The drawing shows a parallel plate capacitor that is moving...

The drawing shows a parallel plate capacitor that is moving with a speed of 32 $\mathrm{m} / \mathrm{s}$ through a $3.6-\mathrm{T}$ magnetic field. The velocity $\overrightarrow{\mathbf{v}}$ is perpendicular to the magnetic field. The electric field within the capacitor has a value of $170 \mathrm{N} / \mathrm{C},$ and each plate has an area of $7.5 \times 10^{-4} \mathrm{m}^{2}$ . What is the magnetic force (magnitude and direction) exerted on the positive plate of the capacitor?

Key Concepts

Relationship between Electric Field and Surface Charge Density in a Capacitor

In a parallel plate capacitor, the electric field between the plates is directly related to the surface charge density (?) by the relationship E = ?/??, where ?? is the vacuum permittivity. This relation can be used to determine the total charge on a plate once the electric field and the plate area are known, since the surface charge density is ? = ??E and the charge Q = ?A. This concept is essential when calculating the net magnetic force on the plate due to the motion of its accumulated charge.

Lorentz Force Law

This concept states that a charged particle moving in a magnetic field experiences a force that is perpendicular to both its velocity and the magnetic field. The mathematical expression for this force is F = q(v × B), where q is the charge, v is the velocity, and B is the magnetic field. In the context of the moving capacitor, each moving charge on the positive plate will feel this magnetic force, and the net force on the plate is determined by the total charge multiplied by the cross product of the velocity and magnetic field vectors.

Transcript

00:01 Hello.

00:02 So in this problem, we have an interesting situation with a parallel plate capacitor traveling through a magnetic field.

00:10 And what we're asked to find is the overall magnetic force that is acting on one particular side of the parallel plate capacitor in this case the positive side.

00:22 Now, it's good to get a visualization of what's happening here and to think about what things we want to calculate and some things we had to think about from previous chapters in order to figure out the the answer.

00:32 So let's just do a quick sketch of what's happening here.

00:35 We have magnetic fields headed this way.

00:38 Overall, so directly to the right will represent those like this.

00:44 Traveling within that field, we have a parallel plate capacitor.

00:51 Positive charges will say on this left side, negative charges here, and then therefore there's an electric field inside the parallel plate capacitor headed to the right.

01:03 Now overall, the velocity or the direction with which the the parallel plate capacitor is traveling is straight downward.

01:10 So it's heading this way.

01:12 Now, just from this, we can answer the first part of the question, which is, well, we want the total magnetic force, but remember, forces have two pieces of information that help define them, the magnitude and the direction, because they're vectors.

01:24 And we can already answer the direction of the magnetic force here on the positive side of this parallel plate capacitor.

01:32 So how do we do that with the right hand rule? well, what we can do is put our pointer finger along the direction of our velocity, which would have our pointer finger facing this way.

01:42 Then we can put our middle finger along the direction of the magnetic field, and that would be this way.

01:48 And if we do that, we see that our thumb is pointing out of the screen.

01:54 And therefore, the magnetic force direction is out of the screen.

02:04 So that is one part of the puzzle.

02:09 Now, we have to figure out the magnitude of this force.

02:12 Well, what is the magnitude of the magnetic force? it's going to be given by the total charge of this parallel plate capacitor on the positive side.

02:21 So the total charge of the plate, i'll write it like this.

02:24 And of course it's going to be multiplied by the speed of the plate, the magnet to the magnetic field through which the plate is traveling, and the sign of the angle between the velocity and magnetic field vectors...

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