A wide, long, insulating belt has a uniform positive charge...

A wide, long, insulating belt has a uniform positive charge per unit area $\sigma$ on its upper surface. Rollers at each end move the belt to the right at a constant speed $v .$ Calculate the magnitude and direction of the magnetic field produced by the moving belt at a point just above its surface. (Hint: At points near the surface and far from its edges or ends, the moving belt can be considered to be an infinite current sheet like that in Problem $28.69 .$

Key Concepts

Uniform Charge Distribution

A uniform charge distribution means that the charge per unit area (or per unit volume in other contexts) is constant over the entire surface (or volume) under consideration. This concept simplifies the analysis of fields since every element of the surface contributes equally, allowing the use of symmetry in deriving the behavior of electric or magnetic fields.

Surface Current Density

When a charged object moves, it gives rise to a current. Surface current density represents the current per unit length (or per unit area) on a surface and is typically calculated as the product of the surface charge density and the velocity of the moving charges. This concept is key in bridging the electric charge distribution to the magnetic effects observed due to motion.

Magnetic Field of an Infinite Current Sheet

In magnetostatics, an infinite current sheet is an idealized system where a uniform current flows over an infinitely large plane. The symmetry of the setup allows one to analytically derive the magnetic field on either side of the sheet, leading to a constant magnitude of the magnetic field that is independent of the distance from the sheet in the near-field region. This approximation is useful for understanding the magnetic fields generated by large, flat current-carrying surfaces.

Ampere's Law

Ampere's Law is a fundamental principle in electromagnetism that relates the integrated magnetic field around a closed path to the total current passing through the path. This law is particularly powerful in systems with high symmetry, such as the infinite current sheet, where it can be used to deduce the magnitude and direction of the magnetic field produced by the moving charges.

Transcript

00:01 Okay, we're looking at question twenty eight point eighty, where we have a y long insulating belt with a uniform charge per unit area and this whole built is moving to the right at constant speed.

00:13 Be so and we want to find the magnetic field.

00:16 So we're given a hint here that the moving belt can be considered to be an infinite current sheet.

00:23 So looking at the formula for the magnetic field from an infinite current sheet, that's this one here.

00:29 So we know we're going to be wanting to use this formula eventually.

00:34 But let's think about it.

00:35 I mean, if you look at this formula, we don't have the value for pi the current.

00:39 So we need to break this down a little bit.

00:43 So think about the amount of of charge from one little length delta x on the belt.

00:51 So don't accept this small amount of length built.

00:54 Still to cue is the amount of charge on that length, and we can express the amount of charge in terms of this length and then the total length of the belt and then thie charged density, which is a signal here, so ah, this gives us the amount of charge.

01:14 Remember, we want the current.

01:15 So when we define current ah, current is really just be amount of charge moving per unit time so we can define i still take you over delta t lt is time.

01:32 So looking at chris, we just found our expression for delta q so let's plug that in here, and this might not be helpful right away.

01:43 But if you just rearrange this a tiny rim of the signal over here, notice that we have this term delta x divided by delta t and this is really just velocity.

01:55 Uh, this is the definition of velocity right here on velocity is something that we're given in the court questions...

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