 'extended backwards' meet close to a single point of focus, or 'focal point'.)
A spherical reflector of radius R reflects all parallel rays close to the center ray (called the 'axial ray') to a focal point which lies at distance R / 2 from the reflector.
Using similar triangles we determine that the focal point for the initial refraction through a circular disk with index of refraction 1.33, when light is incident from a vacuum, lies at distance 4 R from the refracting surface, where R is the radius of the disk. The rays therefore encounter the 'back surface' of the disk before reaching this focal point, and are refracted once more is such a way that they focus at distance (2 - n) / (2n - 2) * R from the 'back surface'.
For a thin lens the focal distance is measured from the center of the lens. By considering the paths of selected rays from an illuminated object on one side of a thin converging lens, we can infer the location, size and orientation of the image formed on the other side the lens.
Using principle rays and similar triangles we obtain the relation among the image distance, object distance and focal distance of a thin lens, and infer the nature (real vs. virtual), magnification and orientation (upright or inverted) of the image.
Waves originating from an in-phase source, traveling along two paths, interfere constructively when the numbers of wavelengths along the two paths differ by a whole number, and destructively when the difference is a whole number plus 1/2.
A plane wave which is normally incident on a barrier containing two or more equally-spaces slits makes the slits into in-phase sources, resulting in a diffraction pattern whose geometric properties relate the spacing of the slits to the wavelength of the original wave.
A hand-cranked generator creates a potential difference, which can create an electrical field which to accelerate fast-moving free electrons in a wire, producing an oriented 'slow drift' of the electrons, which we experience as an electrical current. The speed of this 'drift' is limited by the fixed atoms in the wire, which tend to randomize the motion of the electrons. The resulting interaction increases the oscillatory energy of the atoms and the average kinetic energy of the electrons, raising the temperature of the wire.
If current flows in series through two wires of the same material, the wires being identical except that one is thinner than the other, electrons in the thinner wire must drift faster than those in the thicker wire, and therefore give up more energy per unit of wire length. So the thinner wire 'resists' the flow of current to a greater extent than the thicker wire.
The same potential difference across a thick and a thin wire results the same electrical field in both, so each free electron experiences the same acceleration in both wires, resulting in the same drift velocity. The thicker wire contains more free electrons, so will have the greater flow of current.
If two wires of different lengths but of the same thickness are held at the same potential difference, the potential change per unit of distance (i.e., the electric field) is greater in the longer wire, resulting in greater acceleration of individual electrons and greater drift velocity. So the shorter wire will have the greater current.
Resistance is defined to be potential difference / current. The greater the current for a given potential difference, the less the resistance to the flow of current. For a given material, fatter and shorter wires result in more current flow and hence have lower resistance than do longer and skinnier wires.
Two resistances in series each get only part of the potential difference across the combination, and so for a given potential difference allow less current flow than either alone. The current in both is the same, so the potential differences are proportional to the resistances. It follows that the resistance of the series is the sum of the two resistances.
Two resistances in parallel each experience the full potential difference applied to the combination, so the same current flows in each resistance as it would in the absence of the other. The consequence is that the reciprocal of the resistance of the combination is the sum of the reciprocals of the individual resistances.
The current in a horizontal strip of metal suspended by a flexible conductor will interact with a vertical magnetic field to deflect the strip from its previous equilibrium position. The new equilibrium position, voltage across the system, dimensions of the system, resistivities, mass densities, force on the strip, current through the strip, and magnetic field are related. In particular we can see how a current segment interacts with a magnetic field.
A coil orientation appropriately in a magnetic field will tend to rotate when a current flows through it. This is the basis of the ammeter (in which the coil is in series with a large resistance, the combination which is in series with a small resistance) and voltmeter (in which the coil is in series with a small resistance). The ammeter is used in series within a branch of a circuit, the voltmeter in parallel across part of the circuit.
A charged capacitor will discharge slowly through a voltmeter. Voltage vs. time measurements reveal that the voltage decreases exponentially, as expected. The time constant of the exponential can be measured, and related to the resistance of the meter and the capacitance of the capacitor.
Changing the magnetic flux through a loop or coil induces a voltage, which at any instant is the rate at which the magnetic flux is changing.
A properly oriented coil whose normal axis rotates with a constant frequency in a magnetic field experiences a flux that changes sinusoidally in time, resulting in a voltage which is also sinusoidal in time and 90 degrees out of phase with the flux.
Coulomb's Law relates the force exerted on one point charge by another to the two charges and their relative positions. The law is an inverse-square law analogous to Newton's Law of Universal Gravitation.
A small displacement of one charge in the presence of another fixed charge, with the displacement either in the direction of the force exerted on the moving charge, results in either negative or positive work being done on the moving charge. The amount of work done, per unit of moving charge, is the potential difference in the field of the fixed charge, between the initial and final positions of the moving charge.
The potential at a point due to a fixed charge q can be defined as the work required per unit of moving charge Q to move Q from a large separation from q to the given point. The work required is V(r) = k q / r.
For a system consisting of three charges we use Coulomb's Law to calculate the net force on each of the charges, and the total potential energy of the system.
The electric field at a point is the force that would be experienced by a 'test charge' Q placed at that point, per unit of test charge. If in Coulomb's Law we regard one of the charges as the 'test charge', then when dividing the force by the 'test charge' to find the field, the 'test charge' will divide out of the expression.
The electric field of a charge can be depicted by representative vectors at various points relative to the charge. The electric field is radial in inverse-square in nature, directed toward the charge if it is negative, away from the charge if it is positive.
The net force experienced by a test charge at a point, due to a system of two fixed charges, is the vector sum of the forces exerted by the individual charges. The electric field at that point is the vector sum of the fields of the two charges.
(The approximate flux of an electric field through a small plane region is the product of the area of the region and the component of the field perpendicular to the region, where the field is approximated by an arbitrary point within the region. The proportional error of the approximation approaches zero as the diameter of the region approaches zero). (To make this a little more precise we have to take into account whether the field is directed outward or inward relative to the region, but that will be accounted for in what follows).
The total flux of a charge Q through a surface S which encloses q is 4 pi k Q. It doesn't matter where the charge is within the enclosed region, and it doesn't matter what the shape of the surface is. However if Q is a point charge at the center of a sphere, then since the field is radial and inverse-square in nature, it's easy to show that the magnitude of the field is constant and directed out of the sphere, and then that the product of the field and the area is indeed 4 pi k Q.
When the electric field at a surface is always of the same magnitude and always perpendicular to and directed outward from the surface, the flux through the surface is equal to the area of the surface, multiplied by the magnitude of the field. (If the field is always directed inward, then the flux is the negative of this product).
The field of a charge uniformly distributed over a long wire, at a point much nearer the wire than either of the wire's ends, is by symmetry radial and uniform over the curved surface of a short cylinder which is coaxial with the line and contains the point. Assuming an appropriate short length, the charge enclosed by the cylinder, and the area of its curved surface, are easily calculated (and are both proportional to the assumed length). When the enclosed charge is multiplied by 4 pi k Q to get the flux, and the result divided by the area of the curved portion of the cylinder, the assumed length divides out and the result is the magnitude of the electric field at the point.
Movement of a test charge perpendicular to an electric field is associated with zero work by the field. Movement along a curve or a surface which is always perpendicular to the field results in no change in the potential energy of the test charge, and hence no change in the electrostatic potential along that curve. Any such curve is therefore characterized by constant electrostatic potential. We call such a curve or surface an equipotential curve or surface.
A field line is actually a curve which at a point is parallel to the field. Field lines follow the direction of the field. Since equipotential curves or surfaces are perpendicular to the field, the field lines and equipotentials are at right angles to one another. Field lines are shown for a few examples of two charges, some with like and some with unlike charges.
The electric field lines tend to move away from a positive charge and toward a negative charge.
Excess positive or negative charge in a conducting object will be distributed on its surface. The interior will contain balanced amounts of positive and negative charge, since any imbalance of charge will create an electric field which will cause charge to migrate; this process very quickly causes all excess charge to migrate to the surface (the effect is that the excess charges get a far away from other excess charges as possible; they move apart until they get to the surface). On a flat conducting plate the charge will be uniformly distributed over the surface of the plate. At a point which is much nearer the to plate than to its edges, a symmetry argument shows that the electric field must be perpendicular to the plate. If we construct a box-shaped rectangular surface four of whose sides are perpendicular to the plate and whose 'ends' are at the same distance from the plate as our point, field lines will only exit through the two 'ends' of the box. Such a box has uniform rectangular cross sections, one of which coincides with the plate. Using any cross-sectional area, if the density of charge on the surface is known the enclosed charge is easily found and the flux through the box is easily calculated. Dividing the flux by the cross-sectional area we obtain the electric field at the ends, which has a magnitude independent of how far the ends are from the plate (we are still assuming that the point is much closer to the plate than to its edges, so that our symmetry argument remains valid). This will be the electric field at the point.
A parallel-place capacitor uses two such plates and a source of potential difference to separate charge, typically taking negative charge from one plate and depositing it on the other. The plates are placed parallel to one another, separated by a uniform distance that is small compared to their dimensions, so that practically all of the region between the points experiences a uniform electric field perpendicular to the plates. Charge is more or less gradually moved from one plate to the other, increasing the electric field, until the potential difference between the plates is equal and opposite to that of the source; this occurs when the product of the electric field and the plate separation is equal to that voltage.
When a capacitor is used to separate the process generally involves a voltage source, which is connected to the two plates and move charge by means of a current through the source and a resistance. As charge builds on the capacitor its voltage, which opposes that of the source, increases. This decreases the voltage across the resistance, resulting in a lesser current, which results in a slower buildup of charge. The result is that the voltage across the capacitor, the charge in the capacitor, and the current through the circuit are all exponential functions of time.
A voltmeter is used to measure the voltage of a discharging capacitor and determine the unknown resistance of the circuit.
When the capacitor discharges through a hand-cranked generator, it discharges much more quickly if the generator is not permitted to turn.
A network of resistances and sources is analyzed using Kirchoff's Laws.
The Bohr model supposes that an electron in a circular orbit around a proton is subject to the condition, that its centripetal acceleration is equal to the Coulomb force, and that its angular momentum is quantized.
The KE and PE of an electron in a circular orbit at a given distance from the proton are easily calculated. Adding the quantization condition we find the allowable energy levels and the expressions for the differences in these levels. The corresponding energies correspond to those observed in the discharge spectrum of hydrogen gas.
The effect of bringing a charged PVC rod close to an aluminum rod are discussed.
A magnetic dipole tends to align with a magnetic field. The current loops in atoms create magnetic dipoles, which in certain materials can partially align with an external magnetic field.
Atomic structure is dictated by the quantization of the orbital, angular momentum, magnetic moment and spin quantum numbers which apply to electron orbitals.
The electrostatic potential which can be overcome by electrons emitted from photoelectric metals by light of a given wavelength reveals the quantization of the light into photons, and allows us to determine the relationship between photon energy and wavelength.
Time dilation follows from the hypothesis that the speed of light is independent of inertial reference frame.
A circular loop of current results in a magnetic field whose magnitude and direction at the center of the loop are easily calculated.
Set 1: Electricity part 1
1-10: Coulomb's Law, Electric Field for point charges.
11-12: Electric field between plates, work on test charge, potential difference
13-18: Electric field, work, potential difference, change in potential energy, potential gradient, energy conservation
19-20: Power
21-23: Electrostatic flux, symmetries
1-5: Conduction model, effect of length and cs area, potential gradient, force on electron, power
1-3: Power, energy, current, voltage, resistance, series and parallel combinations of resistances
4-6: Short-term behavior of a capacitor in series with resistance and source
1-3: Magnetic field due to a current segment, magnetic field of a plane coil
4-9: Flux (sunbeam then electrostatic then magnetic)
10-11: Changing flux, average voltage (not quite correct...)
12-13: Rotating loop, resistance, power required to maintain rotation (consider also self-induction)
14-16: Force on and path of electron in magnetic field, crossed electric and magnetic fields
17-18: Force on a conductor in a magnetic field
1-3: Pressure column, PE then plug analysis to predict outflow velocity
4-9: Bernoulli's Law and applications
10-11: Temperature gradient, thermal conductivity
12-13: Specific heat, calorimetry
14-16: Gas behavior (P, V, n T)
17-18: Kinetic theory 1 dim, 3 dim
19-20: Thermal energy in, out, work done, efficiency
waves/set_56_problem_number_1.htm
1-8: Frequency, wavelength, period, propagation velocity
9-10: Mathematical representation of standing and traveling harmonic wave
11-13: Frequencies, wavelengths, nodes, antinodes, propagation velocity in a standing wave
14: The 12-tone scale
15-16: Snapshots of wave in space or time with various given conditions
Power and energy in a wave
Interference
19-20: Doppler shift
21: Interference
22: Image formation
23-24: Refraction
1-2: Photoelectric effect
3: Momentum of photon
Wavelength of electron beam
5-6: Uncertainty in velocity of confined electron, confined proton
7-9: Bohr model of hydrogen atom
10: Interaction of photon and hydrogen atom
11: Kelvin temperature at which KE_ave > binding energy
12-13: Energy equivalent of mass
14: Mass defect and energy
15-19: Alpha, beta, gamma decay
20: Energy in Fusion
21-23: Binding energy per nucleon, possibility of given decay, decay sequences (actual tables of elements)
24-26: Time dilation, length contraction, mass increase
Set 1: Electricity part 1
1-10: Coulomb's Law, Electric Field for point charges.
11-12: Electric field between plates, work on test charge, potential difference
13-18: Electric field, work, potential difference, change in potential energy, potential gradient, energy conservation
19-20: Power
21-23: Electrostatic flux, symmetries
1-5: Conduction model, effect of length and cs area, potential gradient, force on electron, power
1-3: Power, energy, current, voltage, resistance, series and parallel combinations of resistances
4-6: Short-term behavior of a capacitor in series with resistance and source
1-3: Magnetic field due to a current segment, magnetic field of a plane coil
4-9: Flux (sunbeam then electrostatic then magnetic)
10-11: Changing flux, average voltage (not quite correct...)
12-13: Rotating loop, resistance, power required to maintain rotation (consider also self-induction)
14-16: Force on and path of electron in magnetic field, crossed electric and magnetic fields
17-18: Force on a conductor in a magnetic field
1-3: Pressure column, PE then plug analysis to predict outflow velocity
4-9: Bernoulli's Law and applications
10-11: Temperature gradient, thermal conductivity
12-13: Specific heat, calorimetry
14-16: Gas behavior (P, V, n T)
17-18: Kinetic theory 1 dim, 3 dim
19-20: Thermal energy in, out, work done, efficiency
waves/set_56_problem_number_1.htm
1-8: Frequency, wavelength, period, propagation velocity
9-10: Mathematical representation of standing and traveling harmonic wave
11-13: Frequencies, wavelengths, nodes, antinodes, propagation velocity in a standing wave
14: The 12-tone scale
15-16: Snapshots of wave in space or time with various given conditions
Power and energy in a wave
Interference
19-20: Doppler shift
21: Interference
22: Image formation
23-24: Refraction
1-2: Photoelectric effect
3: Momentum of photon
4: Wavelength of electron beam
5-6: Uncertainty in velocity of confined electron, confined proton
7-9: Bohr model of hydrogen atom
10: Interaction of photon and hydrogen atom
11: Kelvin temperature at which KE_ave > binding energy
12-13: Energy equivalent of mass
14: Mass defect and energy
15-19: Alpha, beta, gamma decay
20: Energy in Fusion
21-23: Binding energy per nucleon, possibility of given decay, decay sequences (actual tables of elements)
24-26: Time dilation, length contraction, mass increase