A rigid wire loop of square shape having side of length L and resistance R is moving along the x-axis with a constant velocity `v_(0)` in the plane of the paper. At `t =0`, the right edge of the loop enters a region of length 3L where there is a uniform magnetic field `B_(0)` into the plane of the paper, as shown in the figure. For sufficiently large `v_(0)` the loop eventually crosses the region. Let x be the location of the right edge of the loop. Let v(x), I(x) and F(x) represent the velocity of the loop, current in the loop, and force on the loop, respectively, as a function of x. Counter-clockwise current is taken as positive.
. Which of the following schematic plot(s) is (are) correct? (Ignore gravity)

Options

A.
B.
C.
D.

Answer

b,d

Explanation

Emf is induced in the loop only at the time of entering the field and at the time of exiting the field. In between there will be a time interval when no emf is induced in the loop and no current flows through it.
Refer part I of figure when loop has already entered the field by a distance x. Let speed of loop at this instant is u. Emf induced in the loop can be written as follows:
` e = BLv`
Current flowing through the loop can be written as follows:
` I = e/R rArr I = (BL)/R v ` ....(1)
Magnetic force acts opposite to velocity or along the negative X-axis, Magnetic force can be written as follows:
`F = - ILB rArr F = - (B^(2)L^(2))/R v` ....(2)
We can now apply Newton.s second law of motion as follows:
`- (B^(2)L^(2))/R v = m (dv)/(dt) rArr - (B^(2)L^(2))/R v = m (dv)/(dx) (dx)/(dt)`
`rArr - (B^(2)L^(2))/R v = m (dv)/(dx) v rArr (dv)/(dx) = - (B^(2)L^(2))/(mR)`
`rArr dv = - (B^(2)L^(2))/(mR) dx rArr underset(v_(0))overset(v) int dv = - (B^(2)L^(2))/(mR) underset(0)overset(x) int dx `
`rArr v - v_(0) = - (B^(2)L^(2))/(mR) x `
` rArr v = v_(0) - (B^(2)L^(2))/(mR) x` ....(3)
We can see that velocity is decreasing linearly with the distance. Above obtained equation is valid only for ` x = 0 " to " x = L `. Afterwards up to `x = 3L` velocity remains constant. Variation in velocity for the interval `x = 3L " to " x = 4L` will be similar to equation (3) although exact equation will be different. We can see that option (b) correctly describes variation of velocity with the distance .
We can see that direction of force acting on loop remains same (along negative X-axis) while entering and exiting the magnetic field, hence option (c) cannot be correct.
` I = (BL)/R (v_(0) - (B^(2)L^(2))/(mR) x)`
We can see that variation of current with the distance is linear. But current is counter clockwise (positive) while entering and clockwise (negative) while exiting the field. Hence option (a) cannot be correct but option (d) is correct.