MASSACHUSETTS INSTITUTE OF TECHNOLOGY
Department of Physics
8.02
Spring 2013
Problem Set 8
Due: Tuesday, April 16 at 9 pm.
Hand in your problem set in your section slot in the boxes outside the door of 32-082 or
26-152 depending on which is your classroom. Make sure you clearly write your name,
section, and table and group number on your problem set.
Week Ten Magnetic Induction
and Energy; DC Circuits
Problem Set 7 Due Tuesday April 9 at 9 pm
W10D1 M/T Apr 8/9
Reading
Inductance & Magnetic Energy,
Course Notes: Sections 11.1-3
W10D2 W/R Apr 10/11
Reading
DC Circuits & Kirchhoff’s Loop Rules;
Course Notes: Sections 7.1-7.5
W10D3 F Apr 12
PS07: PhET: Building a Circuit 7.1-7.5, 7.10
Week Eleven Exam 3
Problem Set 8 Due Tuesday April 16 at 9 pm
W11D1 M/T Apr 15/16
Patriot’s Day Holiday
W11D2 W/R Apr 17/18
Exam 3 Review
W11D3 W/R April 19

No Class
Problem 1: Inductor
An inductor consists of two very thin conducting cylindrical
shells, one of radius a and one of radius b, both of length h.
Assume that the inner shell carries current I out of the page,
and that the outer shell carries current I into the page,
distributed uniformly around the circumference in both
cases. The z-axis is out of the page along the common axis
of the cylinders and the r-axis is the radial cylindrical axis
perpendicular to the z-axis.
a) Use Ampere’s Law to find the magnetic field
between the cylindrical shells. Indicate the direction
of the magnetic field on the sketch. What is the
magnetic energy density as a function of r for a < r < b?
b) Calculate the inductance of this long inductor recalling that U B 
1 2
LI and using
2
your results for the magnetic energy density in (a).
c) Calculate the inductance of this long inductor by using the formula
F = LI = ò B ×dA and your results for the magnetic field in (a). To do this
open surface
you must choose an appropriate open surface over which to evaluate the magnetic
flux. Does your result calculated in this way agree with your result in (b)?
Problem 2: Self Inductance, Energy, Induced Electric Fields, and Faraday’s Law
A wire is wrapped N = 100 times around a cylinder of non-magnetized material of radius
a = 3.0cm and length l = 2.5 ´101 cm . At t  0 , the current through the wire is increased
according to
I(t) = bt,
0 < t < 2s
where b = 2.0 ´10-1 A ×s-1 . Assume the direction of the current through each turn is
clockwise when seen from above. After t ³ 2s the current remains constant. A small
wire loop of radius r = 1.0cm is placed on the solenoid’s axis at the mid point of the
solenoid with the normal to the plane of the loop pointing along the solenoid’s axis. The
loop has resistance R = 5.0 W .
a) After t ³ 2s , use Ampere’s Law to find the direction and magnitude of the
magnetic field within the solenoid. (Ignore edge effects).
b) Sketch the magnetic field lines everywhere including edge effects.
c) What is the self-inductance of the solenoid?
d) The rate of doing work against the back emf is dW / dt = -e I , where the emf is
given by e = -L(dI / dt) . How much work is done against the back emf in order
to reach a steady current after t  2s in the solenoid?
e) The stored energy in a magnetic field is equal to U magnetic 
1
2 0

B 2 dV . How
all space
much energy is stored in the magnetic field of the solenoid? Do you expect your
result to agree or disagree with your answer in part e)? Briefly explain your
reasoning.
f) The changing magnetic flux inside the solenoid produces an induced tangential
electric field. During the interval, 0  t  2s , find an expression for the induced
electric field as a function of distance from the central axis of the solenoid.
Problem 3 Mutual Inductance of Two Rings
Consider the circuits shown in the figure below, two coplanar, concentric rings, a small
ring C2 of radius R2 and a much larger ring C1 of radius R1 with current I1 . What is the
mutual inductance? You may assume that R2 << R1 , so that you may neglect the variation
of the magnetic field B1 over the interior of the small ring.
Problem 4 Straight Wire and Rectangular Loop of Wire
At t = 0 , there is a current I 0 through a very long wire. (You may assume this is an
infinite wire). No current is flowing in a rectangular loop of wire with dimensions length
l and width w, and has a resistance R . The side of the square loop closest to the wire is
a distance s away from the wire. At t = 0 , the current in the wire is decreased according
to
ì(1- a t)I0 , 0 < t < 1/ a
I(t) = í
ïî0, t > 1/ a
a) In what direction does the induced current in the rectangular loop flow?
b) What total charge passes a given point in the loop during the time this current
flows?
Problem 5: Alternating-Current Generator
An N-turn rectangular loop of length a and width b is rotated at a frequency f in a
uniform magnetic field B which points into the page, as shown in the figure below. At
time t = 0, the loop is vertical as shown in the sketch, and it rotates counterclockwise
when viewed along the axis of rotation from the left.
a) Make a sketch depicting this “generator” as viewed from the left along the axis of
rotation at a time t shortly after t = 0, when it has rotated an angle  from the
vertical. Show clearly the vector B , the plane of the loop, and the direction of the
induced current.
b) Write an expression for the magnetic flux  B passing through the loop as a
function of time for the given parameters.
c) Show that an induced emf
 appears in the loop, given by
  2 fNBab sin(2 ft )   0 sin(2 ft )
d) Design a loop that will produce an emf with  0  120 V when rotated at 60
revolutions/sec in a magnetic field of 0.40 T. (Describe the dimensions of the
loop and the number of wrappings.)
Problem 6
A pair of conducting rails separated by a distance w is inclined at an angle q to the
horizontal plane. The lower ends of the rails are connected by a resistor with resistance
R . A conducting bar of mass m slides along the rails under the influence of gravity.
There is a constant external magnetic field of magnitude B 0 that points vertically
upwards. Assume that there is no friction between the bar and the rails. Use g for the
gravitational constant. The resistance of the bar, rails, and the contact between the bar
and rails is negligibly small.
a) If the bar moves down the rails at a speed v , determine the induced current that
flows through the resistor. Indicate the direction of the current. You may assume
the resistance R is large enough to ignore self-inductance.
b) If the bar moves down the rails at a speed v , what is the magnitude and direction
of the induced force on the bar?
c) Using Newton’s Second Law, derive a differential equation for the velocity v(t)
of the bar as a function of time. Think about choosing the easiest coordinate
system for describing the motion down the inclined plane.
d) Solve the equation in part c) for the velocity as a function of time.
e) What is the terminal velocity of the bar?
f) What is the rate at which electrical energy is being dissipated through the resistor?
g) What is the rate of work done by gravity on the bar? The rate at which work is
done is F × v . How does this compare to your answer in (f)? Why?
Problem 7: Pressure Transducers
A common way of measuring small pressure changes (for example, in car transmission
systems, airplane cabin pressure monitors, and Parenchymal catheter tips, used, for
example, to measure intercranial pressure) is to use piezoresistive materials, that is,
materials that change resistance when a pressure is applied to them. Often, four identical
piezoresistors are connected as shown below, in such a way that when a pressure is
applied Ra and Rd increase by some fraction and Rb and Rc decrease by about the same
fraction. The potential difference Vin = V2 -V1 and Vout = V3 -V1
a) Determine an expression for Vout / Vin in terms of Ra , Rb , Rc , and Rd .
b) If a 1 V battery is connected across points 1 and 2 such that Vin = V2 -V1, what
voltage will be measured at Vout = V3 -V1 if the system is unstrained and all four
resistors have the same resistance R ?
c) An unfortunate problem with piezoresistors is that they are also temperature
dependent. If a temperature change causes the resistance of all four resistors to
increase by 5%, what is the new output voltage at Vout = V3 -V1?
d) Starting from the condition where all four resistors have resistance R , a pressure
is applied that causes Ra and Rd to increase by 1% and Rb and Rc to decrease by
1%. What now is measured at Vout = V3 -V1?
Problem 8 Design an Ohmmeter.
You own a micro-ammeter that reads 50 m A at full scale deflection, and the coil in the
meter movement has a resistance of 20 W . By adding two resistors, R1 and R2 , and a
1.5 V battery you can convert this into an ohmmeter. When the two outgoing leads of
this ohmmeter are connected together, the meter should register 0 ohms by giving fullscale deflection. When the leads are connected across an unknown resistance Ru , the
deflection will indicate the resistance value if the scale is appropriately marked. In
particular, we want half-scale deflection to indicate 15 W . What values of R1 and R2 , are
required, how should the connections be made, and where on the ohm scale will the
marks be (with reference to the old micro-ammeter calibration) for 5 W and 50 W ? Note
that there is more than one way to choose these resistors. Depending on your values for
the resistors, do you think your design is a feasible design? HINT: Can you make any
approximations that make your calculations far easier?