Capacitors, Inductors, and RC and RL Circuits Homework Set
Problem 1. You have two flat metal plates of area 1.0 m2 with which to construct a parallel plate
capacitor. If the capacitance of the device is to be 1.0 F, what must be the separation between
the plates? Could this actually be constructed?
Write the general expression for capacitance of an object in terms of its area A and the separation
distance, d.
Perform algebraic manipulation and calculate the separation distance.
Look up and write below, the average radius of a hydrogen atom in meters
Compare and discuss the plate separation distance versus the radius of the hydrogen atom.
Capacitors, Inductors, and RC and RL Circuits Homework Set
Problem 2. Find the equivalent capacitance in the circuit below. Assume that C1= 10.0 μF, C2=
5.0 μF and C3 = 4.0 μF.
Find the equivalent capacitance between C1 and C2
Find the equivalent capacitance between C1/C2 and C3
Capacitors, Inductors, and RC and RL Circuits Homework Set
Problem 3. In the figure, the battery has a potential difference of 20 V. Find
a) the equivalent capacitance of all capacitors
Redraw the circuit with the two 3.0 μF capacitors with their centers aligned. Perform the
same re-arrangement for the two 2.0 μF capacitors.
Capacitors, Inductors, and RC and RL Circuits Homework Set
Find the equivalent capacitance of the two 4.0 μF capacitors
Use this equivalent capacitance along with the capacitances of the two 2.0 μF capacitors
to find the equivalent capacitance in the upper parallel junction
Find the equivalent capacitance of the two 3 μF capacitors.
Use the above information to calculate the total capacitance.
Capacitors, Inductors, and RC and RL Circuits Homework Set
b) the charge stored on that equivalent capacitance
Find the total charge stored in the circuit. (Hint: Q=CV)
c) the potential across and the charge stored on capacitor C1
Use facts about series networks and charge, to calculate voltage on and charge stored on
C1
d) the potential across and the charge stored on capacitor C3
Calculate the potential difference across the upper parallel network.
Find the amount of charge stored on the equivalent capacitance of the two 4 μF
capacitors.
Use facts about series networks and charge, to calculate voltage on and charge stored on
C3
Capacitors, Inductors, and RC and RL Circuits Homework Set
Problem 4. A 3.0 MΩ resistor and a 1.00 μF capacitor are connected in series with an ideal
battery of E=4.00 V. At 1.00 s after the connection is made, what are the rates at which
a) the charge of the capacitor is increasing?
Write the relationship for the charge of a charging capacitor in terms of the voltage
applied V, the capacitance C, the resistance R, and the time t.
Take the derivative of this expression with respect to time,
dq
.
dt
Use the values from the problem and calculate the charging rate.
Capacitors, Inductors, and RC and RL Circuits Homework Set
b) the energy is being stored in the capacitor?
Write the relationship for the energy U in a capacitor as a function of charge, q and
capacitance C.
Take the derivative of this expression with respect to time,
then
dU
. (Hint: if x=f(t) and
dt
d 2
dx
x = 2x
dt
dt
Use the values from the problem and calculate the energy storage rate. (Hint: the value
of the derivative was calculated previously.)
c) the thermal energy is appearing in the resistor?
The rate at which the capacitor is charging is the same as the current. Write an
expression for the power dissipated in a resistor as a function of current and resistance.
Insert the numerical values from the problem and solve.
Capacitors, Inductors, and RC and RL Circuits Homework Set
d) the energy is being delivered by the battery?
Write an expression for the power in the battery as a function of voltage and current.
Insert the numerical values from the problem and solve.
Capacitors, Inductors, and RC and RL Circuits Homework Set
Problem 5. The figure below shows the circuit of a flashing lamp, like those attached to barrels
at highway construction sites. The fluorescent lamp L (of negligible capacitance) is connected in
parallel across the capacitor C of an RC circuit. There is current through the lamp only when the
potential difference across it reaches the breakdown voltage VL; in this event, the capacitor
discharges completely through the lamp and the lamp flashes briefly. Suppose that two flashes
per second are needed. For a lamp with a breakdown voltage of VL=72.0 V, a 95.0 V ideal
battery, and a 0.150 μF capacitor, what should the
Write an expression for the voltage across the capacitor in terms of the capacitance C, the
resistance R, the emf E, and the time t.
Calculate the time between flashes in seconds.
Perform algebraic manipulation and solve for the resistance.
Capacitors, Inductors, and RC and RL Circuits Homework Set
Problem 6. The circuit below shows a capacitor C, two ideal batteries, two resistors and a switch
S. Initially S has been open for a long time. If it is then closed for a long time, by how much
does the charge on the capacitor change? Assume C=10.0 μF, E1= 1.0 V, E2=3.0 V, R1=0.20 Ω
and R2=0.4 Ω.
Explain what happens to a capacitor which has been attached to a battery for a long
time.
Use your explanation to calculate the charge on the capacitor while the switch is open.
Capacitors, Inductors, and RC and RL Circuits Homework Set
Once the switch has been closed a long time, redraw below the equivalent circuit.
(Hint: is current flowing through the capacitor?)
From the equivalent circuit, use the loop rule to calculate the current below.
Capacitors, Inductors, and RC and RL Circuits Homework Set
Take a partial loop traversal across E2 and R2 and calculate the potential difference
across them.
This potential difference is the same as the voltage across the capacitor. Calculate the
charge on the capacitor.
Calculate the change in charge from switch closed to switch open.
Capacitors, Inductors, and RC and RL Circuits Homework Set
Problem 7. In the circuit below, E=10 V, R1 = 5 Ω, R2= 10 Ω, and L= 5.0 H. For the two
separate conditions (I) switch S has just closed and (II) switch S closed for a long time, calculate:
a) the current i1 through R1
Condition I
How much current is the inductor allowing to
flow through it?
Condition II
How much current is the inductor allowing to
flow through it?
Capacitors, Inductors, and RC and RL Circuits Homework Set
Condition I
Condition II
Based on this logic, draw the resultant circuit
below.
Based on this logic, draw the resultant circuit
below.
Use the loop rules to solve for i1
Use the loop rules to solve for i1
Capacitors, Inductors, and RC and RL Circuits Homework Set
b) the current i2 through R2
Condition I
Based on the previous logic, use the loop rules
to solve for i2
Condition II
Based on the previous logic, use se the loop
rules to solve for i2
c) the current i through the switch
Condition I
Sum the currents i1 and i2.
Condition II
Sum the currents i1 and i2.
Capacitors, Inductors, and RC and RL Circuits Homework Set
d) the potential difference across R2
Condition I
Condition II
Based on the current in the inductor, estimate
the voltage across the inductor.
Based on the current in the inductor, estimate
the voltage across the inductor.
The voltage across R2 can be found as the
difference in the voltage of the battery and the
voltage across the inductor. Find that
difference.
The voltage across R2 can be found as the
difference in the voltage of the battery and the
voltage across the inductor. Find that
difference.
Capacitors, Inductors, and RC and RL Circuits Homework Set
e) the potential difference across L
Condition I
Based on the logic in the previous section,
estimate the voltage across the inductor.
f) the rate of change
Condition II
Based on the logic in the previous section,
estimate the voltage across the inductor.
di2
dt
Write an expression for voltage across an inductor with its inductance L and the rate of change of
current though it (di/dt).
Capacitors, Inductors, and RC and RL Circuits Homework Set
Condition I
Based on the logic in the previous section and
the above expression, calculate the rate of
change of i2.
Condition II
Based on the logic in the previous section and
the above expression, calculate the rate of
change of i2.
Capacitors, Inductors, and RC and RL Circuits Homework Set
Problem 8. A coil with an inductance of 2.0 H and a resistance of 10 W is suddenly connected to
a resistanceless battery of E=100 V. At 0.10 s after the connection is made, what are the rates at
which
a) the energy is stored in the magnetic field?
Write the expression for the energy stored in an inductor of inductance L with current i.
Take the derivative of this expression with respect to time,
dU
. (Hint: if x=f(t) and then
dt
d 2
dx
x = 2x )
dt
dt
Write the expression for current in an inductor as a function of the emf E, the resistance R,
the inductance L and the time t. Also write its derivative with respect to time.
Capacitors, Inductors, and RC and RL Circuits Homework Set
Use the numerical values of the problem and find the current and its rate of change.
Apply these values in the rate of change of energy expression.
Capacitors, Inductors, and RC and RL Circuits Homework Set
b) the thermal energy is appearing in the resistance?
Write an expression for energy dissipation in a resistance in terms of its resistance R
and the current through it i.
Use the numerical values from the previous section to calculate the energy dissipation.
Capacitors, Inductors, and RC and RL Circuits Homework Set
c) the energy is being delivered by the battery?
Write an expression for power supplied by a battery in terms of its voltage V and the
current through it i.
Use the numerical values from the previous section to calculate the power supplied by
the battery.
Demonstrate from these values that the conservation of energy is obeyed.