Circuits with Capacitors and Inductors - AP Physics C Electricity and Magnetism

1. 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 $\omega = 1/\sqrt{LC}$.
   Answer: $q(t) = Q_{max} \cos(\omega t)$ 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, $q(t) = Q_{max} \cos(\omega t)$.
2. 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?
   Answer: They are out of phase by $\pi/2$ radians ($90^\circ$). 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.
3. 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?
   Answer: $2T$ 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 $T_{new} = 2\pi\sqrt{(4L)C} = 2(2\pi\sqrt{LC}) = 2T$.
4. 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?
   Answer: $U_{L,max} = \frac{Q_{max}^2}{2C}$ 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 $U_{C,max} = \frac{Q_{max}^2}{2C}$.
5. 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?
   Answer: $T = 2\pi\sqrt{LC}$ 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 $T = 2\pi\sqrt{LC}$.
6. 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?
   Answer: $L\frac{dI}{dt} + \frac{q}{C} = 0$ 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 $L\frac{dI}{dt} + \frac{q}{C} = 0$.
7. 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?
   Answer: $f/4$ 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 $f_{new} = \frac{1}{2\pi\sqrt{(2L)(8C)}} = \frac{1}{2\pi\sqrt{16LC}} = \frac{1}{4} \left( \frac{1}{2\pi\sqrt{LC}} \right) = \frac{f}{4}$.
8. 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?
   Answer: $3.0$ C 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 $Q_{max} = I_{max}\sqrt{LC} = (0.5 \text{ A})\sqrt{(4.0 \text{ H})(9.0 \text{ F})} = (0.5 \text{ A})\sqrt{36 \text{ s}^2} = (0.5 \text{ A})(6 \text{ s}) = 3.0$ C.
9. 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?
   Answer: $50$ rad/s 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 $\omega^2 = 2500$ s$^{-2}$. Taking the square root gives the angular frequency $\omega = 50$ rad/s.
10. 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?
    Answer: The current flowing through the inductor is zero. 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.
11. Which of the following statements provides the best description of the energy transformations in an ideal LC circuit?
    Answer: Energy is transformed back and forth between the electric field of the capacitor and the magnetic field of the inductor. 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.
12. 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?
    Answer: $t = T/8$ 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 $t = T/8$.
13. 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 $I = dq/dt$?
    Answer: $I(t) = -\omega Q_{max} \sin(\omega t)$ Current is the time derivative of charge, $I(t) = \frac{dq}{dt}$. Taking the derivative of the given expression for charge gives $I(t) = \frac{d}{dt}(Q_{max} \cos(\omega t)) = -\omega Q_{max} \sin(\omega t)$.
14. In an ideal LC circuit, which of the following quantities oscillate sinusoidally with time?
    Answer: Charge on the capacitor and current in the inductor only. 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 $\cos^2$ or $\sin^2$, which is not a simple sinusoid, and they oscillate at twice the frequency of the charge and current.
15. 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?
    Answer: $1.0$ A The maximum current $I_{max}$ is related to the maximum charge $Q_{max}$ by $I_{max} = \omega Q_{max}$. 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, $I_{max} = (100 \text{ rad/s})(10 \times 10^{-3} \text{ C}) = 1.0$ A.
16. 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?
    Answer: $20$ $\mu$J 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, $U_L = U_{total} - U_C = 50 \mu J - 30 \mu J = 20 \mu J$.
17. 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?
    Answer: $2I_{max}$ 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 $I_{new,max} = 2I_{max}$.
18. 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?
    Answer: $\omega = \frac{1}{\sqrt{LC}}$ 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 $\omega = \frac{1}{\sqrt{LC}}$.
19. 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?
    Answer: It increases because the total energy of the circuit increases. 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.
20. 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 $C$?
    Answer: $0.125$ F 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 $C = 1/8 = 0.125$ F.

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