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

1. What is the formula for the maximum current in an LC circuit?

   $I_{max} = \sqrt{2E/L}$. Maximum current when all energy is magnetic.

2. What happens to the total energy in an ideal LC circuit?

   It remains constant. No energy loss in ideal LC circuit, only conversion between forms.

3. What is the formula for the resonant frequency of an LC circuit?

   $f_0 = \frac{1}{2\pi\sqrt{LC}}$. Derived from setting $X_L = X_C$ at resonance.

4. What is the expression for the current in an LC circuit?

   $I(t) = -\omega Q_0 \sin(\omega t + \phi)$. Current is negative derivative of charge with respect to time.

5. What is the formula for the maximum charge on a capacitor in an LC circuit?

   $Q_{max} = \sqrt{2E/C}$. Maximum charge when all energy is electric.

6. In an LC circuit, when is energy purely magnetic?

   When capacitor is fully discharged. All energy stored in inductor's magnetic field.

7. What is the phase relationship between voltage and current in a purely capacitive circuit?

   Current leads by $90^\circ$. Current flows before voltage builds up across capacitor.

8. What is the expression for voltage across an inductor in an LC circuit?

   $V_L = L \frac{di}{dt}$. Voltage across inductor depends on rate of current change.

9. State one condition for resonance in an LC circuit.

   $\omega L = \frac{1}{\omega C}$. Inductive and capacitive reactances are equal at resonance.

10. What happens to the current in an LC circuit at resonance?

    Current is maximum. At resonance, impedance is minimum so current reaches maximum value.

11. What is the formula for the natural frequency of an LC circuit?

    $\omega_0 = \frac{1}{\sqrt{LC}}$. Angular frequency without the $2\pi$ factor.

12. What is the relationship between charge and current in an LC circuit?

    $I = \frac{dq}{dt}$. Current is the time derivative of charge.

13. Identify the effect of increasing capacitance on the resonant frequency of an LC circuit.

    Decreases resonant frequency. Higher C makes $f_0 = \frac{1}{2\pi\sqrt{LC}}$ smaller.

14. What is the phase relationship between voltage and current in a purely inductive circuit?

    Current lags by $90^\circ$. Current cannot change instantly through an inductor.

15. Find the maximum energy stored in a 2 F capacitor with 10 V potential in an LC circuit.

    $E = \frac{1}{2}CV^2 = 100 \text{ J}$. Use $E = \frac{1}{2}CV^2$ with $C = 2$ F, $V = 10$ V.

16. What is the total impedance of an LC circuit at resonance?

    Zero. Equal and opposite reactances cancel at resonance.

17. What is the expression for voltage across a capacitor in an LC circuit?

    $V_C = \frac{q}{C}$. Voltage is proportional to stored charge.

18. Find the resonant frequency if $L = 1 \text{ H}$ and $C = 4 \text{ F}$.

    $f_0 = \frac{1}{2\pi\sqrt{LC}} = 0.0796 \text{ Hz}$. Apply resonance formula with $L = 1$ H and $C = 4$ F.

19. Which component stores energy in a magnetic field in an LC circuit?

    Inductor. Magnetic field in inductor coil stores magnetic energy.

20. Which component opposes changes in voltage in an LC circuit?

    Capacitor. Capacitor voltage cannot change instantaneously.

21. Which component stores energy in an electric field in an LC circuit?

    Capacitor. Electric field between capacitor plates stores electric energy.

22. In an LC circuit, when is energy purely electric?

    When current is zero. All energy stored in capacitor's electric field.

23. Determine the time period of oscillation for an LC circuit with $L = 1 \text{ H}$ and $C = 1 \text{ F}$.

    $T = 2\pi\sqrt{LC} = 2\pi$ seconds. Use $T = 2\pi\sqrt{LC}$ with given L and C values.

24. What is the phase angle between voltage and current in an LC circuit at resonance?

    Zero. Voltage and current oscillate in phase at resonance.

25. Which quantity oscillates between the capacitor and inductor in an LC circuit?

    Energy. Energy alternates between electric and magnetic forms.

26. Find the angular frequency of an LC circuit if $L = 2 \text{ H}$ and $C = 0.5 \text{ F}$.

    $\omega = \sqrt{\frac{1}{LC}} = 1 \text{ rad/s}$. Use $\omega = \frac{1}{\sqrt{LC}}$ with given values.

27. What is the formula for inductive reactance in an LC circuit?

    $X_L = \omega L$. Inductive reactance increases with increasing frequency.

28. Identify the effect of increasing inductance on the resonant frequency of an LC circuit.

    Decreases resonant frequency. Higher L makes $f_0 = \frac{1}{2\pi\sqrt{LC}}$ smaller.

29. What is the expression for the charge on a capacitor in an LC circuit?

    $q(t) = Q_0 \cos(\omega t + \phi)$. Cosine function represents oscillating charge with amplitude $Q_0$.

30. What is the formula for capacitive reactance in an LC circuit?

    $X_C = \frac{1}{\omega C}$. Capacitive reactance decreases with increasing frequency.

31. Which component opposes changes in current in an LC circuit?

    Inductor. Lenz's law: inductor opposes current changes.

32. State the expression for the total energy stored in an LC circuit.

    $E = \frac{1}{2}CV^2 + \frac{1}{2}LI^2$. Sum of electric energy in capacitor and magnetic energy in inductor.

33. Identify the formula for the time constant in an LC circuit.

    $\tau = \sqrt{LC}$. Characteristic time scale for LC oscillations.

34. What is the energy conversion in an LC circuit?

    Electrical to Magnetic and vice versa. Energy oscillates between electric field and magnetic field.

35. Find the energy stored in a 1 H inductor with 5 A current in an LC circuit.

    $E = \frac{1}{2}LI^2 = 12.5 \text{ J}$. Use $E = \frac{1}{2}LI^2$ with $L = 1$ H, $I = 5$ A.

36. What is the phase difference between voltage and current at resonance in an LC circuit?

    Zero. Voltage and current are in phase when $X_L = X_C$.

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