Q: How much thermal energy is deposited in the wire after the capacitor is fully discharged?

$$U_{\text{thermal}} = \frac{Q_0$^2}{2C}$$

Q: A flat circular loop (radius $$r = 0.1 \text{m}) $$connects the sides of a parallel plate capacitor with gap $$d = 10$$^{-4}$$ \text{m}. $$The breakdown electric field is $$E_{\text{breakdown}} = 3 \times 10^6$ \text{V/m}. If $$the magnetic field increases as $$B = 2e$$^{3t}$$, what is the value of B$ when the gap breaks down?

$$B = \frac{E_{\text{breakdown}} d}{3 \pi r^2$}$$

Q: A rectangular loop (width $$0.3 \text{m}$$, height $$0.1 \text{m}) $$carrying $$I_1 = 3$$ \text{A}$$ is placed above a long wire carrying I_2$ = 22 \text{A}. $$What is the net force on the loop?

$$F_{\text{net}} = \frac{\mu_0$ I_1$ I_2$ \ell}{2 \pi} \left(\frac{1}{r_1$} - \frac{1}{r_2$}\right)$$

Q: For the heat pump cooling the container, what is the coefficient of performance (COP) for cooling at 70% Carnot efficiency, with $$T_L = 283 \text{K}$$ and $$T_H = 318 \text{K}$$?

$$COP = 0.70 \times \frac{T_L}{T_H - T_L}$$

Question 1

A capacitor with $$C = 2 \ \mu	ext{F} is $$charged to $$Q = 3 	imes 10$$^{-4}$$ 	ext{C}$$ and then discharged through a wire of length $$\ell = 3 	ext{m}$$, radius $$r = 1 	imes 10$$^{-5}$$ 	ext{m}$$, and resistivity $$ho = 9.7 	imes 10$$^{-8}$$ \ \Omega	ext{m}. $$What is the resistance $$R of $$the wire?

Accepted Answer

$$R = \frac{\rho \ell}{\pi r^2$}$$

Question 2

For the RC discharge circuit described above, what is the time $$t_1$$ for the capacitor voltage to drop from $$150 	ext{V} to$$ $$75 	ext{V}$$?

Accepted Answer

$$t_1 = RC$$ \ln 2$$

Question 3

During the RC discharge, what is the voltage across the resistor at time $$t_1$$ when the capacitor voltage is $$75 	ext{V}$$?

Accepted Answer

$$V_R(t_1$) = 75 	ext{V}$$

Question 4

How much thermal energy is deposited in the wire after the capacitor is fully discharged?

Accepted Answer

$$U_{	ext{thermal}} = \frac{Q_0$^2}{2C}$$

Question 5

A flat circular loop (radius $$r = 0.1 	ext{m}) $$connects the sides of a parallel plate capacitor with gap $$d = 10$$^{-4}$$ 	ext{m}. $$The breakdown electric field is $$E_{	ext{breakdown}} = 3 	imes 10^6$ 	ext{V/m}. If $$the magnetic field increases as $$B = 2e$$^{3t}$$, what is the value of B$ when the gap breaks down?

Accepted Answer

$$B = \frac{E_{	ext{breakdown}} d}{3 \pi r^2$}$$

Question 6

In a coaxial cable with inner radius $$r_1$$ and outer radius $$r_2$$, carrying current $$+I in $$the inner cylinder and $$-I in $$the outer, what is the magnetic field $$B(r)$$ for $$r_1$$ \u003c r \u003c $$r_2$$?

Accepted Answer

$$B(r) = \frac{\mu_0$ I}{2 \pi r}$$

Question 7

What is the magnetic energy stored per unit length in the coaxial cable between $$r_1$$ and $$r_2$$?

Accepted Answer

$$\frac{U}{\ell} = \frac{\mu_0$ I^2$}{4 \pi} \ln \left(\frac{r_2$}{r_1$}\right)$$

Question 8

A rectangular loop (width $$0.3 	ext{m}$$, height $$0.1 	ext{m}) $$carrying $$I_1 = 3$$ 	ext{A}$$ is placed above a long wire carrying I_2$ = 22 	ext{A}. $$What is the net force on the loop?

Accepted Answer

$$F_{	ext{net}} = \frac{\mu_0$ I_1$ I_2$ \ell}{2 \pi} \left(\frac{1}{r_1$} - \frac{1}{r_2$}ight)$$

Question 9

In the circuit with a fully charged capacitor ($$C = 13 	imes 10$$^{-6}$$ 	ext{F}$$), what is the energy stored in the capacitor if the voltage across it is $$0.74 	ext{V}$$?

Accepted Answer

$$U = \frac{1}{2} C V^2$

Question 10

A cubical container ($$L = 3 	ext{m}) $$with walls and roof of thickness $$d = 0.1 	ext{m}$$ and thermal conductivity $$\kappa = 0.1 	ext{W}/	ext{m}\cdot	ext{K} is $$kept at $$10^\circ	ext{C}$$ while the outside is $$45^\circ	ext{C}. $$What is the heat leak rate through the walls and roof?

Accepted Answer

$$\dot{Q}_L = \kappa A \frac{\Delta T}{d}$$

Question 11

For the heat pump cooling the container, what is the coefficient of performance (COP) for cooling at 70% Carnot efficiency, with $$T_L = 283 	ext{K}$$ and $$T_H = 318 	ext{K}$$?

Accepted Answer

$$COP = 0.70 	imes \frac{T_L}{T_H - T_L}$$

Question 12

A charge $$-Q of $$mass $$m is $$released from rest at $$y = R$$ inside an infinite cylinder of radius $$R$$ and charge density $$\rho. $$What is the electric field inside the cylinder at radius $$r \u003c R$$?

Accepted Answer

$$E(r) = \frac{\rho r}{2 \$$epsilon_0$$}$$

Question 13

What is the speed $$v of $$the charge $$-Q$$ when it reaches the origin, given the cylinder's charge density $$\rho$$, radius $$R$$, and mass $$m$$?

Accepted Answer

$$v = R \sqrt{\frac{Q \rho}{2 m \$$epsilon_0$$}}$$

Question 14

Why does the spherical shell exert no force on the charge $$-Q as it $$moves from $$y = R to $$the origin?

Accepted Answer

The electric field inside a uniformly charged spherical shell is zero by Gauss's law.

Question 15

In the RC discharge problem, what is the temperature rise $$\Delta T of $$the wire due to the complete discharge, given mass $$m$$ and specific heat $$c_p$$?

Accepted Answer

$$\Delta T = \frac{U_{	ext{thermal}}}{m c_p}$$

Question 16

In the heat pump problem, why is the COP for cooling always less than the COP for heating?

Accepted Answer

Because $$COP_{	ext{heating}} = \frac{T_H}{T_H - T_L}$$, which is always 1 unit larger than $$COP_{	ext{cooling}} = \frac{T_L}{T_H - T_L}.$$

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