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

1. What is the energy stored in an inductor $L$ with current $I$?

   $\frac{1}{2}LI^2$. Magnetic field energy in inductor.

2. If resistance $R$ doubles in an LR circuit, what happens to the time constant $\tau$?

   Halved. Time constant inversely proportional to resistance.

3. Calculate energy stored in an inductor with $L=4\,H$ and $I=2\,A$.

   $16\,J$. $U = \frac{1}{2} \times 4 \times 2^2 = 16\,J$

4. Find the phase angle $\theta$ if $R=5\,\Omega$ and $X_L=12\,\Omega$.

   $\theta = 67.38^\circ$. $\theta = \tan^{-1}(\frac{12}{5}) = 67.38°$

5. What is the current $I(t)$ at $t=\tau$ in an LR circuit during charging?

   $I(t) = 0.632I_0$. At one time constant, $1-e^{-1} = 0.632$

6. What is the effect of $L$ on the phase angle $\theta$ if $L$ increases?

   Increases. Higher $L$ increases $X_L$, increasing phase angle.

7. What is the time-dependent current $I(t)$ in an LR circuit during charging?

   $I(t) = I_0(1 - e^{-\frac{t}{\tau}})$. Exponential rise to steady-state current.

8. If $L$ is doubled, what happens to the initial rate of change of current in an LR circuit?

   Halved. Initial slope $\frac{dI}{dt} = \frac{V}{L}$ decreases.

9. If $R$ increases, what happens to the power dissipated in an LR circuit?

   Increases. Power dissipation $P = I^2R$ increases with $R$.

10. Identify the formula to calculate the impedance $Z$ in an LR circuit.

    $Z = \sqrt{R^2 + (X_L)^2}$. Impedance combines resistance and inductive reactance.

11. Predict the phase difference between current and voltage in an LR circuit.

    Current lags voltage. Inductor causes current to lag voltage.

12. Determine the impedance $Z$ if $R=3\,\Omega$ and $X_L=4\,\Omega$.

    $Z = 5\,\Omega$. $Z = \sqrt{3^2 + 4^2} = 5\,\Omega$

13. Determine $X_L$ if frequency $f$ is doubled in an LR circuit.

    Doubles. Reactance linearly proportional to frequency.

14. What is the power factor when $R = X_L$ in an LR circuit?

    $pf = 0.707$. When $R = X_L$, $\cos(45°) = 0.707$

15. What is the phase angle $\theta$ when $X_L = 0$ in an LR circuit?

    $\theta = 0^\circ$. No inductive reactance means purely resistive.

16. Calculate the power factor $pf$ in an LR circuit.

    $pf = \cos(\theta)$. Cosine of phase angle between voltage and current.

17. State the formula for the time constant $\tau$ in an LR circuit.

    $\tau = \frac{L}{R}$. Time constant determines how quickly current changes.

18. What happens to $X_L$ when $L$ is halved in an LR circuit?

    Halved. Reactance directly proportional to inductance.

19. Determine the total power $P$ dissipated in an LR circuit.

    $P = I^2R$. Only resistor dissipates power as heat.

20. Find $Z$ if $R=6\,\Omega$ and $X_L=8\,\Omega$ in an LR circuit.

    $Z = 10\,\Omega$. $Z = \sqrt{6^2 + 8^2} = 10\,\Omega$

21. Calculate $I(t)$ at $t=2\tau$ during discharging if $I_0 = 10\,A$.

    $I(t) = 1.35\,A$. $I = 10 \times e^{-2} = 1.35\,A$

22. Which component causes a phase shift between voltage and current in an LR circuit?

    Inductor. Inductors oppose current changes, creating phase lag.

23. In an LR circuit, what happens to $\theta$ if $R$ is much larger than $X_L$?

    $\theta \rightarrow 0^\circ$. When $R >> X_L$, circuit becomes resistive.

24. Determine the voltage across the inductor $V_L$ at time $t$ in an LR circuit.

    $V_L = L \frac{dI}{dt}$. Inductor voltage opposes current changes.

25. What is the phase angle $\theta$ in an LR circuit?

    $\theta = \tan^{-1}(\frac{X_L}{R})$. Phase angle shows current lagging behind voltage.

26. In an LR circuit, what is the effect of increasing inductance $L$ on the time constant?

    Increases time constant. Higher $L$ means slower current changes.

27. Calculate the time constant $\tau$ for $L=3\,H$ and $R=6\,\Omega$ in an LR circuit.

    $\tau = 0.5\,s$. $\tau = \frac{3}{6} = 0.5\,s$

28. Find the inductive reactance $X_L$ if $L=2\,H$ and $f=50\,Hz$.

    $X_L = 628.32\,\Omega$. $X_L = 2\pi \times 50 \times 2 = 628.32\,\Omega$

29. Identify the effect of a higher frequency on inductive reactance $X_L$ in an LR circuit.

    Increases. Reactance proportional to frequency.

30. What happens to impedance $Z$ if resistance $R$ decreases in an LR circuit?

    Decreases. Lower resistance reduces total impedance.

31. If $\theta = 45^\circ$, what is the relationship between $R$ and $X_L$?

    $R = X_L$. At 45°, resistive and reactive parts are equal.

32. In an LR circuit, what happens to current $I(t)$ as time $t \rightarrow \infty$ during charging?

    Approaches $I_0$. Current reaches steady-state value over time.

33. What is the time-dependent current $I(t)$ in an LR circuit during discharging?

    $I(t) = I_0 e^{-\frac{t}{\tau}}$. Exponential decay from initial current.

34. What is the total current $I$ if $V=120\,V$ and $Z=15\,\Omega$ in an LR circuit?

    $I = 8\,A$. $I = \frac{V}{Z} = \frac{120}{15} = 8\,A$

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