AP Physics C Electricity and Magnetism Quiz: Circuits With Capacitors And Inductors

20 questions · exam conditions

0:00

Circuits With Capacitors And InductorsQuestion 1 of 20

An ideal LC circuit has a total electromagnetic energy of $50$ $\mu$ J. At a certain instant, the energy stored in the electric field of the capacitor is $30$ $\mu$ J. What is the energy stored in the magnetic field of the inductor at that same instant?

80

\mu

50

\mu

30

\mu

20

\mu

← Back to quizzes

AP Physics C Electricity and Magnetism Quiz

AP Physics C Electricity and Magnetism Quiz: Circuits With Capacitors And Inductors

Practice Circuits With Capacitors And Inductors in AP Physics C Electricity and Magnetism with focused quiz questions that help you check what you know, review explanations, and build confidence with test-style prompts.

What this quiz covers

This quiz focuses on Circuits With Capacitors And Inductors, giving you a quick way to practice the rules, question types, and explanations that matter most for AP Physics C Electricity and Magnetism.

How to use this quiz

Try each quiz question before looking at the correct answer. Use the explanations to review missed ideas, then come back to similar questions until the pattern feels familiar.

All questions

Question 1

$80$ $\mu$ J
$50$ $\mu$ J
$30$ $\mu$ J
$20$ $\mu$ J (correct answer)

Explanation: In an ideal LC circuit, total energy is conserved. The total energy

U_{total}

is the sum of the energy in the capacitor

U_C

and the energy in the inductor

U_L

. Therefore,

U_L = U_{total} - U_C = 50 \mu J - 30 \mu J = 20 \mu J

Question 2

In an oscillating ideal LC circuit, the total energy is constant. At the instant the charge on the capacitor is at its maximum positive value, which of the following statements is true?

The energy stored in the inductor's magnetic field is at its maximum value.
The current flowing through the inductor is at its maximum value.
The current flowing through the inductor is zero. (correct answer)
The energy is equally shared between the capacitor and the inductor.

Explanation: When the charge on the capacitor is maximum, all the energy of the circuit is stored in the capacitor's electric field. Therefore, the energy stored in the inductor is zero. Since inductor energy is given by

\frac{1}{2}LI^2

, the current

I

must be zero at this instant.

Question 3

In an ideal LC circuit, the charge on the capacitor $q(t)$ and the current in the inductor $I(t)$ oscillate sinusoidally. What is the phase difference between the charge oscillations and the current oscillations?

They are in phase ( $0$ radians).
They are out of phase by $\pi/2$ radians ( $90^\circ$ ). (correct answer)
They are out of phase by $\pi$ radians ( $180^\circ$ ).
They are out of phase by $\pi/4$ radians ( $45^\circ$ ).

Explanation: If the charge is given by

q(t) = Q_{max} \cos(\omega t)

, the current is

I(t) = dq/dt = -\omega Q_{max} \sin(\omega t)

. The sine and cosine functions are out of phase by

\pi/2

radians. The current leads the charge by

\pi/2

radians.

Question 4

An LC circuit is oscillating. At the instant when the capacitor stores its maximum energy, the plates of the capacitor are pulled apart, doubling their separation distance. How does the maximum current $I_{max}$ in the subsequent oscillations compare to its initial value?

It increases because the total energy of the circuit increases. (correct answer)
It decreases because the total energy of the circuit decreases.
It remains the same because the charge on the capacitor is conserved.
It decreases because the capacitance decreases, reducing the stored charge.

Explanation: At the instant of maximum energy in the capacitor, the current is zero, so the capacitor is electrically isolated. Pulling the plates apart does positive work on the system, which increases the stored potential energy

U_C = Q^2/(2C)

because C decreases while Q remains constant. This increased total energy is then conserved. Since the new maximum inductor energy

\frac{1}{2}LI_{new,max}^2

must equal this new, larger total energy, the new maximum current must be greater than the initial maximum current.

Question 5

In an ideal LC circuit, a capacitor with capacitance $C$ is initially charged to a maximum charge $Q_{max}$ . The circuit then oscillates. By conservation of energy, what is the maximum energy $U_{L,max}$ stored in the inductor during one cycle?

$U_{L,max} = \frac{Q_{max}^2}{2C}$ (correct answer)
$U_{L,max} = \frac{1}{2} L Q_{max}^2$
$U_{L,max} = \frac{Q_{max}^2}{2L}$
$U_{L,max} = 0$

Explanation: The total energy in an ideal LC circuit is conserved. It oscillates between the capacitor's electric field and the inductor's magnetic field. The maximum energy stored in the inductor must equal the maximum energy initially stored in the capacitor, which is

U_{C,max} = \frac{Q_{max}^2}{2C}

Question 6

An ideal LC circuit has a capacitor with capacitance $C$ initially charged to a potential difference $V_{max}$ . The maximum current is $I_{max}$ . If the capacitance is changed to $4C$ and the capacitor is again charged to the same initial potential difference $V_{max}$ , what is the new maximum current?

$I_{max}/2$
$I_{max}$
$2I_{max}$ (correct answer)
$4I_{max}$

Explanation: By energy conservation, the maximum energy in the capacitor equals the maximum energy in the inductor. Initially,

\frac{1}{2}CV_{max}^2 = \frac{1}{2}LI_{max}^2

. With the new capacitance,

\frac{1}{2}(4C)V_{max}^2 = \frac{1}{2}LI_{new,max}^2

. This means

4(\frac{1}{2}CV_{max}^2) = \frac{1}{2}LI_{new,max}^2

, so

4(\frac{1}{2}LI_{max}^2) = \frac{1}{2}LI_{new,max}^2

. This simplifies to

4I_{max}^2 = I_{new,max}^2

, so

I_{new,max} = 2I_{max}

Question 7

An ideal LC circuit consists of an inductor with inductance $L$ and a capacitor with capacitance $C$ . Applying Kirchhoff's loop rule to this circuit results in which of the following differential equations, where $q$ is the charge on the capacitor and $I$ is the current?

$L\frac{dI}{dt} + \frac{q}{C} = 0$ (correct answer)
$L\frac{dI}{dt} - \frac{q}{C} = 0$
$I + C\frac{dq}{dt} = 0$
$\frac{dI}{dt} + LCq = 0$

Explanation: According to Kirchhoff's loop rule, the sum of potential differences around a closed loop is zero. The potential difference across the inductor is

V_L = L\frac{dI}{dt}

and across the capacitor is

V_C = \frac{q}{C}

. Therefore,

V_L + V_C = 0

, which gives

L\frac{dI}{dt} + \frac{q}{C} = 0

Question 8

The charge $q$ on the capacitor in an ideal LC circuit is described by the differential equation $L\frac{d^2q}{dt^2} + \frac{q}{C} = 0$ . If the capacitor has its maximum charge $Q_{max}$ at $t=0$ , which of the following is a valid solution for $q(t)$ ? Let $\omega = 1/\sqrt{LC}$ .

$q(t) = Q_{max} \sin(\omega t)$
$q(t) = Q_{max} e^{-\omega t}$
$q(t) = Q_{max} (1 - \cos(\omega t))$
$q(t) = Q_{max} \cos(\omega t)$ (correct answer)

Explanation: The given differential equation is for simple harmonic motion. The general solution is

q(t) = A\cos(\omega t) + B\sin(\omega t)

. The initial condition

q(0) = Q_{max}

gives

A=Q_{max}

. At

t=0

, the charge is maximum, so the current

I = dq/dt

must be zero.

I(0) = -A\omega\sin(0) + B\omega\cos(0) = B\omega = 0

, which implies

B=0

. Thus,

q(t) = Q_{max} \cos(\omega t)

Question 9

In an ideal LC circuit, the capacitor is fully charged at $t=0$ . The period of oscillation for the charge is $T$ . At which of the following times is the energy stored in the capacitor first equal to the energy stored in the inductor?

$t = T/8$ (correct answer)
$t = T/4$
$t = T/2$
$t = T$

Explanation: Let

q(t) = Q_{max} \cos(\omega t)

. The energy in the capacitor is

U_C = \frac{q^2}{2C} = U_{total} \cos^2(\omega t)

. The energy is split equally when

U_C = U_L = U_{total}/2

. This occurs when

\cos^2(\omega t) = 1/2

, or

\cos(\omega t) = \pm 1/\sqrt{2}

. The first positive time this occurs is when

\omega t = \pi/4

. Since

\omega = 2\pi/T

, we have

(2\pi/T)t = \pi/4

, which gives

t = T/8

Question 10

An inductor with inductance $L=2.0$ H and a capacitor with capacitance $C$ are connected in an ideal LC circuit. The charge on the capacitor oscillates with a period of $\pi$ seconds. What is the value of the capacitance $C$ ?

$0.125$ F (correct answer)
$0.25$ F
$0.50$ F
$2.0$ F

Explanation: The period of an LC circuit is

T = 2\pi\sqrt{LC}

. We are given

T = \pi

s and

L = 2.0

H. Plugging these values in:

\pi = 2\pi\sqrt{(2.0)C}

. Dividing by

2\pi

gives

1/2 = \sqrt{2C}

. Squaring both sides gives

1/4 = 2C

, so

C = 1/8 = 0.125

Question 11

The behavior of the charge $q$ in an ideal LC circuit is described by the differential equation $\frac{d^2q}{dt^2} + 2500q = 0$ , where $q$ is in coulombs and $t$ is in seconds. What is the angular frequency of the oscillation?

$2500$ rad/s
$\sqrt{2500/\pi}$ rad/s
$50$ rad/s (correct answer)
$25$ rad/s

Explanation: The standard form of the differential equation for simple harmonic motion, which applies to an LC circuit, is

\frac{d^2q}{dt^2} + \omega^2 q = 0

. By comparing the given equation to this standard form, we can identify that

\omega^2 = 2500

^{-2}

. Taking the square root gives the angular frequency

\omega = 50

rad/s.

Question 12

In an ideal LC circuit with inductance $L = 4.0$ H and capacitance $C = 9.0$ F, the maximum current observed is $I_{max} = 0.5$ A. What is the maximum charge $Q_{max}$ stored on the capacitor?

$1.5$ C
$3.0$ C (correct answer)
$6.0$ C
$18$ C

Explanation: By conservation of energy, the maximum energy in the inductor equals the maximum energy in the capacitor:

\frac{1}{2}LI_{max}^2 = \frac{1}{2}\frac{Q_{max}^2}{C}

. Solving for

Q_{max}

gives

Q_{max} = I_{max}\sqrt{LC} = (0.5 \text{ A})\sqrt{(4.0 \text{ H})(9.0 \text{ F})} = (0.5 \text{ A})\sqrt{36 \text{ s}^2} = (0.5 \text{ A})(6 \text{ s}) = 3.0

Question 13

An LC circuit oscillates with a natural frequency $f$ . If the inductance is doubled and the capacitance is increased to eight times its original value, what is the new frequency of oscillation?

$f/16$
$16f$
$f/4$ (correct answer)
$4f$

Explanation: The natural frequency of an LC circuit is given by

f = \frac{1}{2\pi\sqrt{LC}}

. Let

L_{new} = 2L

and

C_{new} = 8C

. The new frequency is

f_{new} = \frac{1}{2\pi\sqrt{(2L)(8C)}} = \frac{1}{2\pi\sqrt{16LC}} = \frac{1}{4} \left( \frac{1}{2\pi\sqrt{LC}} \right) = \frac{f}{4}

Question 14

Which of the following statements provides the best description of the energy transformations in an ideal LC circuit?

Energy is continuously dissipated as thermal energy due to the resistance of the inductor and connecting wires.
The circuit's total energy oscillates between a maximum value and zero with a frequency determined by L and C.
Energy is transformed back and forth between the electric field of the capacitor and the magnetic field of the inductor. (correct answer)
A constant source of energy, such as a battery, must be present to sustain the oscillations in the circuit.

Explanation: In an ideal LC circuit with no resistance, the total electromagnetic energy is conserved. This energy oscillates between being stored as electric potential energy in the capacitor's electric field and as magnetic potential energy in the inductor's magnetic field.

Question 15

In an ideal LC circuit, which of the following quantities oscillate sinusoidally with time?

Charge on the capacitor, current in the inductor, and total energy in the circuit.
Energy in the capacitor and energy in the inductor only.
Total energy in the circuit only.
Charge on the capacitor and current in the inductor only. (correct answer)

Explanation: In an ideal LC circuit, the charge on the capacitor and the current in the inductor vary sinusoidally as energy is transferred between the two components. The total energy of the circuit is conserved and remains constant, so it does not oscillate. While the individual energies of the capacitor and inductor do oscillate, their functional form is proportional to

\cos^2

\sin^2

, which is not a simple sinusoid, and they oscillate at twice the frequency of the charge and current.

Question 16

An LC circuit consists of a $10$ mH inductor and a $10$ $\mu$ F capacitor. The capacitor is initially charged to a maximum charge of $10$ mC. What is the maximum current in the circuit?

$0.1$ A
$1.0$ A (correct answer)
$10$ A
$100$ A

Explanation: The maximum current

I_{max}

is related to the maximum charge

Q_{max}

I_{max} = \omega Q_{max}

. The angular frequency is

\omega = 1/\sqrt{LC} = 1/\sqrt{(10 \times 10^{-3} \text{ H})(10 \times 10^{-6} \text{ F})} = 1/\sqrt{10^{-4} \text{ s}^2} = 100

rad/s. Therefore,

I_{max} = (100 \text{ rad/s})(10 \times 10^{-3} \text{ C}) = 1.0

Question 17

An ideal LC circuit consists of a capacitor with capacitance $C$ and an inductor with inductance $L$ . What is the angular frequency $\omega$ of the electromagnetic oscillations in the circuit?

$\omega = \frac{1}{\sqrt{LC}}$ (correct answer)
$\omega = \sqrt{\frac{L}{C}}$
$\omega = \sqrt{\frac{C}{L}}$
$\omega = \sqrt{LC}$

Explanation: The differential equation for an ideal LC circuit is

L\frac{d^2q}{dt^2} + \frac{1}{C}q = 0

, which is the equation for simple harmonic motion,

\frac{d^2q}{dt^2} + \omega^2 q = 0

. Comparing these gives

\omega^2 = \frac{1}{LC}

, so the angular frequency is

\omega = \frac{1}{\sqrt{LC}}

Question 18

An ideal LC circuit contains an inductor of inductance $L$ and a capacitor of capacitance $C$ . What is the period $T$ of the oscillations in this circuit?

$T = \frac{\sqrt{LC}}{2\pi}$
$T = \frac{1}{2\pi\sqrt{LC}}$
$T = 2\pi\sqrt{\frac{L}{C}}$
$T = 2\pi\sqrt{LC}$ (correct answer)

Explanation: The angular frequency of an LC circuit is

\omega = \frac{1}{\sqrt{LC}}

. The period

T

is related to the angular frequency by

T = \frac{2\pi}{\omega}

. Substituting the expression for

\omega

gives

T = 2\pi\sqrt{LC}

Question 19

An LC circuit has an oscillation period of $T$ . If the inductance $L$ is quadrupled while the capacitance $C$ remains the same, what is the new period of oscillation?

$T/2$
$2T$ (correct answer)
$4T$
$T$

Explanation: The period of an LC circuit is given by

T = 2\pi\sqrt{LC}

. If the inductance becomes

4L

, the new period

T_{new}

will be

T_{new} = 2\pi\sqrt{(4L)C} = 2(2\pi\sqrt{LC}) = 2T

Question 20

In an ideal LC circuit, the charge on the capacitor is given by $q(t) = Q_{max} \cos(\omega t)$ . Which expression represents the current $I(t)$ in the circuit, assuming $I = dq/dt$ ?

$I(t) = -\omega Q_{max} \sin(\omega t)$ (correct answer)
$I(t) = \omega Q_{max} \sin(\omega t)$
$I(t) = \omega Q_{max} \cos(\omega t)$
$I(t) = -\frac{Q_{max}}{\omega} \sin(\omega t)$

Explanation: Current is the time derivative of charge,

I(t) = \frac{dq}{dt}

. Taking the derivative of the given expression for charge gives

I(t) = \frac{d}{dt}(Q_{max} \cos(\omega t)) = -\omega Q_{max} \sin(\omega t)