2015
NOTE: Not all of these problems are assigned, and those that are are not in sequential order.
QP#1
and let the device of interest be modeled by a square loop of wire of area A located a distance R
from the power line. R
A1/2 so that the magnetic field due do I(t) can be approximated as a
constant at any given time over the whole area A. Assume that the power line lies in the plane of A
to maximize its e⇤ect.
A
R
I(t)
What is the emf E induced in the loop due to I(t), in terms of the parameters given?
The power line runs at 2 kV (RMS) at 60 Hz. The typical rms power drawn by the locomotive is
1.5 MW (i.e., 1.5 ⇥ 106 watts). In the design, R was about 30 cm, and A is typically no bigger than
10 cm ⇥ 10 cm (i.e., 10 2 m2 ).
What is the numerical estimate of the emf E induced in a typical device due to the power line, for
the values given?
QP 3
At the Caltech Athletics Department 1978 awards dinner, Mary Mack (B.S. ’78), three-time NCAA
Western Division jump rope champ, was presented a deluxe jumping rope whose outer layer was a
braid of copper and brass threads. It sparkled, but it also conducted electricity from one end to the
other. Hence, when jumping in the Earth’s magnetic field (B ⌃ 0.5 ⇥ 10 4 T), Miss Mack generated
a voltage V between the ends of the rope. (Also, Miss Mack could easily do 5 jumps per second.)
To determine whether it was safe to jump without insulating gloves, first consider the induced
voltage V . In particular, write
V ( , t) = Vmax · f ( ) · g(t)
where Vmax is the maximum value of V ( , t), is the direction she faces with
(magnetic) north, = ⌅/2 being due west, etc., and t is the time.
= 0 being due
a) What is Vmax ? Express your answer first algebraically in terms of B (the field of the Earth), l
(the length of the rope), and , ⇤, or T (the angular frequency, frequency, or period of the motion,
respectively [ = 2⌅⇤ = 2⌅/T ]) and assuming that the motion is uniform and the rope forms a
semicircle. Write down reasonable numerical values for the relevant parameters to find a numerical
estimate, in Volts, of the potential generated.
b) What is g(t) in terms of the parameters of the problem?
c) What is f ( )?
2
Held tightly in sweaty hands, with full hand contact, the rope-to-two-hands (in series) resistance
was only 500⇥ in total. Compared to that contact resistance, the rope and her internal resistances
were negligible.
d) Estimate the largest possible peak current induced by the Earth’s field that passed through her
arms when she first tried out this rope.
e) Taking account of the fact that the Earth’s field in Pasadena has a vertical component as well
as a horizontal component, was there an orientation for which Miss Mack would have generated
essentially no current while she jumped? If so, what would have been the relation of her hands, e.g.,
the straight line that connected her two hands, to the direction of the B field?
f) What were the color of Miss Mack’s buttons?
3
QP 4
B
θ
v
x
figure 1
A conducting bar is sliding at velocity v to the right on a V-shaped conducting rail, as shown
in figure 1. There is a uniform magnetic field B out of the page. The rails are frictionless and
resistanceless. The bar has a resistance ⇥ per unit length. The half-angle of the V is .
(a) What is the current I as a function of time (taking the position of the bar to be x = 0 at time
t = 0)?
(b) What is the direction of the current?
(c) What is the magnitude and direction of the force required to maintain a constant velocity versus
time?
4
QP 5
In a standard design of an AC voltage transformer, two wire coils are wound around the central
post of an iron “core” (which is really more of a frame) as shown in the sketches below. Iron is chosen
for its outstanding magnetic permeability. Note that in the sketches, the dashed lines denote edges
that are not visible from the outside in that particular view.
When an AC current passes through the first of the coils, it induces electric fields in the iron which,
in turn, generate currents in the iron. These currents are undesirable because they lead to Ohmic
heating and consequent power loss and because, by Lenz’s Law, they reduce the desired magnetic
field.
To reduce the e⇤ects of these undesired currents the iron is laminated. That means that it is
actually made up of thin sheets of iron that are insulated from each other with a layer of varnish.
5
a) Of the four lamination geometries, A, B, C, or D, suggested by the accompanying sketches,
which one would be most e⇤ective at minimizing the currents induced in the iron?
b) Of the three lamination geometries, A, B, or C, suggested by the accompanying sketches, which
one would be least e⇤ective at minimizing the currents induced in the iron?
c) (for thought, not for points — ) So what’s with geometry D? To answer parts a) and b) above,
you have to distiguish between eddy currents with diameters of order 1 cm and those restricted to
be less than one lamination thickness. Would there be eddy currents in D? Where? How big? What
limits their magnitude?
6
QP 6
r
2
r
N turns
1
d
figure 2
An iron toroid of rectangular cross section has an inner radius r1 , an outer radius r2 , and a thickness
d, as shown in figure 2. A large number of turns, N , of wire are wound uniformly around the iron
core, which has magnetic permeability µ. (You should ignore the resistance of the wire.)
(a) A steady current I is flowing in the coil, which generates a magnetic field B in the toroid. Derive
an expression for B within the iron toroid as a function of the radial distance from the symmetry
axis (r1 < r < r2 ) in terms of the quantities given.
(b) Using the result of part (a), obtain an expression for the self-inductance L of this coil. You
should obtain a result of the form L = N 2 · L0 , where L0 is the single-turn inductance. What is L0
in terms of the properties of the toroid given above (i.e., µ, r1 , r2 , and d)?
QP 7
Consider a long air-core solenoidal coil with N turns and total inductance L0 . A constant current
I is flowing in the coil. Answer the following in terms of the quantities L0 , N , and I.
(a) What is the magnitude of the magnetic flux
B
through each turn?
(b) What is the total energy stored in the coil?
Now a rod of soft iron with magnetic permeability µm is inserted into the solenoid, completely
filling its interior volume. The current through the coil is held fixed at the value I. Answer the
following in terms of the quantities L0 , N , and I (as above) and µm .
(c) What is the new magnetic flux
0
B
through each turn?
(d) What is the new value of inductance L?
(e) What is the total energy now stored in the coil? If the energy is di⇤erent, discuss the origin of
the increase or decrease in terms of the principle of conservation of energy.
7
QP 8
The formula derived in ZAP! and used in Experiment 10 for the op-amp amplifier circuit shown
below is correct in the limit that the “open loop” gain, G, of the op-amp itself is enormous.
+
-
R2
Vin
Vout
R1
Open loop gain G is defined by
Vpin 6
(+)
Vground = G(Vpin 3
( )
Vpin 2 )
where the four voltages are defined in the next figure.
(In actuality, the op-amp Vground is slightly ambiguous until some sort of feedback is established.
Another one of the op-amp’s undrawn inputs can be used to adjust that value to agree exactly with
some other reference voltage. This would allow, for example, adjusting the output voltage of your
101⇥ amplifier to zero when the amplifier input voltage is zero.)
(+)
Vpin
3
+
(-)
Vpin
2
-
Vpin 6
Vground
Your op-amp has a value G ⌦ 5 ⇥ 105 for DC applications. For slowly varying sinusoidal inputs
this value is maintained, but at some high frequency it begins to drop — eventually reaching values
that aren’t big at all.
(The frequency at which a significant decline in G sets in is different for different models of op-amp.
However, there are always engineering trade-offs, and better high frequency response is not always the
sole desideratum; in some cases it is actually undesirable.
Hence, it may be of interest to know how the amplifier circuit above behaves for G = ↵.)
8
a) Find the formula for the amplification factor Vout /Vin , for finite G, in terms of R1 , R2 , and G.
You should assume that no current flows in or out of the + or op-amp inputs even though voltages
are applied.
This last idealization concerning op-amp input currents is not precisely valid either. The current
that actually flows in or out of the + or
op-amp inputs can be roughly characterized as follows.
The real inputs behave as if there were a large resistance Rin to ground which precedes an ideal
op-amp that allows no input current flow. This is represented in the following figure.
R in
+
real
-
+
ideal
-
R in
b) Find the formula for the amplification factor Vout /Vin of the original amplifier circuit in terms
of R1 , R2 , and Rin , assuming G = ↵.
(In contrast to all the previous real-world modifications to ideal behavior explored in this quiz,
modern designs have made this one pretty irrelevant. In particular, your op-amp has Rin ⌃ 1012 ⇥.)
c) Describe in words (a few is OK, not more than four sentences) what happens to Vout in the
amplifier circuit with a real op-amp when the + and
inputs are inadvertently reversed, i.e., as
shown below.
(no electrical
connection
here)
+
-
Vout
Vin
9
QP9
X
R
L
Y
C=1 µf
Z
figure 4
In your laboratory kit you find an unlabeled inductor. Let R be its internal resistance and L be
its inductance. You quickly determine R to be 35 Ohms using your ohmmeter. Curiosity drives you
to go to the help lab where you find a 1000 Hz signal generator and a 1 microfarad capacitor. Your
a⌅nity for the smell of melting solder then drives you to construct the circuit shown in figure 4.
Using your AC voltmeter, you measure the RMS voltage between points X and Z to be 10.1 volts.
The RMS voltage between Y and Z is then measured to be 15.5 volts.
(a) What is the RMS current in the circuit?
(b) What are the two values of L (in henries) that are consistent with these data?
(c) For each value of L from part (b) predict the voltmeter reading between X and Y . Therefore,
one can make this final measurement to deduce the correct value of L.
10
QP 10
Consider a battery connected to an inductor through a switch, as shown. The point of this problem
is to begin to figure out what actually happens when the switch is opened.
The instant the switch is opened, it actually becomes a capacitor, with capacitance C, and C ⌦ 100
pF = 10 10 F. (It is, after all, two pieces of metal, separated a small distance by an insulator.) The
largest resistance in the circuit is Rint , the internal resistance of the battery, and Rint ⌦ 0.6 ⇥, while
the unloaded voltage of the battery is Vo , with Vo ⌦ 3 V. The inductor has an inductance L, with
L ⌦ 0.02 H. (While there certainly are other resistances, capacitances, and inductances in the circuit,
these others are negligible in magnitude compared to the ones just described.)
Let t = 0 be the time of opening the switch. The steady-state behavior established before t = 0
serves as initial conditions on the t ⇧ 0 system. In particular, if Q(t) is the charge on the capacitor
(i.e., the open switch), then Q(0) = 0. And, if I(t) is the current in the circuit, then I(0) = Vo /Rint .
[In your answers for parts a) and b), please use the symbols Vo , Rint , C, and L rather than their
numerical values.]
a) What is the di⇤erential equation that governs the t-dependence of Q(t) for t ⇧ 0? (Hint: draw
the e⇤ective circuit diagram and follow the voltage drops and rises all the way around á là Kirchho⇤’s
loop rule.)
b) The answer to part a) can be cast into the form of the equation for the RLC circuit (i.e., without
any Vo ) by a change in variables that involves shifting Q(t) by a time-independent constant. What
is that constant shift (in terms of the parameters of this problem)?
c) If there were absolutely no resistance at all in this circuit (e.g., Rint ⇤ 0), it would oscillate
indefinitely as t
↵. (Imagine, however, that I(0) were still some finite initial value.) What would
be the period of those oscillations (in seconds, using the numerical values as provided)?
d) Taking into account the actual value of Rint , estimate the decay time of the current, i.e., the
time it takes to drop roughly to 1/e of its t = 0 value (in seconds, using the numerical values as
provided).
11
QP 11
+
R1
v
v
IN
OUT
L
C
i
R
2
figure 5
Recall that a well-designed circuit with an op-amp can be analyzed using two properties of ideal
op-amps:
1. The current into the + and – inputs is 0.
2. The + and – inputs are at the same voltage relative to ground.
Consider the circuit shown in figure 5. An AC signal is applied at vIN . This circuit forms a
frequency-dependent amplifier. It has the property that the input impedance is very large. In other
words, no matter how large a voltage is applied at vIN , very little current is drawn through the input.
(a) Draw a phasor diagram for the two resistors, the capacitor, and the inductor, indicating i, vR1 ,
vR2 , vC , vIN , and vOU T on the diagram.
(b) What is the ratio of voltage amplitudes VOU T /VIN for this circuit?
(c) In the region of frequency where
output voltage vOU T ?
L > 1/( C), does the input voltage vIN lead or lag the
(d) What is the ratio of voltage amplitudes VOU T /VIN for values of the frequency which are very
large? What is the ratio for very small? (e) At what frequency 0 do you expect the signal to be a
maximum? (Note: it is not necessary to di⇤erentiate to write down this answer!) What is the value
of VOU T /VIN at = 0 ?
12
QP 12
Traditional electric guitar design goes back well before the invention of op-amps and transistors,
but it does include built-in combinations of capacitors and variable resistors that allow the player to
adjust the volume and tone of the output with knobs on the face of the guitar. The initial electrical
signal is the voltage induced in the “pick-up” coil by the oscillatory motion of the magnetized strings.
“Volume” is altered by feeding the signal through a variable resistor. Tone control is achieved with
a variety of variable R filters.
a) If we consider this as a system with a specified vin (t) and a desired vout (t), sketch a circuit
that could serve as a low-pass filter using a single capacitor C and a single variable resistor R, i.e., it
would pass from “in” to “out” all frequencies well below some adjustable point and seriously attenuate
signals well above that point. I.e., draw the appropriate connections to a C and an R in place of the
“?” box in the following diagram.
?
vin(t)
vout (t)
For historical reasons, the industry standard for such variable resistors is a 250 k⇥ pot, i.e., 0 ⌅
R ⌅ 250 k⇥.
b) What is the minimum value of C that ensures that the roll-over frequency, i.e., the point where
the filter gives Vout /Vin = 1/2 for the respective amplitudes of sinusoidal voltages, can be varied to
include the range of 1000 to 10,000 Hz when combined with a 250 k⇥ pot? (Don’t forget to distinguish
between the angular frequency in radians per second and the oscillation cycle frequency ⌃ in Hz.)
While the considerations above give a reasonable estimate of the appropriate C, real guitars are
not wired this way. A relevant consideration is that the pick-up is not an ideal voltage source. While
it is true that the vibrating magnetized strings induce voltages in the pick-up coil, the coil’s voltage
output is influenced by the actual current and its time derivative. Just like the battery and power
supply internal resistances that you measured, a magnetic pick-up coil has an internal resistance that
can be as high as 10 k⇥ — simply because it is an enormous length of very fine wire. Furthermore,
it has its self-inductance, which, at a sizeable fraction of a Henry, can have a significant influence on
the circuit at audio frequencies.
Hence a better model of the pick-up coil is a voltage source vin (due to the action of the strings)
in series with the coil’s self-inductance L and its internal resistance Rint . A typical hook up is
then described by the diagram below, where Rpot is the 0 to 250 k⇥ variable resistor and C is the
accompanying capacitor.
L
v in
R int
R pot
vout
C
c) Assuming an input voltage of angular frequency , find the formula for the voltage amplitude
ratio Vout /Vin for the circuit above in terms of L, Rint , Rpot , C, and .
13
QP 13
A variable inductor (inductance Ladj ) can be used as a dimmer for lights in an AC circuit, and it’s
far more e⌅cient than using a variable resistor. Consider a 100 W bulb plugged into a wall socket
(which provides 60 Hz 120 VRMS
AC ).
First, consider an adjustable resistor as a dimmer, as indicated below.
Radj
VAC
Rbulb
If Radj is adjusted so that the bulb runs at 25 W, what is the total (time-averaged) power being
drawn from the wall socket? (You should assume that the bulb is approximately “Ohmic” over the
range 25 to 100 W, i.e., has a fixed resistance Rbulb such that it runs at 100 W when connected
directly into the wall socket.)
And compare to using an adjustable Ladj , wired in series with the bulb, as shown below.
L adj
VAC
Rbulb
For what value of Ladj will the bulb run at 25 W?
For that value of Ladj , what is the total (time-averaged) power being drawn from the wall socket?
14
QP 14
Consider a 10:1 step-down transformer that plugs into a 60 Hz, 120 V (rms) wall socket and
provides 12 V (rms, 60 Hz) for low voltage applications. The primary (high voltage) and secondary
(low voltage) coils are wound around a common iron core with a winding number ratio of 10:1. The
relevant inductances are L2 = L = 0.010 H, L1 = 100L = 1.0 H, and (mutual) M = 10L = 0.10
H so that L1 /M = 10, L2 /M = 1/10, and L1 L2 = M 2 . The resistances of the coils themselves are
negligible and are to be ignored in the following, but the output may or may not be connected to a
load resistance R — depending on whether the switch is closed or open. The transformer input and
output voltages are ⇤in (t) = Vin cos t and ⇤out (t) = Vout cos( t + ⌥out ) with Vin = (120 V) 2 and
= 2⌅ ⇥ 60 sec 1 .
i1(t)
vin(t)
i2(t)
L1
R
L2
vout(t)
M
With the switch open, the behavior of this system is governed by the equations
⇤in (t)
⇤out (t)
1
L1 di
dt = 0
1
M di
dt = 0
While with the switch closed,
⇤in (t)
⇤out (t)
1
L1 di
dt
2
L2 di
dt
2
M di
dt = 0
1
M di
dt = 0
⇤out (t) = i2 R
Consider first the situation of the transformer plugged into the wall socket but no connection made
to the transformer output, i.e., the switch is open in the accompanying circuit diagram. Hence, there
is no current in the secondary coil, i.e., i2 (t) = 0, and the simpler first set of equations apply.
What is the phase angle between the voltage across the primary coil, vin (t), and the current that
flows in the primary coil, i1 (t)? (Give a numerical value in degrees.)
What is the instantaneous power delivered to the primary coil as a function of time? (Express your
answer in terms of the parameters given.)
Consider now closing the switch and connecting the transformer output to some device which, for
simplicity, is modeled as an R = 100 ⇥ load resistance.
What is the time-averaged power dissipated in the secondary circuit?
What is the time-averaged power delivered by ⇤in ?
15
QP 15
At the mid-point of a long, thin, straight wire that carries a steady current I is a capacitor made
of two parallel, circular plates of radius R and separation D, with D ⌥ R, as shown. The current
charges the capacitor.
r
R
I
I
D
a) What is the magnitude of the magnetic field B(r) half way between the plates and at a distance
r from the axis that passes through their centers? (Give the answer for the whole range 0 < r < ↵.)
b) Sketch B(r) versus r for 0 ⌅ r ⌅ 2R. On the same graph use a dotted line to represent the
magnitude of the magnetic field a distance r from the wire at a location along the wire that is far
from the capacitor. (Please make the diameter of your dots about double the thickness of your first
line so that there is no ambiguity if there are values of r for which the two magnetic fields have the
same value. Also, please take some care in making the sketch. If the function in question is linear,
don’t give it curvature and vice versa. If it is concave down, don’t draw it concave up. And if it is
di⇤erentiable, don’t draw it with a kink.)
16
QP 16
A straight wire carries a current I(t) along the negative z-axis from -↵ to the origin. There the
wire ends, and a charge q(t) builds up on a small metal sphere centered at the origin. The goal of
this problem is to compute the resulting magnetic field B at points P and P 0 using the fact that this
situation has rotational symmetry about the z-axis and using the Ampère-Maxwell law, i.e., including
the displacement current.
Both P and P 0 are located a distance R in the y-direction from the z-axis. P is at z = 0, and the
line from the origin to P 0 makes an angle with the z-axis as shown.
a) What is the electric flux
“centered” at the origin?
E (t)
that passes through the upper (z > 0) hemisphere of radius R
b) Use the z > 0 hemisphere to deduce the magnitude B of the magnetic field at P in terms of
I(t) and R, using the Ampère-Maxwell law.
c) Would you get the same answer if you used the lower, z < 0 hemisphere? Explain your answer.
z
P’
θ
P
R
y
I(t)
polar cap
z>0 hemisphere
θ
r
d) Evaluate the magnitude of the magnetic field B at P 0 . The following geometrical fact should
prove useful: The part of the surface area of a sphere of radius r (i.e., a sphere with 4⌅r2 total area)
that forms the “polar cap” within an angle of the polar axis is 2⌅r2 (1 cos ). (Note that this cap
area is zero for = 0 and half of the total for = 90o .)
17
QP 17
For the Earth, the Sun’s gravitational attraction dominates by far over the e⇤ects of radiation
pressure. Comets, however, have tails consisting of small particles of condensed dust and ice.
(a) Assume that the dust grains in a comet’s tail are reflecting spheres and that they all have the
same density, ⇧. Show that particles with radius less than some critical value will be blown out of
the solar system. You may approximate the spheres as reflecting disks oriented toward the sun in
considering the e⇤ect of incident radiation.
(b) Estimate the critical size numerically, assuming the density ⇧ = 1 gm/cm3 and the power
output of the sun WSun = 3.8 ⇥ 1026 Watts. You may need the following constants: G = 6.7 ⇥ 10 11
N m2 /kg2 and the mass of the sun M = 2 ⇥ 1030 kg.
QP 18
A microwave communications link consists of two identical antennae. Each antenna works symmetrically in send or receive mode.
a) Assume antenna 1 radiates 1 W/m2 into a beam of circular cross section with a diameter equal
to the diameter of the parabolic reflector, D = 1 m. Calculate Erms and Brms for the radiated beam.
D =1m
L = 10 km
b) Assume the parabolic reflector falls o⇤ antenna 1, which still radiates the same power but now
emits this power isotropically, i.e., uniformly in all directions. How much power is received at Station
2?
18
QP 19
Adrienne
Beatrice
Chloe
v=0.99c
Axel
figure 7
Brad
Clyde
Photograph taken by camera at the station
(a) A high-speed (v = 0.99c) express train is traveling past the Pasadena station. Axel, Brad, and
Clyde are standing in the station. Axel and Clyde are at the ends of the station platform; Brad is
in the center. (Note that all observers are wearing high-priced, accurate [and precise] Swiss watches
that have been synchronized in their own rest frame.) The proper length of the train is 100 m. What
is the length of the train if measured in the rest frame of the station?
(b) Axel measures the time it takes for the train to pass him, front to back. What does he measure?
(c) Three observers are riding the train, Adrienne, Beatrice, and Chloe. Adrienne and Chloe are
at the ends of the train; Beatrice is at the center. Each of them notices the time on Brad’s watch as
they pass him, along with the time on their own watch. Beatrice notes that both on her own watch
and on Brad’s watch t = t0 = 0. What time does Adrienne observe on her watch and on Brad’s as
she passes him?
(d) What time does Brad observe for Beatrice’s passage?
QP 20 (The saga of QP15 continues....)
(a) Suddenly, as if the participants lived in a Physics text gone terribly wrong, two huge lightning
bolts hit the front and back of the train. Axel and Clyde record the times of the bolts, and they are
simultaneous in the station frame. (All participants survive, as this is the Hollywood version.) A
little bit later, Brad sees the bolts hitting simultaneously and faints instantly. How much later?
(b) Denise (the “Fourth Woman”) happens to be standing in the train at such a location as to
pass Brad at the instant he faints. Does she also see both bolts of lightning simultaneously? Is
she standing in the center of the train? How do you know? What does she conclude about the
simultaneity of the bolts hitting the train in her frame?
19
QP 21 (Following long established tradition, we escape to the realm of Fantasy & Science Fiction
to elucidate issues in Special Relativity.)
Sir Bevis and Count Rumpkopf were to joust to the death in the lists at Canterbury in the Spring
tourney. The wizard Merlin approached Bevis with an o⇤er. In return for rights to all Bevis’ lands,
Merlin would provide enchanted oats for Bevis’ horse that would allow the horse to run at relativistic
speed. Merlin explained the advantage as follows. The knights’ lances were of equal length when at
rest. If they approached each other at relativistic speed, in Bevis’ own rest frame, Rumpkopf’s lance
would be genuinely shorter. Having the longer lance, Bevis could pierce Rumpkopf’s chest and then
jump out of the way of Rumpkopf’s approaching lance. (Of course, unbeknownst to Bevis, Merlin
made the same deal with Rumpkopf.)
Let Lo be the rest length of each knight’s lance. They charge at each other with equal speed v 0
relative to the ground. Let v be the speed at which Rumpkopf approaches Bevis as determined in
Bevis’ rest frame.
(In Bevis’ rest frame, Merlin and the ground itself approach Bevis at speed v 0 where
v = (c2 /v){1 [1 (v/c)2 ]1/2 }.)
0
(Express answers in terms of Lo and v.)
a) What is the length of Rumpkopf’s lance in Bevis’ rest frame?
20
Consider carefully the following possible events. Event A: Bevis’ lance pierces Rumpkopf’s chest.
The horses continue running at a constant speed, and (event B:) Rumpkopf’s lance pierces Bevis’
chest.
b) What is the distance between event A and event B in Bevis’ rest frame? (Hint: If the answer is
not immediately obvious, consider the above sketch, which is meant to represent an instant in Bevis’
rest frame.)
c) What is the time between event A and event B in Bevis’ rest frame?
d) How does the distance of part b) divided by the time of part c) compare to the speed of light, c?
(Is it greater than c, less than c, or is it not determined without numerical values for the parameters
of the problem?)
e) Does Sir Bevis have time after seeing his lance pierce Count Rumpkopf to jump o⇤ his horse
and avoid death? (You may assume that his nerves are excellent and his reaction time is essentially
zero.)
f) What is the time between event A and event B in Merlin’s rest frame (in which the horses have
equal [but opposite] speeds)?
21
QP 22
In an electron-positron collider, the electron is accelerated to an energy of 5.11 MeV. The electron
mass is 511 keV/c2 .
(a) What is the velocity (expressed as v/c), kinetic energy, and momentum of the electron?
A positron is accelerated in the opposite direction to the same energy and collides with the electron.
(b) What is the maximum mass of a new particle X created by the collision [e+ + e
is the new particle’s momentum in this case?
X] ? What
(c) What is the velocity (expressed as v/c) of the positron in the electron’s rest frame? [Careful!
This calculation requires some precision.]
(d) What is the total energy in this frame? Discuss why most accelerators now use colliding beams
as opposed to fixed targets.
A new particle, the Techion, is created with a rest mass 3 MeV and total energy 6 MeV in the
laboratory frame. In the lab frame, the Techion travels 2 m between where it is created and where
it decays.
(e) What is the proper decay time (i.e., the decay time in its own rest frame) for the Techion?
22
QP#23##
!
You!have!a!solenoid!whose!length!L!is!large!compared!to!its!radius!R.!!The!solenoid!
lies!along!the!x:axis!from!x=:L/2!to!+L/2.!!There!is!a!current!Is!flowing!in!the!
solenoid,!producing!a!magnetic!field!which,!in!its!interior,!points!along!the!!
+x:direction.!!!
!
A!small!metal!ring,!of!radius!r!<!R!is!moving,!at!constant!speed!v,!toward!the!
solenoid.!!The!ring!is!centered!on!the!x:axis!and!is!oriented!so!that!its!plane!is!
perpendicular!to!the!axis.!!As!time!goes!on,!the!ring!will!enter!and!eventually!leave!
the!solenoid.!!Assume!that!at!time!t=0!the!ring!is!at!the!center!of!the!solenoid.!See!
the!drawing!below:!
!
!
!
ring
!
!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!
solenoid
x:axis
!
v
!
!
Consider!the!current!Iring!induced!in!the!ring.!!For!definiteness,!define!Iring!as!positive!
if!the!magnetic!field!it!creates!at!the!ring's!center!points!along!the!positive!x:axis.!
!
You!may!assume!that!the!electrical!resistance!of!the!ring!is!sufficiently!large!that!the!
magnetic!flux!passing!through!the!ring!due!to!its!own!magnetic!field!is!always!
negligible!compared!to!the!the!flux!produced!by!the!magnetic!field!of!the!solenoid.!
!
a)!(2!points)!Make!a!graph!of!Iring!versus!time.!!Have!the!graph!run!from!times!well!
before!the!ring!enters!the!solenoid,!until!well!after!it!leaves.!!NOTE:!Although!you!do!
not!need!to!quantitatively!calculate!Iring,!your!graph!should!be!accurate!in!all!its!
qualitative!features.!!!
!
b)!(2!points)!Make!a!graph!of!the!force!vs.!time!that!must!be!applied!to!the!ring!by!
some!external!agent!in!order!to!maintain!the!speed!v!of!the!ring!constant.!!Be!sure!to!
indicate!the!sign!and!direction!of!the!force.!!!!
!
c)!(2!points)!Make!a!graph!of!the!Joule!heating!occuring!in!the!ring!as!a!function!of!
time.! How!would!this!graph!change!if!the!ring!was!replaced!by!another!which!had!
twice!the!electrical!resistance?.!
!
!
QP#24#
#
A"Physics"1"student"connects"a"power"supply"to"a"rectangular"loop"of"wire.""The"
power"supply"is"sitting"on"a"table"and"the"wire"loop"stands"vertically"as"indicated"in"
the"figure.""Unknown"to"the"student,"there"is"a"large"uniform"magnetic"field"B0"
oriented"perpendicular"to"the"loop,"but"only"over"its"upper"portion.""In"the"diagram,"
the"magnetic"field"points"out"of"the"page."
"
"
"
"
gravity,"g
L
power"supply
"
"
B0"pointing"out"of"
page"inside"dashed
region
table"top
"
"
"
The"total"mass"of"the"power"supply"and"loop"is"M"and"the"width"of"the"loop"is"L,"as"
shown.""The"wire"loop"has"resistance"R."
"
Note"that"the"power"supply"is"capable"of"keeping"the"current"constant"at"any"preset"
value.""It"does"this"by"adjusting"the"voltage"it"delivers.""In"this"sense,"it"differs"from"
an"ordinary"battery"which"supplies"a"constant"voltage,"not"current."
"
a)"The"student"begins"to"increase"the"current"supplied"by"the"power"supply.""At"
some"point,"to"her"surprise,"the"power"supply"lifts"off"the"table"and"begins"to"move"
upward!""(Don't"worry"about"the"power"cord!)"By"carefully"adjusting"the"current,"
she"is"able"to"maintain"the"power"supply"and"loop"at"a"constant"height"above"the"
table.""Determine"the"magnitude"and"direction"(clockwise"or"counterLclockwise"in"
the"diagram)"of"the"current"I0"and"the"voltage"V"across"the"power"supply"terminals."
"
b)""With"the"current"set"to"the"value"determined"above,"the"power"supply"and"loop"
are""floating""at"constant"height.""The"student"now"gives"the"power"supply"a"slight"
nudge"and"it"begins"to"move"upward"at"constant"speed"v0.""The"power"supply"
maintains"the"current"at"the"value"I0,"but"the"voltage"V"across"its"terminals"changes."
Find"this"new"voltage,"expressing"your"answer"in"terms"of"the"variables"given.""Is"the"
voltage"larger"or"smaller"than"when"the"system"was"floating"at"rest?"
"
c)"With"power"supply"drifting"upward"as"in"b),"how"much"electrical"power"is"being"
delivered"by"the"supply?""Show"that"this"power"matches"the"Joule"heating"in"the"wire"
plus"the"increasing"gravitational"potential"energy"of"the"system."
"
d)""It"turns"out"that"a"second"student,"hiding"behind"a"curtain,"is"in"control"of"the"
magnetic"field.""This"second"student"now"increases"the"magnetic"field"by"an"amount"
ΔB.""The"power"supply"and"loop"begins"to"accelerate"upward"(the"current"remains"
constant"at"I0).""Find"the"upward"acceleration"a."
"
e)"Throughout"this"problem"the"magnetic"field"appears"to"be"the"only"source"of"an"
upward"force.""So,"when"the"power"supply"moves"upward"it"would"seem"that"the"
magnetic"field"is"doing"work"against"gravity.""But"magnetic"fields"can"do"no"work!"
Show"that"in"fact"the"magnetic"field"is"doing"no"actual"work.""What"then"is"doing"the"
work?"
"
QP 25
Two identical square loops of wire, of side length a and resistance R, are glued onto a nonconducting substrate. The loops are placed right next to one another, the two adjacent sides being
almost, but not quite touching. See the drawing below.
There are two sources of AC magnetic field available to you. One magnetic field source just
barely covers the left-hand wire loop, while the other magnetic field barely covers the right-hand
wire loop. The fields are uniform over the regions they cover and point perpendicular to the
plane of the loops. (Nevermind that such a field configuration would be very hard to create!)
You may assume that the left-hand field source injects essentially no flux into the right-hand loop
and, similarly, the right-hand field injects no flux into the left-hand loop. Importantly, however,
the two field regions do very slightly overlap; the edges of the two loops which are nearly
touching experience BOTH magnetic fields.
Throughout this problem you may ignore the self- and mutual inductances of the loops.
!!
! !
left%wire%loop
left%field%region
right&wire&loop
field&overlap&region
right&field&region
!
!
a) Suppose only the left-hand magnetic field is non-zero, and it is varying periodically:
BL(t) = B0 cos(ωt). Find the magnitude of the electrical current flowing in each.
b) Find the magnitude of the net force on the system under the conditions of part a).
Now suppose the right-hand magnetic field is also turned on. It has the same frequency and
amplitude as the left-hand field, but is phase-shifted relative to it: BR(t) = B0 cos(ωt + φ).
c) Find the net force acting on the system under these new conditions. For what phase shift is the
force maximized? Does the force depend on time? If so, how?
Hint: Some useful trig identities: cos(ωt+φ) = cos(ωt) cosφ - sin(ωt) sinφ;
sin(ωt+φ) = sin(ωt) cosφ + cos(ωt) sinφ
!
!
1!
QP 26
A 'transformer' is nothing more than two inductors coupled by a mutual inductance M. An ideal
transformer maximizes M by making sure that all of the magnetic field lines which pass through
one inductor also pass through the other. Typically the transformer consists of two coils of wire
(the inductors) wound on a single toroidal iron core. Let the 'primary' winding have Np turns and a
self-inductance Lp and the 'secondary' winding have Ns turns and a self-inductance Ls. You may
assume the coils to have zero resistance.
a) A sinusoidal voltage of amplitude Vp is applied to the primary winding. Determine the
amplitude of the voltage Vs across the secondary winding.
b) Find the mutual inductance M of this ideal transformer. Express your results in terms of the
variables given above.
Now suppose a sinusoidal voltage V0 cos(ωt) is applied to the primary and a 'load' resistor R is
connected across the secondary. Currents will flow in both the primary and secondary circuits.
Let the amplitude of these currents be Ip and Is , respectively.
0
If the load resistance R in the secondary circuit is not too large (how large need not concern us
here), the transformer acts as a perfect conveyor of electrical power from the primary to the
secondary. In other words, IpVp = IsVs.
c) Find Is and Ip in terms of the voltage applied to the primary and the number of turns, Np and Ns,
in the primary and secondary windings.
d) Find the peak power dissipation in the load resistor R. By what factor, if any, does it differ
from the power dissipation that would occur if the transformer were omitted and the resistor
connected directly to the voltage source?
Finally, suppose a resistor R0 is inserted into the primary circuit, in series with the primary
winding. (We assume these resistances are both sufficiently small that the idealized conditions set
out above remain correct.)
0
e) For this new circuit again find the peak power dissipation in the load resistor R. Express your
result in terms of the drive voltage amplitude V0, Np, Ns, and R0. Prove that the power is
maximized when Ns/Np = (R/R0)1/2. Evaluate this maximum power. This problem illustrates the
concept of impedance matching.!
QP 27
An ac voltage V0 sin(ωt) is applied to an inductor L and capacitor C which are connected in
parallel as shown in this drawing:
a) Find the current flowing through the inductor and the capacitor as functions of time.
b) Show that the total current supplied by the ac source vanishes when ω = ω0 = (LC)-1/2.
c) Find the energy stored in the capacitor and the inductor as functions of time. Show that when
ω = ω0 the sum of these two energies is independent of time. How much work is the ac source
doing at this frequency?
Now a resistor R is inserted into the circuit as shown in this diagram:
d) What current is supplied by the ac source at ω << ω0 ? What about ω >> ω0?
e) Make a qualitative sketch of the current supplied by the source as a function of ω, covering the
range from ω << ω0 to ω >> ω0. You may assume R is "small".
NOTE: You do NOT need to find the exact analytical solution for the current in this second
circuit in order to successfully answer parts d) and e).
!
!
1!
QP 28
At t = 0 the switch shown in the RC circuit below is closed. (Prior to this moment, the capacitor
is completely discharged.)
R1
VB
C
R2
a) What is the voltage VC across the capacitor at t = 0?
b) What is the voltage VC across the capacitor for t → ∞ ?
Let I1, I2, and IC denote the current flowing in R1, R2, and the capacitor C. Let VC denote the
voltage across the capacitor.
c) Find the relation between VC and IC and the relation between I1, I2, and Ic.
d) Write down Kirchoff's Law for the loop consisting of the battery, R1 and R2, and for the loop
containing only the capacitor and R2.
e) Find the differential equation which governs the capacitor voltage VC at all t > 0. You do not
need to solve this equation.
f) What is the exponential time constant τ for this circuit (in terms of R1, R2, and C)?
!
QP 15 QP 29
In this problem we will revisit the magnetic and electric field inside a charging capacitor. You
At the mid-point
of a worked
long, thin,
wire on
that
carries
a steady
current
I is aof
capacitor
mademetal
have already
on straight
this in QP15
HW
#7. The
capacitor
consists
two circular
of two parallel, circular plates of radius R and separation D, with D ⇥ R, as shown. The current
plates of radius R separated by a distance D.
charges the capacitor.
r
R
I
I
D
To refresh your memory, the capacitor is being charged by a steady current I. Charge is building
uniformly
on the
plates.
The magnitude
of the
electric
field
the at
plates
is E = It/πR2ε0
a) What up
is the
magnitude
of the
magnetic
field B(r) half
way
between
thebetween
plates and
a distance
t ispasses
time and
R is the
radius
of the(Give
plates.
magnetic
magnitude,
< R, is B =
r from the where
axis that
through
their
centers?
the The
answer
for thefield
whole
range 0 < for
r <r⇤.)
µ0Ir/2πR2.
b) Sketch B(r) versus r for 0
r
2R. On the same graph use a dotted line to represent the
is the capacitance
C? r from the wire at a location along the wire that is far
magnitude a)
of What
the magnetic
field a distance
from the capacitor. (Please make the diameter of your dots about double the thickness of your first
Findisthe
UE stored
thevalues
electric
at time
t. Express
line so thatb)there
noenergy
ambiguity
if thereinare
of field
r forinside
which the
thecapacitor
two magnetic
fields
have theyour
same value.answer
Also, please
take
some
care in
making
the sketch.
If the the
function
question
in terms
of the
current
I, the
capacitor
dimensions,
time t,inand
ε0. is linear,
don’t give it curvature and vice versa. If it is concave down, don’t draw it concave up. And if it is
differentiable,
don’t
draw it vector
with a iskink.)
c) The
Poynting
defined as S = (E × B)/µ . In what direction does S point in the gap of
0
the capacitor?
d) Find the magnitude S of the Poynting vector in the gap at the edge of the capacitor, i.e. at r =
R.
e) Using the Poynting vector, find the total amount of energy flowing into (or out of) the gap of
the capacitor.
f) Using the results of b) and e), show that the time rate of change of UE is exactly matched by the
total energy flux implied by S.
!
!
1!
16
QP 30
a) Use the Poynting vector to estimate the magnitude of the electric and magnetic fields in the
light beam which emanates from a laser pointer. Assume the laser pointer puts out 1 mW of
power and has a beam diameter of 1 mm.
An electromagnetic wave carries energy and the Poynting vector gives the flux of that energy.
The wave also carries momentum. It may not surprise you to learn that the momentum flux
(momentum per unit time per unit area) in a light beam is just S/c, where S is the Poynting vector
and c is the speed of light.
b) With what force, in newtons, does the laser pointer in part a) push back on your hand as you
use it?
c) Assuming you are in outer space, far from anything except that silly laser pointer in your hand,
how long would it take for you to accelerate yourself from rest up to a speed of 0.1c, with c the
speed of light? (This is slow enough that you may take a non-relativistic approach to to the
problem.) Your mass, including your spacesuit, is 100 kg. Is this a practical way to explore the
universe?!