Electron in circular motion in B field problem check

[nissanztt90]

Homework Statement

An electron of mass M_e, with charge -e, is in a circular orbit in the xy-plane. There is a uniform magnetic field B in the positive z direction. It is moving at constant velocity V.

Working in CGS units...

Find R in terms of M_e, -e, v, and B

Find the angular frequency $$\omega$$.

Find the current I with the given parameters.

Homework Equations

F= (2IqV)/(rc²)

B= (2I)/(rc)

And i believe F = mv²/r = mr$$\omega$$ ?

The Attempt at a Solution

My thought was to solve B= (2I)/(rc) for I, and substitute that quantity for the I in F= (2IqV)/(rc²) to get F = (B(-e)v)/c, and set that equal to the centripetal force mv²/r.

Through that subsitution i found r = M_evc / B(-e)

For the angular frequency i solved mv²/r = mr$$\omega$$ for r=v$$\omega$$, substituded that into the equation for radius, and solved for $$\omega$$ = B(-e) / M_ec

For the current, i used I = Brc/2, and substituted the value i found for r into this equation, for I = M_evc² / -e

I am somewhat confused, because i thought both the force on the moving electron and the centripetal force pointed in. Any guidance and/or clarification would be greatly appreciated.

 

[rl.bhat]

Centripetal force mv^2/r = mω^2/r

 

[nissanztt90]

Whoops. Actually its F = mv2/r = mrw^2, though r = vw is still correct.

 

[rl.bhat]

Yes.
And there is only one force is acting on the election. The same force can be expressed in two different ways.

 

[nissanztt90]

So the magnetic field is the cause of the centripetal force?

 

[ideasrule]

Yes, it is.

 

[nissanztt90]

Great, thanks! So my analysis is correct?

 

[ideasrule]

Maybe I'm being dense today, but I don't understand the analysis. Where did you get F= (2IqV)/(rc^2) and B= (2I)/(rc)? What does the "c" mean? The "standard" equation for F is qvB. This, along with F=mv^2/r and mw^2r, should be enough to answer the first and second questions. As for the third question, what current is it asking for? Is it the current produced by the lone electron going around in circles?

 

[nissanztt90]

The F equation i used is for a test charge, in CGS units. "c" stands for the speed of light in this case. The B equation is the magnetic field this moving test charge produces.

So far as i can understand, u₀, the permeability of free space, = (4*pi) / c, if that helps. The books I am following tends to work in CGS units, and not SI.

The test charge, which is an electron in this case, would be moving in the +x direction, and would experience the F i mentioned, in the -y direction, with the B i also mentioned, in the +z direction. This is straight from the textbook "Electricity and Magnetism" by Edward M Purcell.

Im not 100% certain on which current they are asking for, though i am just about 99% certain its the current produced by the lone electron going in circles, since a charge needs to be moving to create a current.