MASSACHUSETTS INSTITUTE OF TECHNOLOGY
Department of Physics
8.02
Spring 2013
Problem Set 9
Due: Tuesday, April 30 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 Twelve Undriven RLC Circuits
W12D1 M/T Apr 22/23
Reading
RC and LR Circuits;
Course Notes: Sections 7.7-7.8, 7.11.3, 11.4-11.6, 11.12.2,
11.13.4-11.13.5
W12D2 W/R Apr 24/25
Reading
Undriven RLC Circuits; Expt. 4: Undriven RLC Circuits
Course Notes: Sections 11.7-11.9, 11.10, 11.13.6
Drop Date Thurs Apr 20
W12D3 F Apr 26
Reading
PS08: Undriven RLC Cirucits
Course Notes: Sections 11.8-9, 11.12-11.13
Week Thirteen EM Waves
Problem Set 9 Due Tuesday April 30 at 9 pm
W13D1 M/T Apr 29/30
Reading
Maxwell’s Equations and Displacement Current; One
Dimensional Wave Equation
Course Notes: Sections 13.1-13.4
W13D2 W/R May 1/2
Reading
Maxwell’s Equations and One Dimensional Wave Equation
Course Notes: Sections 13.5-13.7
W13D3 F May 3
PS09: Maxwell’s Equations; Displacement Current; EM
Waves
Course Notes: Sections 13.9, 13.11, 13.12
Reading
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Problem 1 Charging a Capacitor (Friday Problem Solving)
Consider the circuit shown below. The capacitor is connected to a DC source of emf e . At time
t = 0, the switch S is closed. The capacitor initially is uncharged, Q(t = 0) = 0 .
Question 1: At t = 0 , what is the current in the circuit?
At this instant, the potential difference from the battery terminals is the same as that across the
resistor. This initiates the charging of the capacitor. As the capacitor starts to charge, the electric
potential difference across the capacitor increases in time. The electric potential differences
across a capacitor are summarized in the figure below.
Question 2: Using Kirchhoff’s loop rule, find the differential equation satisfied by the charge
Q(t) on the capacitor.
Question 3: Using your differential equation that you found in Question 2, explain in your own
words how that the charge on the capacitor behaves as a function of time.
We shall solve the differential equation
dQ
Q
R=edt
C
you found in Question 2 by the method of separation of variables. The first step is to separate
terms involving charge and time, (this means putting terms involving dQ and Q on one side of
the equality sign and terms involving dt on the other side),
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dQ
1
= dt
æ
Qö R
çè e - C ÷ø
Þ
dQ
1
=dt
Q - Ce
RC
Question 4: Integrate both sides of the above equation to find an expression for the charge Q(t)
on the capacitor as a function of time. You will be setting up definite integrals with limits of
integration that cover the time interval [0,t] and the corresponding charge interval [0,Q(t)].
Question 5: If you haven’t already done so you can now exponentiate both sides of your result
from Question 4 using the fact that exp(ln x) = x to yield to find an expression for Q(t) .
Question 6: What is the maximum value of the charge on the capacitor?
Question 7: Make a plot of the charge Q(t) as a function of time t . Label all appropriate values
on your plot.
Question 8: After a very long time, how does the electric potential difference across the
capacitor compare to the electric potential difference across the battery?
Question 9: Find an expression for the current I(t) through the circuit as function of time t .
Question 10: Make a plot of the current I(t) as a function of time t . Label all appropriate
values on your plot.
Time Constant:
The current in the charging circuit decreases exponentially in time,
I (t ) = I 0 e-t / RC .
This function is often written as I (t ) = I 0 e-t /t where t = R C is called the time constant. The
time constant t is a measure of the decay time for the exponential function. This decay rate
satisfies the following property:
I (t + t ) = I (t ) e- 1 ,
which shows that after one time constant t has elapsed, the current falls off by a factor of
e-1 = 0.368, as indicated in the figure below left.
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Current for Charging Capacitor
Potential Difference across Charging Capacitor
Similarly, the electric potential difference across the capacitor (see figure below) can also be
expressed in terms of the time constant t :
DVC (t) = e (1- e-t / t ) .
Notice that initially at time t = 0 , DVC (t = 0) = 0. After one time constant t has elapsed, the
potential difference across the capacitor plates has increased by a factor (1 - e-1 ) = 0.632 of its
final value:
DVC (t ) = e (1- e-1 ) = 0.632 e .
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Problem 2: LC Circuit (Friday Problem Solving)
Consider a circuit consisting of a coil with inductance L , a resistor with resistance R , a
capacitor with capacitance C , an applied dc voltage source, e , and two switches S1 and S 2
connected as shown in the figure below.
Question 1: Switch S1 is closed and switch S 2 is left open. After a very long time interval time
Dt >> RC what is the charge on the capacitor?
Now reset your clock so that at t = 0 switch S1 is then opened and switch S 2 is closed,
disconnecting the voltage source, leaving the coil and the charged capacitor connected together.
Question 2: At t = 0 , what are the initial values of the current I(t = 0) and charge on the
capacitor Q(t = 0) ?
Question 3: What differential equation does the charge Q(t) on the capacitor satisfy?
A general form for the solution to the SHO equation is given by the sinusoidally oscillating
function
Q(t) = Acos(w 0t + f )
where w 0 is the angular frequency , f is a phase constant and A is the amplitude determined by
the initial conditions.
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Question 4: What are f and A ?
Question 5: What is the angular frequency and period of oscillation of the charge on the
capacitor?
Question 6: Based on your answers to Questions 4 and 5, what are the expressions for charge on
the capacitor and current in the circuit as functions of time?
Question 7: A plot of Q(t) vs. t with e = 4 V , Ce = 2 nC and T = 2 ms . What is the value of
the self-inductance L ?
Question 8: What is the current in the circuit when the magnetic energy stored in the coil is
exactly equal to the electric energy stored in the capacitor? Use the values given in Question 7
and your answer to Question 7.
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Problem 3: Capacitor discharge You are tasked with designing a capacitor (capacitance C)
that will hold its charge for a long time. You want to make it using a dielectric with dielectric
constant  and resistivity . That resistivity is a problem – even though it is high, it still allows
charge to flow from one plate to the other over time. So you want to maximize the resistance
between the plates.
(a) Compare the resistance of the dielectric between the plates of a parallel plate, cylindrical
and spherical capacitor, each with capacitance C but with whatever dimensions you want.
HINT: Calculate RC and solve for R.
(b) Air also acts as a dielectric in capacitors ( ~ 1) and has a resistivity that we might need
to be concerned about. In fact, changes in the resistivity of air with humidity explains
why 8.02 demos are better in spring semester than in the fall semester. In the summer
Boston is a pretty humid place, and the resistivity of the air can drop to 5 x 1012 -m. In
the winter, when it’s nice and dry, the resistivity of the air can rise to 5 x 1013 -m.
Consider the parallel plate capacitor that we used to demonstrate what happens when you
pull the plates of a capacitor apart. Compare the length of time it takes for the potential
to drop 10% after charging to 10 kV in winter and summer.
Problem 4: Car Fires You should start this problem by watching the movie
http://web.mit.edu/8.02t/www/materials/ProblemSets/PS09_Static.avi
In order to ignite a flammable mixture (like gas vapor) you need a spark that supplies the
minimum ignition energy. For gas this is around 1 mJ (this is approximate – it changes with
additives, for example)
(a) Let’s check out the idea that electrostatic discharge ignited the gasoline in this case. Her
capacitance is probably about 100 pF. What potential would she have needed to be at to
ignite the gasoline? Thinking about previous calculations, is this feasible?
(b) She seems to be wearing rubber soled shoes. How long would she have needed to wait
before touching the nozzle to be safe? Rubber has a resistivity of ~ 1013 -m. Is it
reasonable to ask her to wait?
(c) In answering part (b) you probably assumed that the dominant discharge channel is
through her shoes. But she is also surrounded by air. How does the air affect the
discharge time, if at all? HINT: You can use the results of problem 3 and assume there
was low humidity that evening.
(d) Another source of static discharge is the car itself. When driving there are a number of
sources of friction (car-air, tire-ground, …) which can build up charge on the car.
Assuming that the car charges to the maximum voltage it can (limited by coronal
discharge – think about the van de Graff). Thinking along the lines of parts (b) and (c)
how long do you need to wait for the car to discharge?
(e) If you are unhappy with that amount of time, what could you do to reduce it? Note that
this is very important for people who want to refuel their cars quickly, like NASCAR
drivers.
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Problem 5: Consider the circuit shown in the figure below. At t = 0 , the switch S1 is closed.
The capacitor with capacitance C initially is uncharged, Q(t = 0) = 0. The battery has
electromotive force e . There are three resistors with resistances R1 , R2 , and R3 as shown in
the figure below.
a) What is the current in each branch of the circuit at t = 0 when the switch S1 is closed?
b) What is the current in each branch of the circuit a very long time ( t = ¥ ) after the switch
S1 is closed?
c) Use the loop laws, current conservation, and the relation between charge and current on
the capacitor, to derive a differential equation for the charge on one of the plates of the
capacitor at a time t after the switch S1 is closed at t = 0 . Express your differential
equation in terms of the charge Q(t) on one of the plates, the time derivative of the
charge dQ(t) / dt , the electromotive force e , the resistances R1 , R2 , and R3 , and the
capacitance C .
d) Based on your results from Problem 1, what is the solution Q(t) of your differential
equation?
e) Show that your solution agrees with your results for the current in each branch of the
circuit at t = 0 when the switch S1 is closed, and for the current in each branch of the
circuit a very long time ( t = ¥ ) after the switch S1 is closed.
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Problem 6: Consider a plane parallel capacitor of plate separation d and plate area A . The
capacitor is fully charged with charge Q on the positive plate. Ignore edge effects. Express your
answers in terms of the given quantities.
a) Derive an expression for the magnitude of the electric field between the capacitor plates.
b) What is the electric potential difference between the plates? Which plate is at a higher
potential?
c) What is the capacitance C ?
A current I flows counterclockwise as seen from above through each turn of a long air core
cylindrical shaped solenoid that has N turns, length h , and radius a . Ignore edge effects.
d) Derive an expression for the magnitude of the magnetic field inside the solenoid.
e) What is the self-inductance L of the solenoid?
The capacitor is now connected to the solenoid to form a resonant circuit without any resistance.
The capacitor is initially charged with charge Q0 on the positive plate, and when the switch is
closed the current in the circuit is found to undergo sinusoidal oscillations.
f) What is the period T of oscillation of the charge on the capacitor plate?
g) A time interval Dt = T / 4 has passed since the switch was closed. What is energy stored
in the capacitor? What is energy stored in the inductor?
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Problem 7: Microwave A resonant cavity of the form shown in the figure below is an essential
part of many microwave oscillators.
It can be regarded as a simple LC circuit.
The inductance is that of a toroid with one turn ( N = 1); the inductor is connected directly to a
parallel-plate capacitor. At one instant in the oscillation, the configuration of the electric and
magnetic fields are shown in the figure below. You may neglect edge effects in the parallel-plate
capacitor.
Find an expression for the resonant frequency of this circuit.
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Problem 8 Opening a Switch on an RL Circuit The LR circuit shown in the figure below
contains a resistor R1 and an inductance L in series with a battery of emf e 0 . The switch S is
initially closed. At t = 0, the switch S is opened, so that an additional very large resistance R2
(with R2 >> R1 ) is now in series with the other elements.
a) If the switch has been closed for a long time before t = 0, what is the steady current I 0 in the
circuit?
b) While this current I 0 is flowing, at time t = 0, the switch S is opened. Write the differential
equation for I (t ) that describes the behavior of the circuit at times t ≥ 0. Solve this equation (by
integration) for I (t ) under the approximation that e 0 = 0 . (Assume that the battery emf is
negligible compared to the total emf around the circuit for times just after the switch is opened.)
Express your answer in terms of the initial current I 0 , and R1 , R2 , and L.
c) Using your results from part b), find the value of the total emf around the circuit (which from
Faraday's law is - LdI / dt ) just after the switch is opened. How reasonable is your assumption in
part b) that e 0 could be ignored for times just after the switch is opened?
d) What is the magnitude of the potential drop across the resistor R2 at times t > 0, just after the
switch is opened? Express your answers in terms of e 0 , R1 , and R2 . How does the potential
drop across R2 just after t = 0 compare to the battery emf e 0 , if R2 = 80R1 ?
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