-29PHY 132
Section 9:
h
Induction:
Example: a generator.
Free charges in the wire feel a force
(
), which makes them flow as a
current.
so,
- definition of magnetic flux, ΦB
- Faraday's law of induction
Ex. 9-1: The magnet moves toward the loop in a way that steadily increases B
from 0 to 2.0 mT over 2.0 sec. What does the meter read during that two
seconds?
Ex. 9-2: If that loop has a resistance of 5.0 x 10-4 Ω (with the meter removed), what is the current?
Transformer: A pair of coils arranged so that same magnetic flux goes through both. Changing
current in the "primary" coil creates a changing magnetic field which induces a voltage in the
"secondary" coil.
Ex. 9-3: Find the emf induced in the secondary coil.
The purpose of a transformer is to change voltages: Since a certain amount of V is induced in the
secondary per turn, more turns there means more volts there. More in section 12.
-30Ex. 9-4: Explain why transformers won’t work on steady DC.
(This device is the main reason the power company makes AC not DC: For long distance power
transmission, low I to reduces losses to heat (= I2R). Low I means high V (P = VI). So, step the
voltage up when the power leaves the generator, then step it back down for people's houses. Use
AC so transformers will work.)
Let's look at Faraday's law on a more fundamental level:
Faraday's law, generalized.
Example:
Ex. 9-5: Find E at the position of the secondary coil in example 9-3, at t = .040 s.
Maxwell's Equations:
The four basic equations of electromagnetism, from which everything else can be derived.
We've already covered two of them: Gauss's law, and Faraday's.
Another one is the magnetic version of Gauss's law. (There's a gravitational version too.):
Gauss's law for magnetism
It follows from this that magnetic field lines are always closed loops. (Otherwise
you could find a surface with a net flux through it, as shown.)
The other Maxwell equation is Ampere's law - sort of. Its original form applies only to static
(unchanging) fields.
I will summarize the thinking which lead Maxwell to add another term to Ampere’s law, making it
work in general.
So, listing them together, Maxwell's equations look like this:
-31-
(If field strength or θ vary continuously,
)
Ex. 9-7: At an instant shortly after the switch is closed, the electric
10
field is increasing at a rate of 5.0 x 10 V/m·s. Each plate of the
2
capacitor has an area of .070 m . Find B inside the capacitor as a
function of r, the distance from the center.
Maxwell's modification of Ampere's law says a changing E field induces a B field, just like a
changing B induces an E in Faraday's law. This suggested the possibility of electromagnetic waves:
A changing E induces a changing B which induces a changing E which induces a changing B which
.... Details in Phy 133.
(The process is set off when a charge accelerates.)
This is now known to be the nature of light, radio waves, etc.
-32Section 10: More Induction/ Fields in Matter
Lenz's law: Direction of induced current is such that the loop's own field opposes change in flux.
Example:
As magnet gets closer, its field gets stronger.
To oppose this change, the loop tries to cancel part of the magnet's field by
pointing its own field the other way. Once you know direction of loop's
field, you get the current direction from the right hand rule.
Example: Needle jumps and returns to zero when
switch is opened or closed.
Field of the loop on the bottom points out of the page. (Right hand rule) Since the switch
has just been opened, this field is going away. The loop on top opposes this change; it tries
to hold up the collapsing field by adding its own field to it. So, its field is out of the page
too, which means the current flows in the direction shown.
You work on this:
Ex. 10-1 A) Does current flow in the
loop? If so, does it go up or down the
side closest to you?
B) Does current flow in the loop? If so,
does it go right or left in the side
closest to you?
C) If current flows in the inner
loop, does it flow up or down the
left side?
D) If current flows in the loop,
does it flow up or down the left
side?
-33Self - Induction:
Faraday's law says an emf is induced in any loop of wire with a changing flux through it. This
includes the loop creating that flux in the first place.
E = -L(dI/dt): A more rapidly changing current creates a more rapidly changing magnetic field,
which induces a larger voltage. L depends on the construction of the coil.
Mutual Induction: Similar, except it’s one coil inducing an emf in another rather than in itself.
Ex. 10-2: The current in a certain coil varies according to I = 25 cos (157t) where I is in A and t in s.
What is the voltage across another nearby coil at t = .0005 s, if the system has a mutual inductance
of 65 mH?
Ex. 10-3: A 5.7 mH coil has a resistance of 3.0 Ω. At a particular instant, .50 A is flowing through
it, increasing at 103 A/s. Find the voltage across the coil.
Electric fields in matter:
Dielectric: An insulating material placed between capacitor plates.
becomes polarized and therefore pulls more charges into
capacitor. Therefore, capacitor can store more charge.
Dielectric constant, κ
Permittivity, ε
Capacitance of a parallel plate capacitor, in general
Ex. 10-4: A snugly fitting sheet of paper is placed between the plates of a capacitor holding .106
μC at 12.0 V when empty. If it remains connected to the 12 V battery, how much charge does it
hold with the paper present?
Dielectric strength: The value of
jump.

E
at which the material "breaks down" - allows a spark to
Ex. 10-5: Find the potential difference which will cause a spark to jump 1 cm through air, between
parallel plates.
-34-
Magnetism in matter:
Magnetic fields come from and are felt by electric currents. In a permanent magnet, these currents
are within its atoms: the movement and spin of the charged particles the atom is made of.
In a non-magnetized substance, these subatomic magnets have
random directions, so when you add them up over the whole object,
their total is zero. But, if you place the object in an external
magnetic field, B0, the field can derandomize things, and now the
subatomic fields add up to something.
In the material,
due to object's magnetization.
How magnetized the object becomes depends on B0 and a property of its material called:
Permeability, μ. μ replaces μ0 in all equations if the field is in some material. (like ε and ε0 for
electric fields)
Different materials react to being in a field in different ways:
1. Diamagnetism: Atoms with no magnetic moment develop one antiparallel to
when placed in
a field. (The field slows down some of the atom's electrons, and speeds up others.)
B < B0 (slightly)
2. Paramagnetism: The torque on atoms which do have a permanent moment (



B ) aligns
them somewhat with . (The atoms try to line up with the field like compass needles.) But,
thermal agitation prevents good alignment.
B > B0 (slightly)
3. Ferromagnetism: A quantum mechanical effect overcomes thermal
agitation, aligning all the atoms within tiny regions called DOMAINS.
An ordinary piece of iron is not a magnet because its many domains
have different, random directions. In an external field, favorably
oriented domains grow, and μ's in domains rotate.
B >> B0
Hysteresis: They can get stuck, so the object is still magnetized when B0 is turned off.
-35Section 11:
LR circuits:
Current decay: At t = 0, the switch is quickly thrown from A
to B. Find I as a function of time.
I will show how to set up and solve the differential equation for
this circuit. The result is:
-t/τ
I = I0e
where
τ = L/R
τ = time constant = time to reach 1/e ( 37%)
of the initial value.
(e = 2.71828...)
Current building up: If you start with no current, then connect to the battery at t = 0, a similar
procedure shows that
-t/τ
I = (E /R)(1 - e )
Ex. 11-1: The switch is left at A for a long time, then suddenly
thrown to B. How long after that until I = 50 mA?
-36RC circuits behave similarly to LR circuits:
In a way somewhat similar to the LR circuit, a differential equation comes from the loop rule.
Integrating this equation solves it for the charge on the capacitor, q:
- t/τ
q = Q0e
Time constant:
τ = RC
I = dq/dt : If you need the current, take the derivative of the function above.
V = q/C : This follows from the definition of capacitance, C = q/V.
Charging up: By similar procedure,
-t/τ
q = CE (1-e )
Find I from dq/dt
Find V from C = q/V
Ex. 11-2: A 10 μF capacitor, originally charged to 500 V, is connected to a 1.0 MΩ resistor. Find:
a. the time constant.
b. the voltage after 40 s.
Ex. 11-3: A 10 μF capacitor is placed across a 9.0 V battery with an internal resistance of .50 Ω.
How long will it take to become 90% charged?
-37Magnetic & Electric Energy:
Consider an inductor whose current is building up from zero. The voltage induced across it is
L di/dt. The power it takes in is du/dt = V i. Combine these and cancel the dt’s:
du = L i di
This is the energy lost by the current is it increases by an amount di. This energy is gained by the
field. Integrating from i = 0 to i = I gives the total energy of the field when the current is I.
U = ½ L I2
Energy in an inductor:
Say, for example, the coil is a solenoid: L = (μ N2 A)/l (from text, ch. 32)
energy = ½ [(μ N2 A)/l] I2 = μ N2 I2 =
volume
Al
2l2
B in a solenoid (sec. 8)
Although this only illustrates it for a particular example, the result turns out to be true in general:
Energy density =
Ex. 11-4: In a region where Earth's field is .7 gauss, what volume contains one joule?
Electric fields: Just as I found the energy in a magnetic field by integrating the energy being stored
in an inductor, the energy in an electric field can be found by integrating the energy being stored in
a capacitor:
2
2
U = ½ C V = Q /(2C)
Divide this by the volume of the capacitor,
2
Energy density = ½ ε E
Ex. 11-5: How much voltage must be applied to a 50 μF capacitor to store 1.0 J in it?
-38LC circuits:
1. Close switch. C starts discharging.
2. C is discharged. L's emf keeps I flowing.
3. L's emf reaches 0, so I reaches 0. C is re-charged.
4. And, so on.
Charges move back and forth like the mass in a mechanical oscillator. (Resistance acts like
friction.)
This can also be looked at in terms of energy. In a mechanical oscillator, the energy changes back
and forth between being kinetic energy and potential energy. Here, it changes back and forth
between being electric energy (in the capacitor's field) and magnetic energy (in the coil's field).
UC + UL =
2
2
U= ½CV +½LI
2
2
= ½ C Vmax = ½ L Imax
On the board, I will
- obtain the differential equation for an LC circuit, from the loop rule.
- point out its similarity to a harmonic oscillator's equation of motion, from last fall.
- write the function for q, the charge in an LC circuit. (Since both are really the same
equation, they have what amounts to the same solution.)
Current can be found from I = dq/dt, and
voltage using C = q/V. (Qmax/C = Vmax.)
Ex. 11-6: The capacitor is charged to 20 V, then the switch is closed
at t = 0. Find:
a. the period of oscillation,
b. the voltage across the capacitor at t = 7.0 ms,
c. the current at t = 7.0 ms.
-39Section 12: Alternating Currents
Δv = ΔVmax sin (ωt + )
i = Imax sin (ωt)
where ω = 2πf
= phase angle
What does "1.0 A AC" mean? It's the effective current; the DC current that would deliver the same
power.
2
It's not the average current. That's zero. But power depends on i not i, and the average value of i
is half the peak:
2
2
Pav = iav R
2
Pav =[Imax sin(ωt)]av R
2
2
(sin ωt)av + (cos ωt)av = 1
2
2
(sin ωt)av + (sin ωt)av = 1 ( sin2 and cos2 have the same average.)
2
2(sin ωt)av = 1
2
(sin ωt)av = ½
2
Pav =(Imax) (½) R
Average power loss in a resistor:
- RMS voltage
I will explain exactly what is meant by RMS = root mean square.
AC meters read RMS values.
Ex. 12-1: An AC voltmeter reads 121 V from a 60 Hz outlet. Write the equation for how v varies
with time. Assume = 0.
Resistors:
Since Δv = iR,
ΔVRMS = IRMSR
-40For Δv = iR to be true at all times, Δv and i must rise and fall together.
Δv and i are in phase.
(A common inconsistency: In Δv = iR, a voltage opposing the current is positive. In E = -L di/dt, a
voltage opposing the current is negative. So for now, drop the minus and say E = L di/dt.)
Inductors (coils):
"Ohm's law-like" relationship
Phase relationship
The current is opposed by the coil's self-induced emf.
XL = ωL: A bigger frequency means more ohms, because a more rapidly changing current induces
more emf (E = -L di/dt). A larger inductance also induces more opposition, by E = -L di/dt.
Capacitors:
As with the other devices, ΔV is directly proportional to I:
ΔVRMS = IRMSXC
where
XC = 1/(ωC)
Capacitive Reactance
Current is opposed by charge already in the capacitor repelling more trying to flow in.
XC = 1/(ωC): The greater the frequency the less ohms: At high f, charge flows in for less time
before flowing out again, making average charge on capacitor less, making less repulsion. The
greater the capacitance, the less ohms, because a bigger capacitor has room in it for more charge.
Phase: Apply loop rule to this:
ΔvC + ΔvL = 0
ΔvC = -ΔvL
So, if ΔvL is a cos, ΔvC is a -cos:
Δv lags i by 1/4 cycle (90 )
To help remembering this: Eli the ice man: ELI ICE
E for the voltage, I for current. Inductor (L): E leads, I follows. Capacitor (C): I leads, E follows.
Ex. 12-2: Find the RMS current for a 6.0 μF capacitor in a household plug. (Assume "household
plug" means 60 Hz, 120 V)
-41Ex. 12-3: Repeat with 600 Hz.
RLC Series Circuit:
Effect of the phase differences:
ΔvC and ΔvL cancel because they’re out of phase. (If ΔVC = 5 V and ΔVL = 4 V, the “total” is 1 V
not 9 V.)
If you include ΔvR,
Δvtotal = ΔvR + ΔvC + ΔvL
= (ΔVR maxsin ωt) + (-ΔVC maxcos ωt) + (ΔVL maxcos ωt)
algebra, trig identities, etc.; divide by 2 to get RMS.
RMS (meter) readings:
-Phasor diagrams
-Impedance
Ex. 12-4: Find I.
-42Power:
ΔVRMS and IRMS were defined such that Pav = I2R.
So,
Pav = I ΔV cos
cos is called the circuit's power factor.
Ex. 12-5: Find the power consumed in the previous example.
Discussion of resonance.
Ex. 12-6: An RLC circuit is used as a radio's tuner. If L = 7.96 μH, what should C be to receive a
station at 1000 kHz?
Transformers:
Ideally, there is no resistance, so the induced emf is the only voltage
across each coil.
Ideally power into primary = power out of secondary, and cos = 1.
I1 ΔV1 = I2 ΔV2
Ex. 12-7: 15 amps flows through a 1000 turn primary plugged into 220 V AC. We want 200 amps
to weld with. How many turns should the secondary have?
-43Phy 132
Review of Sections 9 - 12
1. One coil is placed around another as shown. The solenoid has a
cross sectional area of.028 m2 and has 700 turns per meter of length.
The second coil has 130 turns. For a short time, the current in the
solenoid increases steadily according to I = (5000 A/s)t. Find the
voltage induced in the second coil during this time.
Ans: -16.0 V
2. In a 35 mH coil with negligible resistance, the voltage varies with time according to E = 15 cos
(140t), where E is in volts and t is in seconds. Find the current through the coil as a function of
time. Assume that I = 0 when t = 0.
Ans: I = -3.06 sin (140 t)
3. A .5 μF capacitor, initially with 80 volts between its plates, is connected to a 2.5 mH inductor at t
= 0. How much time will it take for the voltage across the capacitor to drop to 55 V?
Ans: 28.7 μs
4. An RMS current of 4.5 A flows through a 3.0 mH coil when it is connected to a 50 V (RMS),
400 Hz AC source. What is the coil's internal resistance?
Ans: 8.16 Ω
5. Short answer, 5 points each:
a. The three types of magnetic behavior are called diamagnetic, paramagnetic and
ferromagnetic. In which kind of material is the net magnetic field weaker than it would be in a
vacuum?
b. The most general form of Faraday’s law is
Explain why, in the term on the right, the derivative and the integral don’t just cancel each
other out.
c. The current in loop A is increasing. Is the current flowing around loop B
clockwise, counterclockwise or zero?
d. Suppose we increase the separation between a capacitor's plates, while keeping their charge
constant. Does the energy in it increase, decrease, or stay the same?
e. You have a capacitor and an inductor, both with negligible internal resistance. If you
connect either of them to a household outlet (120 V, 60 Hz), the 20 A fuse in the circuit will
blow after a moment. If connected to 120 V DC instead,
i. would the capacitor still blow the fuse?
ii. would the inductor still blow the fuse?
-44Section 13:
From Phy 131, recall the equation of state for an ideal gas:
PV = nRT
P = absolute pressure, V = volume, T = absolute temperature.
n = number of moles (1 mole = 6.02 x 1023 molecules)
R = the gas law constant, 8.314 in SI units.
- The First Law of Thermodynamics. (A version of conservation of energy)
-Internal energy, Eint: Add up all forms of energy possessed by the system’s molecules, relative to
its center of mass.
Eint can be changed in two ways: if heat flows or if work is done:
-Work, W, done on a gas:
dW = F cosθ dx
= - F(A/A)dx
= - (F/A)(A dx)
F/A = pressure
cosθ = -1
A = piston’s area
A dx = change in volume of gas
So, dW = -P dV
Ex. 13-1: Find the work done:
Ex. 13-2: Find the formula for the work done in an isothermal (constant temperature) process.
-Heat, Q: From Phy 131, heat is energy which flows due to a temperature difference. If a body's
temperature is changing, Q = m c ΔT. (m = mass, c = specific heat capacity.)
With a gas, it's more convenient to do the specific heat on a per mole basis, rather than per gram:
Q = n C ΔT
C = molar heat capacity, or "molar specific heat"
-45C is measured in
or
CV = Constant volume heat capacity, CP = Constant pressure heat capacity.
Relationship between ΔEint and ΔT:
Apply the laws of mechanics and statistics to the motion of the gas molecules. (kinetic theory:
see text) Result: For a monatomic (1 atom per molecule) ideal gas,
Eint = (3/2)nRT
The situation is similar for any kind of gas: ΔEint = n (a constant)ΔT. It can be shown that this
constant is CV.
- Expressions involving CV, CP and R
- At constant P, some of the energy added goes into work; only part goes into raising T. So, more
heat is needed per degree than at constant V.
Ex. 13-3: How much heat is needed to raise 2 moles of helium by 100 C at (a) constant volume, (b)
constant pressure.
Thermodynamic Processes:
An adiabatic process means no heat flow. (Q = 0)
An isothermal process means ΔT = 0. (And therefore ΔEint = 0.)
An isobaric process means constant pressure.
-Equations for the graphs of each process on a P-V diagram.
Ex. 13-4: Compression stroke in a diesel engine: Air at atmospheric pressure and 20 C is
compressed adiabatically to 1/20 of its original volume. Find the final temperature in C.
The same principle is also involved with refrigerators and other heat pumps: A substance is
compressed, raising its temperature. It then flows through tubing on the outside of the refrigerator,
radiating heat into the room. Next, it expands, lowering its temperature, and goes through tubing
inside the refrigerator, absorbing heat. It then returns to the compressor. Thus, heat is pumped from
inside the fridge to the outside. (This is also aided by the refrigerant's changes of state.)
Ex. 13-5: .004 m3 of steam at 250 C and 20 atmospheres (absolute) enters a cylinder and pushes the
piston until the temperature is 105 C. If no heat flows (adiabatic process), find the work done.
5
Ex. 13-6: Two liters of gas is at 4.0 x 10 Pa. Find the work done on it as it expands to seven liters
(a) isobarically, (b) isothermally.
-46Section 14: Heat Engines & the Second Law
- Definition of Entropy, S, for a reversible process.
Entropy measures the randomness or "disorder" of a system.
"Disorder" means a more likely state, "order" means a less likely state. For example, for 100
pennies in a box, 50 heads and 50 tails has more entropy than 100 heads, because 50/50 corresponds
to many more ways of arranging the pennies. (The more fundamental definition of entropy is S = N
k ln(W), where W is how many ways the system can be arranged in the state you're talking about.)
If you shake the box, 50/50 is the most likely to come up because it corresponds to the most ways of
arranging the pennies, and each way of arranging them is equally likely. So, if you start with all
heads, shaking them will often give 50/50, but if you start with 50/50, shaking won't give all heads:
Second law of thermodynamics: ΔS 0 (Entropy doesn't spontaneously decrease.)
Apply that to heat flow in particular: If a cold pan is placed on a hot stove, the thermal energy is
unevenly distributed, just like an uneven number of heads and tails. As the heat flows, making
things even out, the system is going into a more likely state:
Second law, restated: Heat doesn't spontaneously flow from low temp to high temp.
- ΔS for an ideal gas.
Ex. 14-1: A 0 C ice cube is placed in an insulated container of fixed volume with 100 moles of
helium at 13.37 C. 50 grams of ice has melted when the helium reaches 0 C. Find the change in
the system's entropy.
Notice that ΔS came out positive, meaning that entropy increased. This means that the second law
allows this to happen spontaneously. If you reversed everything, the ice freezing as the helium
warmed up, you'd get a negative ΔS, so that doesn't happen without outside help. (Heat flows only
from hot to cold, as stated above.)
Irreversible process:
(One way of describing a reversible process is that it has a well defined path on a P-V diagram.)
It can be shown that ΔS between two states is independent of the path taken anyhow. So:
To find ΔS for an irreversible process, calculate ΔS for any reversible process between the same two
states.
Ex. 14-2: In an insulated container, a gas
undergoes a free expansion from an volume
of Vi to Vf. Find the change in its entropy.
-47-
(Notice the process is irreversible: You can't return to the initial state without outside help; work is
done if you push the piston down. There is no clearly defined P or V during the expansion, so no
path on the P-V diagram.)
Heat Engines. (Devices which convert heat into mechanical work.)
Any heat engine must do three things: Take in heat, do work, and then give off heat. Here's an
example, designed to have those things happen one at a time, rather than to be practical:
P-V diagram:
AB: Input heat. (Constant volume process):
BC: Gas does work. (Adiabatic process):
CA: Return to original state, by removing heat. (At point C, the gas can't push the piston any
further because it's reached atmospheric pressure. But it's still at an elevated temperature, T2. To
repeat the cycle, it must get back to point A. In a practical engine, the hot gas flows out the exhaust,
but if you
weren't in a hurry, you could just pack the cylinder in ice and contract it back to Vo. It's heat, not
gas, you have to get rid of.)
Net work per cycle done BY engine = area enclosed on PV diagram.
-48-
Ex. 14-3: Find the net work per cycle.
Energy flow through any kind of engine:
- Definition of efficiency.
Example: If fuel containing 1000 J is used to put out 300 J of work the efficiency is .30, or 30%.
- Conservation of energy.
Conservation of energy (the first law) says the efficiency of anything must be 100%. With other
kinds of devices (levers, electrical transformers, etc.), percent efficiencies do commonly make it
into the 90's. But the best heat engines (electric generating plants) are around 35% efficient; your
car is more like 15%. This is because the second law, which has nothing to say about those other
devices, imposes a lower maximum on the efficiency of a heat engine.
- Maximum efficiency of a heat engine. (Carnot efficiency.)
Ex. 14-4: Steam enters a cylinder at 250 C, and leaves at 105 C. What is the maximum possible
efficiency of this engine?
(Friction, shortcomings of the design, etc, cause further reductions in efficiency below this
theoretical maximum.)
Another way to think of it: Consider an engine and an air conditioner whose reservoirs are a cool
room and the warmer air outside. The engine, rather than electricity, drives the air conditioner. If the
engine doesn't waste enough heat into the room, its heat output to the room is less than the air
conditioner's heat output to the outside, meaning a net heat flow from cold to hot. With no energy
input to make this happen, 2nd law is violated. So, engines can't be more efficient than the formula.
-49-
The Carnot Cycle has this maximum possible efficiency:
Ex. 14-5: The engine's true efficiency is 14%. If its power output is 100,000 watts, find the energy
taken in each second and the energy given off as waste heat.
What's so special about heat? Other forms of energy can, in principle, be converted into each other
with an efficiency of 100%.
Being changed into heat "degrades" energy, because thermal energy is associated with disorganized
motion. To convert heat into 100% mechanical energy would mean a spontaneous increase in
organization - like shaking pennies, and getting all heads.
-50Review for Final Exam
The test has eight parts, each worth 25 points. The best seven you do will be counted. (Prefect
score = 175.)
The questions below were picked for being things people often need more work on, not for
similarity to the actual test. You should review the whole course, not just the topics on this review
sheet.
1. A thin spherical shell of radius 2.00 cm has 5.00 pC of charge uniformly distributed over its
surface. E is electric field’s magnitude at a point which is a distance r from the center. Sketch a
graph of E as a function of r from r = 0 to beyond r = 2.00 cm.
2. Two current carrying loops of wire lie in the yz plane. The
outer one has a radius of 2.236 m and carries 1.60 A. The inner
one has a radius of 1.00 m and carries 1.00 A in the opposite
direction. At what point(s) along the x axis does B = 0? (Other
than x = ± )
Ans: .577 m & -.577 m
3. The picture shows the cross sectional view of a solenoid.
The uniform magnetic field inside is growing stronger
3
according to B = .17t , where B is in teslas and t is in
seconds. At t = 2.5 s, what is E, the magnitude of the
induced electric field, just outside the solenoid?
Ans: .478 V/m
4. Three moles of an ideal gas is compressed isothermally at 100 C to one fifth of its initial volume.
Find:
a. the work done on the gas,
b. the change in its internal energy,
c. the heat flow into the system.
Ans: 15.0 kJ, 0, -15.0 kJ
5. In a vacuum,
a. an electron is released from rest, then accelerates through 1000 V. What is its final speed?
b. an electron is released from rest far from an oil drop with a charge of +e and a radius of 3.00
μm. With what speed does it strike the oil drop (which remains stationary)?
Ans: 1.87 x 107 m/s, 1.30 x 104 m/s
-51-
6. Find at the origin due to the charged semicircle. Charge is
distributed along it according to the function dq = sinθ dθ, where
θ is in radians.
Ans:
7. To determine the inductance of a coil, you first measure its
internal resistance, and get 20 ohms. You then connect it and an
AC ammeter to a 50 V, 60 Hz source, and see a current of 1.60
A. What is L?
Ans: 63.7 mH
8. Short answer, 5 points each:
a. Assume the inductor has no resistance. Which point (A or B) will be at the higher potential
when the current is
i. increasing?
ii. decreasing?
b. The first law of thermodynamics (conservation of energy) says no device can have an
efficiency over 100 %. If the device is a heat engine, there is a lower maximum due to another
basic principle. What principle is that?
c. i. Which one of Kirchhoff's Laws is derived from conservation of energy? (Refer to them by
name, not something like "first law": Not all books list them in the same order).
ii) Which one is derived from conservation of charge?
d. When a certain capacitor is charged to 4 V and then connected to an inductor, the circuit
oscillates at 4000 Hz. What would the frequency be if the capacitor had been charged to 16 V
instead?
e. What is the more common name for a coulomb per second?