The magnetic field B, at all points within a circular region...

The magnetic field $\overrightarrow{B}$, at all points within a circular region of radius $R$, is uniform in space and directed into the plane of the page as shown in $\textbf{Fig. P29.63}$. (The region could be a cross section inside the windings of a long, straight solenoid.) If the magnetic field is increasing at a rate $dB/dt$, what are the magnitude and direction of the force on a stationary positive point charge $q$ located at points $a$, $b$, and $c$? (Point a is a distance r above the center of the region, point $b$ is a distance $r$ to the right of the center, and point $c$ is at the center of the region.)

Transcript

00:01 Okay, so we're doing chapter 29 problem 63.

00:07 So in this problem we have some points here separated by distance r, so here's a point a, c, and b.

00:14 We have c is at the origin here, and a is along the y -axis, and b is along the x -axis.

00:23 Okay, so this is all in a circle of magnetic field of radius big r.

00:30 So this whole thing in the xy plane has a magnetic field pointing into the page.

00:43 Okay.

00:47 So it also tells us that the time derivative of the magnetic field is greater than zero, which means we're increasing, which means we also have an increasing flux with respect to time.

01:01 Okay.

01:03 So using kausen's law, we can know that the surface align integral, e .0.

01:12 Is going to be the negative change in flux with respect to time.

01:23 So from that, we now know that e times the circumference in a gaussian surface equals a times b b t.

01:41 Oops, sorry, a times d b d t.

01:47 In the area of that gaussian shell is pi r squared because it's a circle at that radius.

01:52 So we now left with e times 2 pi r equals pi r squared d b d t.

02:00 We arrange that.

02:02 We can find out the magnitude of our electric field as one over one half r db d t.

02:14 Cool.

02:15 So the only reason we were able to do this is because we know that the electric field is exactly always tangential to our galsing surface.

02:25 So what that is saying is this is our origin.

02:29 Here's our gaussian surface at radius r.

02:32 The electric field is always tangential here...

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