Q: A loop of wire with resistance $$R = 5 \, \Omega$$ and area $$A = 0.01 \, m^2$ moves at v = 10$$ \, m/s$$ into a region with a uniform magnetic field B = 0.06$$ \, T$$. What is the magnitude of the induced emf in the loop as it enters the field?

$$\mathcal{E} = B v A = 0.06 \times 10 \times 0.01 = 0.006 \, V$$

Q: A sinusoidal radio wave with frequency $$f = 100 \, MHz$$ and electric field amplitude $$E_0 = 100$$ \, V/m$$ is traveling in the positive z-direction. What is the amplitude of the magnetic field?

$$B_0$$ = \frac{$$E_0$$}{c} = \frac{100}{3 \times $$10^8$$} = 3.33 \times $$10^{-7}$$ \, T$$

Question 1

A loop of wire with resistance $$R = 5 \, \Omega$$ and area $$A = 0.01 \, m^2$ moves at v = 10$$ \, m/s$$ into a region with a uniform magnetic field B = 0.06$$ \, T$$. What is the magnitude of the induced emf in the loop as it enters the field?

Accepted Answer

$$\mathcal{E} = B v A = 0.06 	imes 10 	imes 0.01 = 0.006 \, V$$

Question 2

When the loop described above enters the magnetic field, what is the direction of the induced current?

Accepted Answer

Clockwise, because the magnetic flux is increasing and Lenz's law opposes the change.

Question 3

A sinusoidal radio wave with frequency $$f = 100 \, MHz$$ and electric field amplitude $$E_0 = 100$$ \, V/m$$ is traveling in the positive z-direction. What is the amplitude of the magnetic field?

Accepted Answer

$$B_0$$ = \frac{$$E_0$$}{c} = \frac{100}{3 	imes $$10^8$$} = 3.33 	imes $$10^{-7}$$ \, T$$

Question 4

A parallel plate capacitor is connected to a battery with voltage $$V = 10 \, V. $$The plates are separated by $$d = 0.01 \, m. $$What is the electric field between the plates?

Accepted Answer

$$E = \frac{V}{d} = \frac{10}{0.01} = 1000 \, V/m$$

Question 5

If the right plate of the capacitor above is moved to increase the distance between the plates at a constant rate, what happens to the current coming out of the battery?

Accepted Answer

There is a current because the capacitance is decreasing and the battery must supply charge to maintain the voltage.

Question 6

A circular capacitor with radius $$r = 0.15 \, m$$ and plate separation $$d = 0.01 \, m is $$connected to a battery. If the plate separation is increased at a rate of $$v = 0.06 \, m/s$$, what is the induced electric field inside the capacitor?

Accepted Answer

$$E = \frac{1}{2 \pi r} \frac{d\Phi_B}{dt}$$, but since $$B = 0$$, $$E = 0$$

Question 7

After one second, while the plate is moving, what is the magnetic field between the plates of the circular capacitor at a distance $$r = 0.05 \, m$$ from its axis?

Accepted Answer

$$B = \frac{\mu_0$ I}{2 \pi r}$$, where $$I is $$the displacement current

Question 8

A circuit contains two capacitors $$C_1 = 2$$ \, \mu F$$ and C_2$ = 1.34 \, \mu F in $$series, connected to a battery of $$V = 30 \, V. $$What is the voltage $$V_1$$ over capacitor $$C_1$$?

Accepted Answer

$$V_1 = Q$$ / $$C_1$$, where $$Q = V \left( \frac{C_1$ C_2$}{C_1$ + C_2$} \right)$$

Question 9

After the switch in the above circuit is closed for a long time, what is the charge $$Q_1 on $$capacitor $$C_1$$?

Accepted Answer

$$Q_1$$ = $$V_1$$ $$C_1$$, where $$V_1 is $$the voltage across $$C_1$$ after the switch is closed

Question 10

A parallel plate capacitor is filled with a dielectric of constant $$k = 5. $$How does the voltage $$V$$ across the capacitor change if the charge $$Q is $$kept constant?

Accepted Answer

$$V$$ increases by a factor of $$k$$

Question 11

A sinusoidal electromagnetic wave has an electric field amplitude $$E_0 = 100$$ \, V/m$$. What is the peak electromagnetic energy density in the wave?

Accepted Answer

$$u = \$$epsilon_0$$ E_0$^2 = 8.85 	imes 10$$^{-12}$$ 	imes 100^2$ = 8.85 	imes 10$$^{-8}$$ \, J/m^3$

Question 12

A loop of wire moves with velocity $$v$$ perpendicular to a uniform magnetic field $$B. $$What is the direction of the induced current if the magnetic flux through the loop is increasing?

Accepted Answer

The current flows in a direction that opposes the increase in flux, as per Lenz's law.

Question 13

A capacitor is charged to voltage $$V$$ and then disconnected from the battery. If the plate separation is increased, what happens to the electric field between the plates?

Accepted Answer

The electric field decreases because the voltage remains constant.

Question 14

A radio wave travels in the positive z-direction. If the electric field vector is in the x-direction, what is the direction of the magnetic field vector?

Accepted Answer

The magnetic field vector is in the y-direction.

Question 15

A capacitor is connected to a battery and then the battery is disconnected. If the plates are moved apart, what happens to the potential difference between the plates?

Accepted Answer

The potential difference increases.

Question 16

A loop of wire moves at $$v = 10 \, m/s$$ through a magnetic field region of width $$b = 0.4 \, m. $$How long does it take for the loop to completely pass through the field?

Accepted Answer

$$t = b / v = 0.4 / 10 = 0.04 \, s$$

Question 17

A parallel plate capacitor with area $$A$$ and separation $$d is $$filled with a dielectric of constant $$k. $$What is the new capacitance?

Accepted Answer

$$C = k \$$epsilon_0$$ \frac{A}{d}$$

Question 18

A capacitor is charged and then disconnected from the battery. If a dielectric is inserted between the plates, what happens to the stored energy?

Accepted Answer

The stored energy decreases.

Electromagnetism and Circuits: Exam Study Guide

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