4. Gauss' Law (10 points)

5. Faraday's Law (10 points)

6. A uniformly charged disk has radius \( R \) and surface charge density \( \sigma \) (shown in Fig. 1). Find (a) the electric potential at a point \( P \) along the perpendicular central axis of the disk (5 points), and (b) prove that the \( x \) component of the electric field at a point \( P \) along the perpendicular central axis of the disk is \[ 2\pi k_e \sigma [1 - \frac{x}{(R^2 + x^2)^{1/2}}]. (5 \text{ points}) \]Note that (b) must be solved by differentiating the result in (a) and we set the potential at infinity to be zeros.

7. Prove that a capacitance of two parallel plates shown in Fig. 2 is \( \varepsilon_0 A / d \) (10 points)

8. Please prove that the field inside an ideal solenoid (Fig. 3) is \( B = \mu_0 n I \); where \( n \) is the number of turns per unit length. (5 points). Please explain what conditions for a solenoid to be ideal are. (5 points)

9. What is electrostatic equilibrium of a conductor? (5 points) Explain why the electric field is zero everywhere inside the conductor when the conductor is in electrostatic equilibrium whether the conductor is solid or hollow. (5 points)

10. Explain why charge resides on conductor's surface when the conductor is isolated and carries a charge in electrostatic equilibrium. (10 points)

1. (10%) Find the magnetic field at the point P (as shown in Figure 1) contributed by the straight wire carrying a constant current I with $\theta1 = \frac{\pi}{3}$ and $\theta2 = \frac{-\pi}{6}$. Assume $\mu_0$ is the permeability of vacuum.Figure 1

2. (10%) A coaxial cable consists of two concentric cylindrical conductors. The region between the conductors is completely filled with polyethylene plastic as shown in Figure 2. Current leakage through the plastic, in the radial direction, is unwanted. The radius of the inner conductor is \( a \), the radius of the outer conductor is \( b \), and the length is \( L \). The resistivity of the plastic is \( \rho \). Calculate the resistance of the plastic between the two conductors.

3. (10%) A long solenoid of radius \( R \) has \( n \) turns of wire per unit length and carries a time-varying current that varies sinusoidally as \( I = I{\max} \sin \omega t \), where \( I{\max} \) is the maximum current and \( \omega \) is the angular frequency of the alternating current source (Figure 3). Determine the magnitude of the induced electric fields inside the solenoid at a distance \( r \) from its long central axis.

4. (10%) A conducting bar of length $\ell$ rotates with a constant angular speed $\omega$ about a pivot at one end. A uniform magnetic field $\vec{\mathbf{B}}_{\text{in}}$ is directed perpendicular to the plane of rotation as shown in Figure 4. Find the motional emf induced between the ends of the bar.

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