Question

In a region of space, a magnetic field points in the +$x$-direction (toward the right). Its magnitude varies with position according to the formula $B_x = B_0 + bx$, where $B_0$ and $b$ are positive constants, for $x \geq$ 0. A flat coil of area $A$ moves with uniform speed $v$ from right to left with the plane of its area always perpendicular to this field. (a) What is the emf induced in this coil while it is to the right of the origin? (b) As viewed from the origin, what is the direction (clockwise or counterclockwise) of the current induced in the coil? (c) If instead the coil moved from left to right, what would be the answers to parts (a) and (b)?

   In a region of space, a magnetic field points in the +$x$-direction (toward the right). Its magnitude varies with position according to the formula $B_x = B_0 + bx$, where $B_0$ and $b$ are positive constants, for $x \geq$ 0. A flat coil of area $A$ moves with uniform speed $v$ from right to left with the plane of its area always perpendicular to this field. (a) What is the emf induced in this coil while it is to the right of the origin? (b) As viewed from the origin, what is the direction (clockwise or counterclockwise) of the current induced in the coil? (c) If instead the coil moved from left to right, what would be the answers to parts (a) and (b)?

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University Physics with Modern Physics

University Physics with Modern Physics

Hugh D. Young 14th Edition

Chapter 29, Problem 11

Instant Answer

Solved by Expert Bruce Edelman

06/17/2019

Step 1

The magnetic flux $\Phi_B$ is given by the product of the magnetic field $B$ and the area $A$ through which it passes. Since the magnetic field varies with position, we have $\Phi_B = A(B_0 + bx)$. Show more…

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In a region of space, a magnetic field points in the +$x$-direction (toward the right). Its magnitude varies with position according to the formula $B_x = B_0 + bx$, where $B_0$ and $b$ are positive constants, for $x \geq$ 0. A flat coil of area $A$ moves with uniform speed $v$ from right to left with the plane of its area always perpendicular to this field. (a) What is the emf induced in this coil while it is to the right of the origin? (b) As viewed from the origin, what is the direction (clockwise or counterclockwise) of the current induced in the coil? (c) If instead the coil moved from left to right, what would be the answers to parts (a) and (b)?

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Transcript

00:01 Okay, so we're doing in chapter 29 problem 11.

00:04 So this says in the region of space, a magnetic field points in the positive x direction.

00:10 So call this positive x -b field, and the magnitude is given as an initial magnitude b -not plus b times x, with b -not and b greater than 0 when x is greater than 0.

00:30 So let's call this our xy plane.

00:40 So our origins over here.

00:42 Cool.

00:43 So it also says a flat coil, so it equals one with an area.

00:50 It moves with uniform speed v from right to left with a plane that always put the indicator in the field.

00:56 So let's draw this here and v is moving this way towards the origin.

01:04 Cool.

01:05 So now it says part a.

01:07 What is the induced ems of this guy? so faraday's law of course says the induced emf is given as a negative change magnetic flux.

01:17 It's a function of time or a rate time.

01:20 So now we got to figure out what is the flux to figure this out.

01:26 So a flux is given as the magnitude of b times the cross -sectional area.

01:35 Okay.

01:36 A b is a function of x, right? so this flux is a flux.

01:39 Changes in a function of x.

01:40 So this is b not plus v times x times the area.

01:47 So now when we take the derivative, defy v d t is a b not that has no x dependence so that's just gone.

02:00 Okay, so then we have a times little b times d x d t but d x d t is just the velocity that this is changing, right? so this is a, b times v.

02:15 But v is in the negative direction.

02:18 So this...

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