Electromagnetism: Axle rolling on train rails

[dmayers94]

Imagine a set of train tracks. Now there is one bridge (a resistor) connecting the two rails in one spot. On a different spot, there is rolling axle that acts as another bridge and it makes good electrical connection with the rails creating a full rectangular circuit. This axle rolls at a constant velocity. The only significant resistance in the circuit is from the resistor with resistance R. There is a uniform magnetic field, B, directed vertically downward and perpendicular to the horizontal rails. Assuming there is no friction on the rails, what horizontal force is necessary to keep the axle moving at a constant speed? Now I know the answer is F = B^2 * length of axle squared * velocity / resistance, but how do I find this? What force acts to slow down the axle in the first place since there is no friction? Thanks for the help.

 

[TSny]

What force acts to slow down the axle in the first place since there is no friction? Thanks for the help.

Hello.
Fill in the blank: The B-field could exert a force on the axle if the axle carries a ______________.

 

[dmayers94]

moving charge?

 

[TSny]

moving charge?

OK. Moving charge in a conductor is a current. Can you see any way that a current would be generated in the axle?

 

[dmayers94]

I think the current comes from the induced voltage from the changing area of the loop.

 

[TSny]

I think the current comes from the induced voltage from the changing area of the loop.

Yes. Maybe you've already seen an example worked out in your text or in class where a rod is slid along parallel rails in a B-field to generate current?

 

[dmayers94]

Nope, I don't recall that. I'm still confused about the quantitative parts of this problem. I found the induced voltage and I found the current. I just don't know how to find the force on the moving axle.

 

[TSny]

I found the induced voltage and I found the current. I just don't know how to find the force on the moving axle.

Good. Almost done. Have you learned how to calculate the magnetic force on a straight, current-carrying wire sitting in a magnetic field?

 

[technician]

What did you get for the induced emf and resulting current?

 

[dmayers94]

induced voltage = -cos(angle between area vector and B vector) * B * dA/dt. The angle is 0 because they are parallel. dA/dt = distance between rails * speed of axle so voltage = Bvd. Just divide the voltage by the resistance to get the current.

 

[dmayers94]

Force on a current carrying wire: F = iL x B where i is current, L is the length of the wire, and B is the magnitude of the magnetic field. To get the direction you must do a cross product. The L vector is in the direction of the current and the B vector is downward, so you ultimately get a force antiparallel to the velocity of the axle.

 

[dmayers94]

I want to get clarification on Lenz's Law while I'm doing a problem like this. The magnetic flux decreases because of the shrinking area, so the induced current must have a direction such that its magnetic field is directed downwards, right? So for a square loop, that means that the current is going clockwise, correct?

 

[TSny]

I want to get clarification on Lenz's Law while I'm doing a problem like this. The magnetic flux decreases because of the shrinking area, so the induced current must have a direction such that its magnetic field is directed downwards, right?

Yes.

So for a square loop, that means that the current is going clockwise, correct?

Not sure what clockwise means. Looking from above? From below?

Anyway see if this helps: http://capone.mtsu.edu/phys2020/Lectures/L12-L18/L17/Current_Loops/current_loops.html

Square loops are similar to circular loops.