A Ushaped piece of wire has negligible resistance A slide wi

A U-shaped piece of wire has negligible resistance A slide wire of length L, mass m, and resistance R is moving on it - there is no friction between them. No-one is pushing the rod - you just see that at this instant it has speed v There is a uniform magnetic field B out of the page. Is a magnetic force acting on the slide wire? If so, which way? If not, why not? At this instant, how much force is acting on the slide wire? (Show how you get this answer) If this force was constant, how far would the wire slide coming to rest? Do you think the wire will slide further, less, or the same as the distance that yc calculated above? Explain why. What hap pens to the kinetic energy of the rod as it slows to a stop?

Solution

For a rod moving along the right, the flux through the area it encloses with the right end will keep changing, hence there will be an EMF induced in the rod.

The flux at any point can be written as: BLx where L is the length of the rod and x is its distance from the left end.

Hence, the EMF = d(flux)/dt = BLdx/dt = BLv

This will result in a current flowing through the wire as: EMF/ Resistance = BLv/R

Now we know that the force acting on rod of length L and carrying a current I is given as: F= ILB

SO we get that the force acting on the given rod would be: vB²L²/R and direction would be towards the left.

The deceleration = Force / Mass = vB²L²/mR

We can write: vdv/dx = -vB²L²/mR

or, dv = -(B²L²/mR)dx

integrating from V to 0 we get:0 - V = -(B²L²/mR)x

or, X = mRV / B²L² is the distance it will travel before coming to a stop

The kinetic energy is dissipated as the heat energy in the resistor of the rod.