ssm A charge of 4.0 ×10^-6 C is placed on a small conducting sphere that is located at the end o

Ssm A charge of 4.0 ×10^-6 C is placed on a small conducting...

Key Concepts

Rotating Charge as a Current Loop

When a charge rotates in a circular path, its motion is equivalent to a current loop. The effective current can be determined by dividing the charge by the period of rotation. This fundamental concept bridges the gap between moving charges and the generation of magnetic fields, which is central to electromagnetism.

Magnetic Dipole Moment

The magnetic dipole moment of a current loop quantifies the strength and direction of its magnetic field. It is calculated by multiplying the current by the area of the loop. This concept is crucial in understanding how magnetic fields are produced by moving charges and is applicable in various settings, from atomic to macroscopic scales.

Period and Angular Velocity Relationship

The period of rotation is the time taken for one complete revolution and is inversely related to the angular velocity. Specifically, the period is given by the relation T = 2?/?. This relationship is fundamental in linking rotational motion with time-dependent phenomena in physics, including the determination of the effective current generated by a moving charge.

Transcript

00:01 This problem asks us to figure out the magnetic moment of a small conducting charged sphere that is being rotated around with an axle.

00:10 So let's just get an idea of what's going on in this problem.

00:12 So we have this charged sphere here.

00:15 It's going to be connected to an insulating rod, and then this rod is going to be rotating around with a given angular velocity.

00:25 Let's say it rotates around like this way, and we have an angular velocity that's given as 150.

00:32 So this charged sphere, this sphere that has charged q is basically going to be mapping out a circular path around this of a radius that is the length of the rod, right? now if we remember the definition of the magnetic moment, it's going to be derived is if we have a current going through a loop like so, and it could be a coil or such with many turns like this.

00:59 The magnetic moment of such a thing, let me get rid of those, magnetic moment of such a thing is given by how many turns multiplied by that current that's going through and then multiplied by the surface area of the path that this, the cross -sectional path that the current is mapping out.

01:14 So it could actually be something even more complicated like this, right, and the current's going around in that direction, but whatever this cross -sectional area would be, that would be the a we put in the magnetic moment.

01:23 But it's much easier in this problem, right? we see that the type of path that this charged sphere is mapping out, is just a circle.

01:31 So the cross -sectional area is going to be given by pi r squared or pi l squared, l being the length of this rod here.

01:39 Now, when we have to think about a magnetic moment, right, we want to have a current.

01:45 But we're only given the charge of this thing.

01:47 So how the heck do we turn that into a current? well, if you think about it, this charge of q0 is going around in a circle for every single period, right? so this charge is actually going around per unit time.

02:04 So that actually gives us a current, right? the definition of current is how much charge goes by per unit time, right? so because we know how fast this is rotating at a constant velocity, we know how long it takes to go around one time.

02:18 And we know how much charge goes around one time.

02:20 It's the charge placed on this small conducting sphere.

02:23 So if we want to figure out the current going through this loop here we just take the charge that goes through which is q0 divided by how much how long it takes going on once and that is a period right and we know for something rotating at an omega rotating at omega the period the time it takes to go around once is given by well how many radians are in a whole circle two pi and then we divide that by omega so we see that that gives us how long it takes to go around once so what does the expression for our magnetic moment look like? let's write that out.

03:05 Down here, magnetic moment is the number of currents.

03:08 It's just one, so we'll leave that out...

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