Chapter 30: Electromagnetic Induction

The square loop shown in$FIGUREP30.53$ moves into a $0.80\mathrm{T}\mathrm{ALC}$magnetic field at a constant speed of$10m/s$.The loop has a resistance of $0.10\mathrm{\Omega }$, and it enters the field at $t=0\mathrm{s}.$

a. Find the induced current in the loop as a function of time. Give your answer as a graph of ii versus $t$from $t=0\mathrm{s}$to $t=0.020\mathrm{s}.$

b. What is the maximum current? What is the position of the loop when the current is maximum?

The L-shaped conductor in $FIGUREP30.54$ moves at $10m/s$across and touches a stationary $L$-shaped conductor in a $0.10T$magnetic field. The two vertices overlap, so that the enclosed area is zero, at $t=0s$. The conductor has a resistance of $0.010$ohms per meter.

a. What is the direction of the induced current?

b. Find expressions for the induced emf and the induced current as functions of time.

c. Evaluate$\in$and$l$at$t=0.10s$.

I A 20-cm-long, zero-resistance slide wire moves outward, on zero-resistance rails, at a steady speed of $10\mathrm{m}/\mathrm{s}$in a $0.10\mathrm{T}$magnetic field. (See Figure 30.26.) On the opposite side, a $1.0\Omega$carbon resistor completes the circuit by connecting the two rails. The mass of the resistor is $50\mathrm{mg}$.

a. What is the induced current in the circuit?

b. How much force is needed to pull the wire at this speed?

c. If the wire is pulled for $10\mathrm{s}$, what is the temperature increase of the carbon? The specific heat of carbon is$710\mathrm{J}/\mathrm{kgK}$.

Your camping buddy has an idea for a light to go inside your tent. He happens to have a powerful and heavy horseshoe magnet that he bought at a surplus store. This magnet creates a $0.20T$field between two pole tips $10cm$apart. His idea is to build the hand-cranked generator shown in FIGURE .He thinks you can make enough current to fully light a $1.0\Omega$lightbulb rated at $4.0W$. That’s not super bright, but it should be plenty of light for routine activities in the tent.

a. Find an expression for the induced current as a function of time if you turn the crank at frequency $f$. Assume that the semicircle is at its highest point at $t=0s$.

b. With what frequency will you have to turn the crank for the maximum current to fully light the bulb? Is this feasible?

56. II Your camping buddy has an idea for a light to go inside your CALC tent. He happens to have a powerful (and heavy!) horseshoe magnet that he bought at a surplus store. This magnet creates a$0.20$T field between two pole tips $10\mathrm{cm}$apart. His idea is to build the hand-cranked generator shown in FIGURE P30.56. He thinks you can make enough current to fully light a$1.0\Omega$lightbulb rated at $4.0\mathrm{W}$. That's not super bright, but it should be plenty of light for routine activities in the tent.

a. Find an expression for the induced current as a function of time if you turn the crank at frequency f Assume that the semicircle is at its highest point at $t=0\mathrm{s}$.

b. With what frequency will you have to turn the crank tor the maximum current to fully light the bulb? Is this feasible?

The 10-cm-wide, zero-resistance slide wire shown in FIGURE is pushed toward the $2.0\Omega$resistor at a steady speed of $0.50m/s$. The magnetic field strength is $0.50T$.

a. How big is the pushing force?

b. How much power does the pushing force supply to the wire?

c. What are the direction and magnitude of the induced current?

d. How much power is dissipated in the resistor?

You’ve decided to make the magnetic projectile launcher shown in FIGURE for your science project. An aluminum bar of length $l$slides along metal rails through a magnetic field$B.$The switch closes at $t=0s$, while the bar is at rest, and a battery of emf ${\mathcal{E}}_{\text{bat}}$starts a current flowing around the loop. The battery has internal resistance r. The resistances of the rails and the bar are effectively zero.

a. Show that the bar reaches a terminal speed $V_{\text{term}}$, and find an expression for $V_{\text{term}}$.

b. Evaluate ${\mathcal{E}}_{\text{bat}}$for$V_{\text{term}}$$=1.0V,r=0.10\Omega ,l=6.0cm$and$B=0.50T.$

FIGURE shows a U-shaped conducting rail that is oriented vertically in a horizontal magnetic field. The rail has no electric resistance and does not move. A slide wire with mass $m$and resistance $R$can slide up and down without friction while maintaining electrical contact with the rail. The slide wire is released from rest.

a. Show that the slide wire reaches a terminal speed $v_{\text{term}}$, and find an expression for$v_{\text{term}}$ .

b. Determine the value of $v_{\text{term}}$if $l=20\mathrm{cm},m=10\mathrm{g},R=$$0.10\Omega$and $B=0.50\mathrm{T}$

An equilateral triangle $8.0\mathrm{cm}$on a side is in a $5.0\mathrm{mT}$ uniform magnetic field. The magnetic flux through the triangle is $6.0\mu \mathrm{Wb}$. What is the angle between the magnetic field and an axis perpendicular to the plane of the triangle?

FIGURE shows a bar magnet being pushed toward a conducting loop from below, along the axis of the loop.

a. What is the current direction in the loop? Explain.

b. Is there a magnetic force on the loop? If so, in which direction? Explain.

Hint: A current loop is a magnetic dipole.

c. Is there a force on the magnet? If so, in which direction?