Chapter 29

A wedding ring (of diameter $2.0\mathrm{cm}$ ) is tossed into the air and given a spin, resulting in an angular velocity of 17 rotations per second. The rotation axis is a diameter of the ring. Taking the magnitude of the Earth's magnetic field to be $4.0\cdot {10}^{-5}\mathrm{T}$, what is the maximum induced potential difference in the ring?

What is the inductance in a series $\mathrm{RL}$ circuit in which $R=3.00\mathrm{k}\mathrm{\Omega }$ if the current increases to one half of its final value in $20.0\mu \mathrm{s}?$

A $100.-V$ battery is connected in series with a $500.-\mathrm{\Omega }$ resistor. According to Faraday's Law of Induction, current can never change instantaneously, so there is always some "stray" inductance. Suppose the stray inductance is $0.200\mu \mathrm{H}$. How long will it take the current to build up to within $0.500\mathrm{\%}$ of its final value of $0.200\mathrm{A}$ after the resistor is con- nected to the battery?

A long solenoid with length $3.0\mathrm{m}$ and $n=290$ turns $/\mathrm{m}$ carries a current of $3.0\mathrm{A}.$ It stores $2.8\mathrm{J}$ of energy. What is the cross-sectional area of the solenoid?

A rectangular conducting loop with dimensions $a$ and $b$ and resistance $R$, is placed in the $xy$ -plane. A magnetic field of magnitude $B$ passes through the loop. The magnetic field is in the positive $z$ -direction and varies in time according to $B=B_0(1+c_1{t}^3),$ where $c_1$ is a constant with units of $1/{\mathrm{s}}^3$ What is the direction of the current induced in the loop, and what is its value at $t=1\mathrm{s}$ (in terms of $a,b,R,B_0,$ and $c_1)?$

A circuit contains a 12.0 -V battery, a switch, and a light bulb connected in series. When the light bulb has a current of 0.100 A flowing in it, it just starts to glow. This bulb draws $2.00\mathrm{W}$ when the switch has been closed for a long time. The switch is opened, and an inductor is added to the circuit, in series with the bulb. If the light bulb begins to glow $3.50\mathrm{ms}$ after the switch is closed again, what is the magnitude of the inductance? Ignore any time to heat the filament, and assume that you are able to observe a glow as soon as the current in the filament reaches the 0.100 - A threshold.

A circular loop of area $A$ is placed perpendicular to a time-varying magnetic field of $B(t)=B_0+at+b{t}^2,$ where $B_0,a,$ and $b$ are constants. a) What is the magnetic flux through the loop at $t=0?$ b) Derive an equation for the induced potential difference in the loop as a function of time. c) What is the magnitude and the direction of the induced current if the resistance of the loop is $R?$

A rectangular wire loop (dimensions of $h=15.0\mathrm{cm}$ and $w=8.00\mathrm{cm}$ ) with resistance $R=5.00\mathrm{\Omega }$ is mounted on a door. The Earth's magnetic field, $B_{\mathrm{E}}=2.6\cdot {10}^{-5}\mathrm{T}$, is uniform and perpendicular to the surface of the closed door (the surface is in the $xz$ -plane). At time $t=0,$ the door is opened (right edge moves toward the $y$ -axis) at a constant rate, with an opening angle of $\theta (t)=\omega t,$ where $\omega =3.5\mathrm{rad}/\mathrm{s}$ Calculate the direction and the magnitude of the current induced in the loop, $i(t=0.200\mathrm{s})$.

A steel cylinder with radius $2.5\mathrm{cm}$ and length $10.0\mathrm{cm}$ rolls without slipping down a ramp that is inclined at ${15}^{\circ }$ above the horizontal and has a length (along the ramp) of $3.0\mathrm{m}.$ What is the induced potential difference between the ends of the cylinder at the bottom of the ramp, if the surface of the ramp points along magnetic north?