# Torque on a Disk in a Magnetic Field (E&M)

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1. Oct 19, 2015

### Anode

Hi, I've been struggling on this homework problem for quite a few hours now and would really appreciate a pointing in the right direction. It's a fairly entertaining problem that I'm getting stuck on in several places.

1. The problem statement, all variables and given/known data

The problem goes:
"An erratic engineer, who lives in a residence infested by small, fast bugs, conceives the device shown as a method of driving a disk. The bugs are taught to climb up the axle of the disk, emerge through a hole O at its center, and run radially outward at constant velocity. At the edge, the bugs fall off and drop into a tray. By means of a seat emf not shown, each bug acquires an electric positive charge as it ascends, and carries the charge with it as it runs across the disk. A magnetic field of flux density varying with radius according to B= 0.1r2 acts upward over the disk.

Suppose that 1000 bugs/sec emerge from O, each carrying charge +0.001 Coulomb, and move out radially. The radius of the disk is 0.5 meters.

a) Find the torque on the disk.

The given answer is 0.00156 n-m.

2. Relevant equations
τ=rFsinθ
F=q(V x B)
τ=IBAsinθ
F=I(ℓ x B)

3. The attempt at a solution
So, I've tried this several ways.
Here's the one I think is in the right direction:

I used both the equation for torque and the one for magnetic force.
So I get:
τ = r * qVB (since θ=90 by right hand rule)
τ = r * qV ∫ 0.1 r2 dr (and integrate from 0 to 0.5m)
τ = r * q * V * (0.5)3/3
τ = 0.5m * q * V * (0.5)3/3

Then, I have a problem. I'm fairly sure that q = 0.001C x 1000, but I could be wrong.
In any case, I'm stuck with V, and as far as I can see there's no way to find the velocity, since I'm given neither V nor time of travel of the bugs.

Am I missing something obvious? Thanks!

**diagram attached**

2. Oct 19, 2015

### Anode

Nevermind. I solved it.
Was missing the fact that current = 1000 bugs/sec x 0.001 C/bug

Using that I was able to use dF=IBdr and plug that into dT=rdF.