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.

5. (10%) The conducting bar illustrated in Figure 5 moves on two frictionless, parallel rails in the presence of a uniform magnetic field directed into the page. The bar has mass \( m \), and its length is \( \ell \). The bar is given an initial velocity \( \vec{v}_i \) to the right and is released at time \( t = 0 \). find the velocity of the bar as a function of time.

6. (10%) Two infinitely long, parallel wires are lying on the ground a distance a meter apart as shown in Figure 6. A third wire, of length L meter and mass m kg, carries a current of I₁ Ampere and is levitated above the first two wires, at a horizontal position midway between them. The infinitely long wires carry equal currents I₂ in the same direction, but in the direction opposite that in the levitated wire. What current must the infinitely long wires carry so that the three wires form an equilateral triangle?Figure 7

7. (10%) In an experiment designed to measure the magnitude of a uniform magnetic field, electrons are accelerated from rest through a potential difference of ΔV Voltage and then enter a uniform magnetic field that is perpendicular to the velocity vector of the electrons. The electrons travel along a curved path because of the magnetic field of B Tesla (Such a curved beam of electrons is shown in Figure 7). What is the radius of the curved beam as shown in Figure 7? Note that the mass of an electron is mₑ kg and an electron contains e Coulomb charges.

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