Physics Final Exam Study Guidance: Solenoids, Gauss' Law, and RLC Circuits

Q1. Use Ampère's Law to find the magnetic field inside a long solenoid with current I and n loops per unit length.

Background

Topic: Magnetostatics – Ampère's Law and Solenoids

This question tests your understanding of how to apply Ampère's Law to determine the magnetic field inside a long solenoid, a classic problem in electromagnetism.

Key Terms and Formulas

• Ampère's Law: $\oint \vec{B} \cdot d\vec{l} = \mu_0 I_{\text{enc}}$

• Solenoid: A coil of wire with many turns, producing a nearly uniform magnetic field inside.

• n: Number of loops per unit length

• I: Current through the solenoid

• $\mu_0$: Permeability of free space

Step-by-Step Guidance

1. Draw a diagram of the solenoid, indicating the direction of current and the magnetic field lines inside and outside the solenoid.

2. Choose an Amperian loop: For a solenoid, select a rectangular path that runs partly inside and partly outside the solenoid, parallel to its axis.

3. Apply Ampère's Law to your chosen path: $\oint \vec{B} \cdot d\vec{l} = \mu_0 I_{\text{enc}}$

4. Argue that the magnetic field outside the solenoid is approximately zero, so only the segment inside contributes to the integral.

5. Express $I_{\text{enc}}$ in terms of the number of turns per unit length (n), the current (I), and the length of the path inside the solenoid.

Try solving on your own before revealing the answer!

Q2. Use Gauss' Law to find the electric field at a distance r from the center of a uniformly charged sphere of total charge Q and radius R. Consider both cases: r < R and r > R.

Background

Topic: Electrostatics – Gauss' Law and Spherical Symmetry

This question tests your ability to apply Gauss' Law to a spherically symmetric charge distribution, distinguishing between points inside and outside the sphere.

Key Terms and Formulas

• Gauss' Law: $\oint \vec{E} \cdot d\vec{A} = \frac{Q_{\text{enc}}}{\varepsilon_0}$

• Uniformly charged sphere: Charge is distributed evenly throughout the volume.

• Q: Total charge

• R: Radius of the sphere

• r: Distance from the center (consider r < R and r > R)

Step-by-Step Guidance

1. Draw a diagram showing the charged sphere and a Gaussian surface (a sphere of radius r centered at the same point).

2. Write Gauss' Law: $\oint \vec{E} \cdot d\vec{A} = \frac{Q_{\text{enc}}}{\varepsilon_0}$

3. For r > R (outside the sphere): All charge is enclosed. For r < R (inside the sphere): Only a fraction of the total charge is enclosed. Express $Q_{\text{enc}}$ in terms of Q, r, and R for each case.

4. Use symmetry to argue that $\vec{E}$ is radial and constant over the Gaussian surface, so $\oint \vec{E} \cdot d\vec{A} = E \cdot 4\pi r^2$.

5. Set up the equation for E in both cases, but stop before solving for the explicit value.

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Q3. An RLC circuit is powered by an alternating source: $v(t) = V_{\text{max}} \cos(\omega t + \phi)$, $i(t) = I_{\text{max}} \cos(\omega t)$. (a) Find the time-dependent voltage across the inductor, $V_L(t)$. (b) Minimize impedance Z with respect to angular frequency and find the resonant frequency. (c) For $V_{\text{max}} = 160$ V, $R = 100\,\Omega$, $C = 3.0$ mF, $L = 1.0$ mH, find $I_{\text{max}}$ at resonance. (d) Draw the phasor diagram.

Background

Topic: AC Circuits – RLC Circuits, Resonance, and Phasors

This question tests your understanding of the behavior of RLC circuits under AC driving voltages, including voltage drops across components, resonance, and phasor representation.

Key Terms and Formulas

• Inductor voltage: $V_L(t) = L \frac{di}{dt}$

• Impedance: $Z = \sqrt{R^2 + (\omega L - \frac{1}{\omega C})^2}$

• Resonant frequency: $\omega_0 = \frac{1}{\sqrt{LC}}$

• Current at resonance: $I_{\text{max}} = \frac{V_{\text{max}}}{Z}$

Step-by-Step Guidance

1. (a) To find $V_L(t)$, differentiate $i(t)$ with respect to time and multiply by L: $V_L(t) = L \frac{di}{dt}$.

2. (b) To minimize Z, set $\frac{dZ}{d\omega} = 0$ and solve for $\omega$. This gives the resonant frequency $\omega_0$.

3. (c) At resonance, $\omega = \omega_0$, so $Z = R$. Plug in the given values for $V_{\text{max}}$, $R$, $C$, and $L$ to set up the calculation for $I_{\text{max}}$.

4. (d) Draw the phasor diagram showing the relative orientation of $V_R$, $V_L$, and $V_C$.

5. Stop before plugging in the numbers for the final calculation of $I_{\text{max}}$.

Try solving on your own before revealing the answer!