Q: A flat surface of area $$A = 3.0 \ \mathrm{cm}^2 is $$placed in a uniform magnetic field $$\vec{B}$$ such that the normal to the surface makes an angle of $$30^\circ$$ with $$\vec{B}. If $$the magnetic flux through the surface is $$0.90 \ \mathrm{mWb}$$, what is the magnitude of the magnetic field?

$$B = \frac{\Phi}{A \cos 60^\circ} = \frac{0.90 \times 10$$^{-3}$$ \ \mathrm{Wb}}{3.0 \times 10$$^{-4}$$ \ \mathrm{m}^2 \times 0.5} = 6.0 \ \mathrm{T}$$

Q: A conducting loop of area $$A = 0.012 \ \mathrm{m}^2 is $$placed in a region where the magnetic field increases at a rate $$\frac{dB}{dt} = 0.020 \ \mathrm{T/s}. If $$the total resistance of the circuit is $$5.0 \ \Omega$$, what is the magnitude of the induced current?

$$I = \frac{\mathcal{E}}{R} = \frac{0.24 \times 10$$^{-3}$$ \ \mathrm{V}}{5.0 \ \Omega} = 48 \ \mu\mathrm{A}$$

Q: A metal rod of length $$L = 0.10 \ \mathrm{m}$$ moves with velocity $$v = 2.5 \ \mathrm{m/s}$$ perpendicular to a magnetic field $$B = 0.60 \ \mathrm{T}. If $$the total resistance of the loop is $$0.030 \ \Omega$$, what is the magnitude of the induced current?

$$I = \frac{L v B}{R} = \frac{0.10 \times 2.5 \times 0.60}{0.030} = 5.0 \ \mathrm{A}$$

Q: A long solenoid has 500 turns per meter and a cross-sectional area of $$4.0 \ \mathrm{cm}^2. If $$the current in its windings is increasing at $$100 \ \mathrm{A/s}$$, what is the magnitude of the induced emf in a wire loop outside the solenoid?

$$\mathcal{E} = \mu_0$ n A \frac{dI}{dt} = (4\pi \times 10$$^{-7}$$) \times 500 \times 4.0 \times 10$$^{-4}$$ \times 100 = 25 \ \mu\mathrm{V}$$

Q: A 1.0 mH inductor has a current that drops from $$1.0 \ \mathrm{A} to$$ $$0 in$$ $$5.0 \ \mathrm{ms}. $$What is the induced voltage across the inductor?

$$\Delta V_L = -L \frac{\Delta I}{\Delta t} = -0.0010 \times \frac{-1.0}{5.0 \times 10$$^{-6}$$} = 2000 \ \mathrm{V}$$

Question 1

A flat surface of area $$A = 3.0 \ \mathrm{cm}^2 is $$placed in a uniform magnetic field $$\vec{B}$$ such that the normal to the surface makes an angle of $$30^\circ$$ with $$\vec{B}. If $$the magnetic flux through the surface is $$0.90 \ \mathrm{mWb}$$, what is the magnitude of the magnetic field?

Accepted Answer

$$B = \frac{\Phi}{A \cos 60^\circ} = \frac{0.90 	imes 10$$^{-3}$$ \ \mathrm{Wb}}{3.0 	imes 10$$^{-4}$$ \ \mathrm{m}^2 	imes 0.5} = 6.0 \ \mathrm{T}$$

Question 2

Which of the following actions will NOT induce a current in a coil according to Faraday’s law?

Accepted Answer

Holding a stationary magnet near a stationary coil

Question 3

A conducting loop of area $$A = 0.012 \ \mathrm{m}^2 is $$placed in a region where the magnetic field increases at a rate $$\frac{dB}{dt} = 0.020 \ \mathrm{T/s}. If $$the total resistance of the circuit is $$5.0 \ \Omega$$, what is the magnitude of the induced current?

Accepted Answer

$$I = \frac{\mathcal{E}}{R} = \frac{0.24 	imes 10$$^{-3}$$ \ \mathrm{V}}{5.0 \ \Omega} = 48 \ \mu\mathrm{A}$$

Question 4

A 500-turn circular coil of radius $$4.00 \ \mathrm{cm} is $$placed in a uniform magnetic field that makes an angle of $$60^\circ$$ with the plane of the coil. If the magnetic field decreases at a rate of $$0.200 \ \mathrm{T/s}$$, what is the magnitude of the induced emf?

Accepted Answer

$$\mathcal{E} = n \frac{dB}{dt} A \cos 30^\circ = 0.435 \ \mathrm{V}$$

Question 5

A circular metallic ring of radius $$r = 10.0 \ \mathrm{cm}$$ and resistance $$2.00 \ \Omega is $$placed in a magnetic field $$B(t) = 1.80 \ \mathrm{T} \ e$$^{-0.100 t}$$, perpendicular to the ring. What is the induced current at t = 1.00$$ \ \mathrm{s}$$?

Accepted Answer

$$I = 2.56 \ \mathrm{mA}$$

Question 6

A 2.0-cm-diameter loop of wire with resistance $$0.010 \ \Omega is $$placed at the center of a solenoid (4.0 cm diameter, 20 cm long, 1000 turns). If the solenoid current increases linearly from 0 to 10 A in 1 s, what is the induced current in the loop during this interval?

Accepted Answer

$$I = \frac{\mu_0$ N A}{L R} \frac{dI_{sol}}{dt} = \frac{(4\pi 	imes 10$$^{-7}$$) 	imes 1000 	imes 3.14 	imes 10$$^{-4}$$}{0.20 	imes 0.010} 	imes 10 = 0.197 \ \mathrm{A}$$

Question 7

A metal rod of length $$L = 0.10 \ \mathrm{m}$$ moves with velocity $$v = 2.5 \ \mathrm{m/s}$$ perpendicular to a magnetic field $$B = 0.60 \ \mathrm{T}. If $$the total resistance of the loop is $$0.030 \ \Omega$$, what is the magnitude of the induced current?

Accepted Answer

$$I = \frac{L v B}{R} = \frac{0.10 	imes 2.5 	imes 0.60}{0.030} = 5.0 \ \mathrm{A}$$

Question 8

Which of the following best describes Lenz’s law?

Accepted Answer

The induced current in a circuit always flows in the direction that opposes the change in magnetic flux.

Question 9

A long solenoid has 500 turns per meter and a cross-sectional area of $$4.0 \ \mathrm{cm}^2. If $$the current in its windings is increasing at $$100 \ \mathrm{A/s}$$, what is the magnitude of the induced emf in a wire loop outside the solenoid?

Accepted Answer

$$\mathcal{E} = \mu_0$ n A \frac{dI}{dt} = (4\pi 	imes 10$$^{-7}$$) 	imes 500 	imes 4.0 	imes 10$$^{-4}$$ 	imes 100 = 25 \ \mu\mathrm{V}$$

Question 10

The energy stored in an inductor with inductance $$L$$ and current $$I is $$given by which of the following expressions?

Accepted Answer

$$U_L = \frac{1}{2} L I^2$

Question 11

A 1.0 mH inductor has a current that drops from $$1.0 \ \mathrm{A} to$$ $$0 in$$ $$5.0 \ \mathrm{ms}. $$What is the induced voltage across the inductor?

Accepted Answer

$$\Delta V_L = -L \frac{\Delta I}{\Delta t} = -0.0010 	imes \frac{-1.0}{5.0 	imes 10$$^{-6}$$} = 2000 \ \mathrm{V}$$

Question 12

In an LC circuit, the charge on the capacitor as a function of time is given by $$Q(t) = Q_0$ \cos(\omega t). $$What is the angular frequency $$\omega of $$the oscillations?

Accepted Answer

$$\omega = \frac{1}{\sqrt{LC}}$$

Question 13

A 1.0 mH inductor is to be used in an LC oscillator with a frequency of $$920 \ \mathrm{kHz}. $$What value of capacitance should be used?

Accepted Answer

$$C = \frac{1}{\$$omega^2$$ L} = 3.0 	imes 10$$^{-11}$$ \ \mathrm{F}$$

Question 14

In an LR circuit, the current decays according to $$I = I_0$ e$$^{-t/	au}$$ after the switch is opened. What is the time constant $$	au$$?

Accepted Answer

$$	au = \frac{L}{R}$$

Question 15

A switch in an LR circuit is moved to disconnect the battery at $$t = 0. If$$ $$L = 2.0 \ \mathrm{mH}$$ and $$R = 200 \ \Omega$$, how long does it take for the current to decay to $$1\% of $$its initial value?

Accepted Answer

$$t = -	au \ln(0.01) = -10.0 \ \mu\mathrm{s} \ln(0.01) = 46 \ \mu\mathrm{s}$$

Electromagnetic Induction and Inductance: Study Notes

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