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Electricity and Magnetism

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Faraday--Voltage in a Loop with Changing Area and Magnetic Field

EANDM-YJMUIR

A circular loop of conducting wire with radius $r_o$ is in a plane oriented perpendicular to the direction of a uniform constant magnetic field $B_o$. As shown in the accompanying figure, placed on this wire loop are two conducting sliders.

One slider is fixed and the other is rotating counter-clockwise with an angular frequency of $\omega$ rad/sec. At the time $t=t_o$ the acute angle between the two sliders is $\theta_0$ and the magnetic field is $B_o$. For times $t \geq t_o$, the magnetic field is changing according to:

$$B=B_o \bigg( \cfrac{t}{t_o}\bigg)$$

What is the magnitude of the voltage for times $t \geq t_o$ measured between points 'a' and 'b'? (assume points 'a' and 'b' are very close together at the center of the loop circle)

Created for Albert.io. Copyright 2016. All rights reserved.

A

$V(t) = \cfrac{B_o r_o ^2 \omega t}{2 \theta_o }$

B

$V(t) = \cfrac{B_o r_o ^2}{2 t_o}\bigg[ \theta_o + \omega t_o \bigg]$

C

$V(t) = \cfrac{B_o r_o ^2 \theta_o }{2 \omega t} $

D

$V(t) = \cfrac{B_o r_o ^2 \theta_o }{2 t_o}$