Q: A real battery with emf $$E = 12\ \mathrm{V}$$ and internal resistance $$r = 1\ \Omega is $$connected to a $$5\ \Omega$$ resistor. What is the terminal voltage across the battery?

$$V_{\text{terminal}} = 12\ \mathrm{V} - I r$$,\ where\ I = \frac{12}{6} = 2\ \mathrm{A} \Rightarrow V_{\text{terminal}} = 12 - 2 \times 1 = 10\ \mathrm{V}$

Q: A long straight wire carries a current of $$10\ \mathrm{A}. $$What is the magnitude of the magnetic field $$5\ \mathrm{cm}$$ away from the wire?

$$B = \frac{\mu_0$ I}{2\pi r} = \frac{4\pi \times 10$$^{-7}$$ \times 10}{2\pi \times 0.05} = 4 \times 10$$^{-5}$$\ \mathrm{T}$$

Q: A circular coil with $$N = 20$$ turns and area $$A = 0.01\ \mathrm{m}^2 is $$placed in a uniform magnetic field $$B = 0.5\ \mathrm{T}$$ perpendicular to the plane of the coil. If the field drops to zero in $$0.1\ \mathrm{s}$$, what is the magnitude of the average induced emf?

$$E = -N \frac{\Delta \Phi_B}{\Delta t} = -20 \times \frac{0.5 \times 0.01}{0.1} = -1\ \mathrm{V}$$

Q: A rectangular current loop of area $$A$$ and $$N$$ turns carries current $$I$$ and is placed in a uniform magnetic field $$B. $$What is the maximum torque experienced by the loop?

$$\tau_{\text{max}} = N I A B$$

Q: The intensity of sunlight at Earth’s distance from the Sun is approximately $$I = 1400\ \mathrm{W/m}^2. $$What is the total power output of the Sun?

$$P = 4\pi r^2$ I$$,\ with\ r = 1.496 \times $$10^{11}$$\ \mathrm{m}$

Q: A $$10\ \mu\mathrm{F}$$ capacitor is charged to $$12\ \mathrm{V}$$ and then connected across a $$2\ \Omega$$ resistor. How long will it take for the voltage across the capacitor to drop to $$3\ \mathrm{V}$$?

$$t = RC \ln \left(\frac{V_0$}{V}\right) = 2 \times 10 \times 10$$^{-6}$$ \ln\left(\frac{12}{3}\right) = 20 \times 10$$^{-6}$$ \ln(4) \approx 27.7\ \mu\mathrm{s}$$

Q: A solenoid has $$n = 1000\ \mathrm{turns/m}$$ and carries a current of $$2\ \mathrm{A}. $$What is the magnitude of the magnetic field inside the solenoid?

$$B = \mu_0$ n I = 4\pi \times 10$$^{-7}$$ \times 1000 \times 2 = 2.51 \times 10$$^{-3}$$\ \mathrm{T}$$

Question 1

A beam of unpolarized light with intensity $$I_0$$ passes through two ideal polarizing filters. The first filter is oriented at $$0^\circ$$ and the second at $$60^\circ$$ relative to the first. What is the final transmitted intensity?

Accepted Answer

$$I = \frac{I_0$}{2} \cos^2$ 60^\circ$$

Question 2

A real battery with emf $$E = 12\ \mathrm{V}$$ and internal resistance $$r = 1\ \Omega is $$connected to a $$5\ \Omega$$ resistor. What is the terminal voltage across the battery?

Accepted Answer

$$V_{	ext{terminal}} = 12\ \mathrm{V} - I r$$,\ where\ I = \frac{12}{6} = 2\ \mathrm{A} \Rightarrow V_{	ext{terminal}} = 12 - 2 	imes 1 = 10\ \mathrm{V}$

Question 3

A long straight wire carries a current of $$10\ \mathrm{A}. $$What is the magnitude of the magnetic field $$5\ \mathrm{cm}$$ away from the wire?

Accepted Answer

$$B = \frac{\mu_0$ I}{2\pi r} = \frac{4\pi 	imes 10$$^{-7}$$ 	imes 10}{2\pi 	imes 0.05} = 4 	imes 10$$^{-5}$$\ \mathrm{T}$$

Question 4

A circular coil with $$N = 20$$ turns and area $$A = 0.01\ \mathrm{m}^2 is $$placed in a uniform magnetic field $$B = 0.5\ \mathrm{T}$$ perpendicular to the plane of the coil. If the field drops to zero in $$0.1\ \mathrm{s}$$, what is the magnitude of the average induced emf?

Accepted Answer

$$E = -N \frac{\Delta \Phi_B}{\Delta t} = -20 	imes \frac{0.5 	imes 0.01}{0.1} = -1\ \mathrm{V}$$

Question 5

A series RLC circuit has $$R = 100\ \Omega$$, $$L = 0.2\ \mathrm{H}$$, and $$C = 50\ \mu\mathrm{F}. $$What is the resonance frequency $$f_0$$?

Accepted Answer

$$f_0$$ = \frac{1}{2\pi \sqrt{LC}} = \frac{1}{2\pi \sqrt{0.2 	imes 50 	imes $$10^{-6}$$}} \approx 50.3\ \mathrm{Hz}$$

Question 6

An electron is moving in a circle of radius $$r in a $$uniform magnetic field $$B. If $$the kinetic energy of the electron is doubled, what happens to the radius of its path?

Accepted Answer

The radius increases by a factor of $$\sqrt{2}$$: $$r' = r \sqrt{2}$$

Question 7

A rectangular current loop of area $$A$$ and $$N$$ turns carries current $$I$$ and is placed in a uniform magnetic field $$B. $$What is the maximum torque experienced by the loop?

Accepted Answer

$$	au_{	ext{max}} = N I A B$$

Question 8

The intensity of sunlight at Earth’s distance from the Sun is approximately $$I = 1400\ \mathrm{W/m}^2. $$What is the total power output of the Sun?

Accepted Answer

$$P = 4\pi r^2$ I$$,\ with\ r = 1.496 	imes $$10^{11}$$\ \mathrm{m}$

Question 9

A $$10\ \mu\mathrm{F}$$ capacitor is charged to $$12\ \mathrm{V}$$ and then connected across a $$2\ \Omega$$ resistor. How long will it take for the voltage across the capacitor to drop to $$3\ \mathrm{V}$$?

Accepted Answer

$$t = RC \ln \left(\frac{V_0$}{V}ight) = 2 	imes 10 	imes 10$$^{-6}$$ \ln\left(\frac{12}{3}ight) = 20 	imes 10$$^{-6}$$ \ln(4) \approx 27.7\ \mu\mathrm{s}$$

Question 10

A solenoid has $$n = 1000\ \mathrm{turns/m}$$ and carries a current of $$2\ \mathrm{A}. $$What is the magnitude of the magnetic field inside the solenoid?

Accepted Answer

$$B = \mu_0$ n I = 4\pi 	imes 10$$^{-7}$$ 	imes 1000 	imes 2 = 2.51 	imes 10$$^{-3}$$\ \mathrm{T}$$

Question 11

A metal bar of length $$0.5\ \mathrm{m}$$ moves at $$3\ \mathrm{m/s}$$ perpendicular to a $$0.2\ \mathrm{T}$$ magnetic field. What is the induced emf across the bar?

Accepted Answer

$$E = v \ell B = 3 	imes 0.5 	imes 0.2 = 0.3\ \mathrm{V}$$

Question 12

A transformer has $$N_p = 100$$ turns on the primary and $$N_s = 500$$ turns on the secondary. If the primary voltage is $$120\ \mathrm{V}$$, what is the secondary voltage?

Accepted Answer

$$V_s = V_p \frac{N_s}{N_p} = 120 	imes \frac{500}{100} = 600\ \mathrm{V}$$

Question 13

A proton enters a region with a uniform magnetic field $$\vec{B}$$ pointing into the page and an electric field $$\vec{E}$$ pointing up. If the proton moves to the right and is undeflected, what is its speed?

Accepted Answer

$$v = \frac{E}{B}$$

Question 14

Two long parallel wires are separated by $$10\ \mathrm{cm}$$ and each carries $$5\ \mathrm{A} in $$the same direction. What is the force per unit length between them?

Accepted Answer

$$\frac{F}{L} = \frac{\mu_0$ I_1$ I_2$}{2\pi d} = \frac{4\pi 	imes 10$$^{-7}$$ 	imes 5 	imes 5}{2\pi 	imes 0.1} = 5 	imes 10$$^{-5}$$\ \mathrm{N/m}$$

Question 15

A $$1\ \mathrm{m}$$ long wire carries $$4\ \mathrm{A}$$ perpendicular to a $$0.3\ \mathrm{T}$$ magnetic field. What is the magnitude of the force on the wire?

Accepted Answer

$$F = I L B \sin 	heta = 4 	imes 1 	imes 0.3 	imes 1 = 1.2\ \mathrm{N}$$

Question 16

A $$10\ \mu\mathrm{F}$$ capacitor and a $$1\ \mathrm{mH}$$ inductor are connected in series. What is the angular frequency of oscillation?

Accepted Answer

$$\$$omega_0$$ = \frac{1}{\sqrt{LC}} = \frac{1}{\sqrt{1 	imes 10$$^{-3}$$ 	imes 10 	imes 10$$^{-6}$$}} = 10,000\ \mathrm{rad/s}$$

Question 17

A $$100\ \Omega$$ resistor is connected to a $$10\ \mathrm{V}$$ battery. How much power is dissipated in the resistor?

Accepted Answer

$$P = \frac{(\Delta V)^2$}{R} = \frac{10^2$}{100} = 1\ \mathrm{W}$$

Question 18

A $$1\ \mathrm{m}$$ long solenoid with $$1000$$ turns and cross-sectional area $$1\ \mathrm{cm}^2$$ carries a current that increases at $$5\ \mathrm{A/s}. $$What is the induced emf across the solenoid?

Accepted Answer

$$L = \mu_0$ n^2$ \ell A = 4\pi 	imes 10$$^{-7}$$ 	imes (1000)^2$ 	imes 1 	imes 0.0001 = 0.126\ \mathrm{H}$$;\ E = -L \frac{dI}{dt} = -0.126 	imes 5 = -0.63\ \mathrm{V}$

Question 19

A $$1\ \mathrm{m}$$ diameter circular loop carries a current of $$2\ \mathrm{A}. $$What is the magnetic field at the center of the loop?

Accepted Answer

$$B = \frac{\mu_0$ I}{2R} = \frac{4\pi 	imes 10$$^{-7}$$ 	imes 2}{2 	imes 0.5} = 2.51 	imes 10$$^{-6}$$\ \mathrm{T}$$

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