In Fig. 32-30, a uniform electric field is directed out of...

In Fig. 32-30, a uniform electric field is directed out of the page within a circular region of radius $R=3.00$ $\mathrm{cm} .$ The field magnitude is given by $E=\left(4.50 \times 10^{-3} \mathrm{~V} / \mathrm{m} \cdot \mathrm{s}\right) t$, where $t$ is in seconds. What is the magnitude of the induced magnetic field at radial distances (a) $2.00 \mathrm{~cm}$ and (b) $5.00 \mathrm{~cm}$ ?

Transcript

00:01 All right, so let's say we have a circular region with a radius of three centimeters.

00:07 And within it, we have an electric field that is given by 4 .5 millivolts per second times t.

00:18 So it's a changing electric field.

00:20 And we want to know what's the magnitude of the induced magnetic field at a distance of r.

00:25 Little r equals two centimeters and then five centimeters.

00:30 So let's do the first part.

00:32 So let's first compute the electric flux of this because the magnetic field is going to be related to this.

00:39 So electric flux is going to be the integral of e over the area of this region.

00:44 But the area conveniently enough is just pi r squared.

00:47 So this will be 4 .5 millivolt per second.

00:54 Or sorry, millivolt per meter per second.

00:58 I should have written that per meter per second.

01:03 So and then times pi r squared, which is 0 .03 meters squared.

01:11 And that's the total flux in this region.

01:19 Although if we're looking at a region that is smaller than this, we want to use a smaller radius.

01:24 So we'll write this as, we'll just write this as pi little r squared for the moment.

01:29 The electric flux enclosed through some surface or some region.

01:34 Of radius r is going to be this all right and our magnetic field according to ampere's law our magnetic field around this region integrated around this region so the magnetic field times the circumference of the loop it's going to equal mu not times the current enclosed and the current enclosed is going to be epsilon not times the change or the time derivative of the flux so this is one over the speed of light squared times the time derivative of the flux and so we can write this as 1 over c squared times 4 .5 millivolts per meter times a second times pi or sorry, this should have meant t here, times pi r squared.

02:19 All right...

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