Solving Torque on a Rotating Loop in a Magnetic Field

[prodigy180]

Hello. I was studying for intro physics on topic of electromagnetism and stumbled upon one problem.

Q. A thin wire of lengh L is made of an insulating material. The wire is bent to form a circular loop, and a positive charge q is distributed uniformly around the circumference of the loop. The loop is then set into rotation with angular speed w around an axis perpendicular to the plane of the loop and passing through its center. If the loop is in a region where there is a uniform magnetic field B directed parallel to the plane of the loop, calculate the magnitude of the magnetic torque on the loop.

--> I drew the diagram and I have the equation to find the torque handy but cannot relate angular velocity w to the equation.

torque t = IBA sin a,
where I= current, B= magnetic field, A=sectional area


Also, what is the difference between a conducting loop with charge q distributed over the loop that is rotating and a stationary loop with current flowing around the loop? (similarly, I cannot figure out how to relate angular velocity w to current I)

Thanks for your immediate help and insight!

K.Kim

 

[doriang101]

Well I'm a little confused. You say that it's an "insulating material". If it's an insulating material there shouldn't be any flow of charge, hence the current would be 0 so the torque would be 0. Unless I'm missing something?

 

[Gza]

To clear up the confusion doriang, you are correct in your statement that there is no charge flowing through the wire (no current). Yet there still is a magnetic dipole caused by the ring due to the fact that charge is rotating on the dialectric ring thus emulating a "real" current.

I hope this helps answer your later question Prodigy. There really is no difference in regards to the problem at hand between the charge being stationary relative to the disk, while the disk rotates, and charge moving through to wire. To get to your other question of relating w to the magnetic dipole you are using to calculate the torque, recall the definition of current:

$$I = \frac {\Delta Q} {\Delta t}$$

This represents an amount of charge Q passing through a cross-sectional area in time delta t. For your problem just make the Q the total Q on the disk given by the problem, and the delta t is simply the amount of time it takes for one full revolution of the disk (the period of the disk (T)), since the total amount of charge will pass through any given cross sectional area in the time it takes for a revolution. To find this time, just use $$\frac {2\pi } {\omega}$$ (i'm assuming you know how to derive this; just start from the definition of angular velocity as a ratio of angular displacement to time, and do some integration.)

 

[prodigy180]

Thanks and answer check

Thank you for your help guys!

For those who might have tried solving the problem above(no numerical figures involved), here is what I got...

torque t=qvRB/2 or t=qw(R)^2B/2
where q=total charge, v=velocity, R=radius, B=mag. field, w=angular velocity

Please leave answers if you have tried...I want to check if I solved it correctly.

Thanks again.

K.Kim