A copper bar of length h and electric resistance R slides with negligible fricti

Question

A copper bar of length h and electric resistance R slides with negligible friction on metal rails that have negligible electric resistance (see figure below). The rails are connected on the right with a wire of negligible electric resistance, and a magnetic compass is placed under this wire (the diagram is a top view). The compass needle deflects to the right of north, as shown on the diagram.

Throughout this region there is a uniform magnetic field B pointing out of the page, produced by large coils that are not shown. This magnetic field is increasing with time, and the magnitude is

B = B0 + bt,

where B0 and b are constants, and t is the time in seconds. You slide the copper bar to the right and at time

t = 0

you release the bar when it is a distance x from the right end of the apparatus. At that instant the bar is moving to the right with a speed v.

(a) Calculate the magnitude of the initial current I in this circuit. (Use the following as necessary: b, x, B0, h, v, and R.)

Imag =

(b) Calculate the magnitude of the net force on the bar just after you release it. (Use the following as necessary: b, x, B0, h, v, and R.)

Fmag =

Explanation / Answer

(a) Here the emf induced in the bar will comprise of two parts viz induced emf due to changing magnetic field and the emf induced due to velocity of the bar

the area of the loop at instant the bar is released (A) = h*x

so the total flux passing through the loop at that instant () = B*h*x

Hence emf induced due to changing magnetic field = d/dt = h*x*dB/dt = h*x*b ( as B = B0 + bt) Moreover this induced emf will produce a clockwise current in the loop

Now the emf induced by the motion of bar (motional emf) = v*B*h = v*B0*h ( At t = 0, B = B0) Also this emf would produce an anti-clockwise current

Therefore magnitude of the initial current (I) = (| v*B0*h -  h*x*b |)/R

(b) Magnitude of net force acting on the bar = I*h*B0 = (| v*B0*h -  h*x*b |)*h*B0/R

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