Summary of Topics from Introductory Problems on Thermal Energy and
Fluids
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:
- 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 fluidsthe 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.
Summary of Topics from Introductory Problems on Waves and Optics
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.Summary of Topics from Introductory Problems on Electricity
and Magnetism
- 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.
Summary
of Topics from Introductory Problems on Modern Physics
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.
Special Relativity:
- `dt ' = `dt * 1 / (1 - v^2
/ c^2) (time dilation): Einstein's Theory of Special Relativity is
based on the assumption that the laws of physics are the same in
all interial reference frames. One consequence of these laws is that the speed
of light in a vacuum is c = 3 * 10^8 m/s,
approx.. If we consider a photon bouncing back and forth in
the vertical direction between two mirrors in a spaceship
which passes at velocity v in the horizontal direction, we
are led to the conclusion that the time between round trips as measured
by an occupant of the spaceship will be different
than the time between round trips as measured from our
frame of referencet. Specifically if in our frame of reference we
measure a time interval `dt ' between two events which
occur at the same location as observed by the occupants of the passing
spaceship while the occupants of the spaceship measure time
interval `dt, our time interval `dt ' will be greater
than `dt by factor 1 / (1 - v^2 /c^2). The contraction
of length and the increase of mass, as well as the relative
nature of simultaneity, also follow (see below).
- `dx ' = `dx * `sqrt(1 -
v^2 / c^2) (length contraction): If a spaceship passes us moving
moving at velocity v relative to our frame of reference, and if as
measured by the occupants of the ship its length in the
direction of its velocity is `dx, then the most accurate measurements in
our frame of reference will determine that the length of the
ship is `dx ' , which is less than `dx by factor
`sqrt(1 - v^2 / c^2).
- m ' = m * 1 / `sqrt( 1 -
v^2 / c^2) (relativistic mass): If a spaceship passes us moving
moving at velocity v relative to our frame reference has mass m,
as measured by the occupants of the spaceship (who could, for example, measured the
acceleration and the net force applied to the spaceship to determine its mass), then any
measurement made in our frame of reference to determine the mass will
indicated that the mass is m ' , which is greater than m by
the factor 1 / `sqrt(1 - v^2 / c^2).
- Relative nature of
simultaneity: If a spaceship passes us moving moving at velocity
v relative to our frame reference, then two events that occur simultaneously
but at different locations as measured by the occupants of the spaceship
will not occur simultaneously in our reference
frame.
- KE = m c^2 / `sqrt(1 - v^2
/ c^2 ) - m c^2. If a spaceship with mass m,
as measured in the frame of reference of the spaceship, passes us moving moving at velocity
v relative to our frame reference, then its kinetic energy will
be m c^2 / `sqrt(1 - v^2 / c^2 ) - m c^2. For velocities v which are small
compared to c, this expression reduces to .5 m v^2. The
expression for kinetic energy can also be written by factoring the m
c^2 out, obtaining m c^2 ( 1 / `sqrt(1-v^2/c^2) - 1). This form
suggests a 'rest energy' of m c^2, to which kinetic
energy is added as velocity increases.
- Correspondence Principle:
The Correspondence Principle states that the results of relativistic mechanics
reduce, at small relative velocities v, to the results
of classical Newtonian mechanics. Stated another way, there must be a smooth
transition from the everyday physics of low relative velocities to the relativistic
physics of high relative velocities.
Brief Notes about Particle
Physics:
- If a particle such as a proton or and electron is accelerated,
usually by means of electromagnetic forces, to a high relativistic velocity then
the possibility exists that the total energy of the particle can be converted
to other particles. This usually happens when the particle
interacts with other particles. The rest mass of
the particles so created can exceed the rest mass of the
original particles as some of the kinetic energy of the
original particles converts to mass. This conversion satisfies the
equation E = m c^2.
- When new particles are created a number of conservation laws apply.
The total energy of the particles involved is the same before as
after collision, provided we include the energy
equivalent E = m c^2 of the rest mass of the particles
as part of the energy. Momentum must be conserved (keeping
in mind that we're talking about relativistic momentum m v * 1 / `sqrt(1-v^2/c^2). Electric
charge must be conserved (for example the total charge after a
collision involving a proton and a neutron must be the same, +1, as before the collision).
Other more subtle conservation laws also apply. These
conservation laws lead to a classification of the hundreds of different
particles observed and of their properties. This classification leads to
a very successful mathematical model which successfully predicts the
nature of particles prior to their being observed. This model is called
the Quark Theory.
- It is believed that protons, neutrons and hundreds
of other particles are in fact combinations of more fundamental particles
called Quarks.