AP Physics C
Electric Current and Circuits
Free Response Problems
1. A battery with EMF ε and internal resistance r is connected to a variable resistance R.
The graph of terminal voltage VAB as a function of the current I is presented on the axes
above.
a. Determine the EMF of the battery.
b. Determine the internal resistance of the battery.
c. Determine the value of resistance R that will produce current I = 5 A.
d. Determine the maximum current that can be produced by the battery.
e. The graph above was obtained by measuring voltage and current with
voltmeter
and ammeter
. On the diagram below
complete the circuit with appropriate connection of those two devices.
2. The circuit show above contains two batteries and three resistors. The internal
resistance of the batteries and the resistance of connecting wires are negligible.
a. Calculate the current in the following resistors:
i. 6-Ω resistor.
ii. 3 – Ω resistor.
iii. 7 – Ω resistor.
b. Find the voltage drop across the following resistors:
i. 6-Ω resistor.
ii. 3 – Ω resistor.
iii. 7 – Ω resistor.
c. Calculate the power delivered by 12 – V battery.
3. An electric circuit contains three resistors and a battery. The battery has a constant EMF
ε and a negligible internal resistance.
a.
b.
c.
d.
e.
Determine the net resistance of the circuit.
Determine the EMF of the battery.
Find the voltage drop across 6 - Ω resistor.
Determine the power dissipated in the circuit.
Find the amount of charge passing through 12 – Ω resistor in 2 minutes.
4. A circuit contains a 36 V battery, four resistors, a switch, and a 5µF capacitor. When the
switch is closed it connects the capacitor with 12- Ω resistor in parallel.
Case 1: Switch is open. The capacitor is disconnected from the circuit.
a. Determine the electric current in the battery.
b. Determine the electric current in the 12–Ω resistor.
c. Determine the voltage drop across 12 – Ω resistor.
Case 2: Switch is closed. The capacitor is connected to the circuit.
d. Determine the charge on the capacitor.
e. Determine the energy stored in the capacitor.
5. The electric circuit includes one battery, two capacitors and two switches. The voltage
produced by the battery is V = 120 V. Each capacitor has a capacitance C1 = 9 µF and C2 =
18 µF. Initially, all the switches are open and the capacitors are uncharged.
a. Determine the charge on capacitor C1 after switch S1 is closed for a long
time.
b. Switch S1 is opened and afterward switch S2 is closed. Determine the
charge on capacitor C1 long time after.
c. For part (b), determine the potential difference across capacitor C1 when
the equilibrium is reached.
d. Switch S2 remains closed, and now S1 is also closed. How much extra
charge flows from the battery?
6. An electric circuit contains a resistor R = 4×106 Ω, a capacitor C, and battery with EMF ε
and negligible internal resistance. Initially, the switch is open and the capacitor is
uncharged. At time t = 0 the switch is closed and the current as a function of time is
presented by the following equation:
𝑡
−8
a.
i(t) = I0𝑒 , where I0 = 15 µA and t is in seconds.
Determine the EMF ε of the battery.
b. Develop an expression for the charge q on the capacitor as a function of
time t > 0.
c. On the exes below draw a graph of the charge q on the capacitor as a
function of time t.
d. Determine the capacitance C of the capacitor.
7. An electric circuit contains a battery with an EMF ε = 5000 V and negligible internal
resistance, a capacitor C = 7 µF, and two resistors R1 = 6×105 Ω, R2 = 15×105 Ω. Initially,
switch S is open and the capacitor is uncharged.
a. Calculate the current through resistor R1 immediately after switch S is
closed.
b. On the axes below, show the relationship between the voltage V across
the capacitor and time t.
c. Calculate the current through resistor R2 when the equilibrium is
reached.
d. Calculate the maximum charge on the capacitor.
e. Calculate the maximum energy stored in the capacitor.
f. On the axes below, graph the current through resistor R2 as a function of
time t when switch s is open.
8. A battery with an EMF ε = 120 V is connected in series with a resistor R = 5×105 Ω and a
capacitor C = 8 µF. The capacitor contains two parallel plate separated by air. Switch S is
closed for a long time.
a. Determine the charge on the capacitor when equilibrium is reached.
b. Determine the energy stored in the capacitor.
The spacing between the plates is doubled in an extremely short time compared to the time
constant.
c. Determine the work that must be done in increasing the separation.
d. Determine the current in the circuit immediately after the increase in
spacing.
After a long time the current stops and equilibrium is reached.
e. Determine the total charge that has passed through the battery.
f. Determine the amount of energy added to the battery.
9. In the circuit shown above, switch S1 is initially closed and 1 µF capacitor is fully charged.
Switch S2 is kept open. Ignore the resistance of connecting wires and the internal
resistance of the battery.
a. Determine the magnitude and the sign of the charge on each plate of 1
µF capacitor.
b. Determine the electric energy stored in 1 µF capacitor.
Next S1 is opened and afterward S2 is closed, connecting 1 µF capacitor to a 1.5x106 Ω
resistor and 3 µF capacitor, which is initially uncharged.
c. Determine the initial current in the circuit after switch S2 is closed.
Equilibrium is reached after a long period of time.
d. Determine the charge on each plate of 3 µF capacitor.
e. Find the energy dissipated in the resistor.
10. A parallel-plate capacitor has two parallel sheets of metal, each with an area of 0.2 m2,
separated by a sheet of mica 1.55 mm thick. The dielectric constant of mica is κ = 7.
Initially, the capacitor is uncharged and connected to a 12 –V battery, a 3x106 Ω resistor,
and an open switch.
a. What is the capacitance of the capacitor?
b. What is the current in the circuit immediately after the switch is closed?
c. What is the time constant in the circuit?
d. What is the charge in the capacitor when it is fully charged?
e. What is the maximum electric energy stored in the capacitor?
After the capacitor is fully charged, it is disconnected from the battery, and then the
sheet of mica is removed from the capacitor.
f. What is the new voltage across the capacitor?
g. What is the new electric energy stored in the capacitor?
11. Two capacitors C1 = 6µF and C2 = 18 µF are connected in series with a resistor R =120 Ω
and a switch S. Capacitor C1 is initially charged to a voltage V =120 V, and capacitor C2 is
initially uncharged. Switch S is then closed at time t = 0.
a. What is the new voltage across capacitor C1 after equilibrium is reached?
b. What is the charge on each capacitor after equilibrium is reached?
c. What is the difference between the initial and final energy stored in the system
of two capacitors?
𝑡
The current as a function of time in the resistor is presented by I= I0𝑒 −𝜏 , where
I0=0.4 A and = 1.2× 10-4 s.
d. Find the power dissipated in the resistor as a function of time.
e. How much energy dissipated in the resistor from the time t = 0 to when
equilibrium is reached?
12. A battery with a voltage of 18 V is connected to a resistor of 1.2×105 Ω and a parallel
plate capacitor of 9 µF filled with air.
a. Determine the energy stored in the capacitor when the switch is closed for a
long time.
The spacing between the plates is quickly increased to three time of its original
value. (The change took time negligible compare to the time constant)
b. How much work is done in increasing the spacing between the plates?
c. Determine the current in the resistor immediately after the spacing is increased.
After a long time the circuit reaches a new static equilibrium.
d. Determine the total charge passed through the battery.
e. Determine the energy added to the battery.