A rectangular conducting loop of width omega and height h is mounted on a non-co

Question

A rectangular conducting loop of width omega and height h is mounted on a non-conducting cart as shown in the figure. The cart is placed on the inclined track at position P_1, which is at height y_0 and is released from rest. It rolls with negligible friction down the ramp and through the area in the dashed box, where a magnetic field B points out of the page. a. What will the speed of the cart be as it exits the ramp? b. How much time will the cart take to pass through the right most side of the area with the magnetic field inside of it? c. What is the induced emf in the coil as it leaves the magnetic field? d. What direction will the current flow around the loop?

Explanation / Answer

a) Let the mass of the cart be 'm'.

PE of the cart is converted to its KE as it rolls down the bottom of the incline.

mgyo = mv2/2

=> v = (2gyo)1/2

b) Time taken to pass through, t = 3w/v = 3w/(2gyo)1/2

c) The area of the loop inside the field reduces as the cart leaves the field.

Induced emf = d/dt = -d(BA)/dt = -B(dA/dt)

Let the length of the loop in the field be 'x' at one instant. Then,

Area in the field, A = hx

dA/dt = h(dx/dt) = h(-v) = -hv

So, induced emf = -B(-hv) = bhv

d) Since the flux is decreasing as the cart is moving out of the field and magnetic field is out of the page, the current in the loop will be in counterclockwise direction.

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