Suppose you hold a square copper loop vertically at the lower edge of a horizont

Question

Suppose you hold a square copper loop vertically at the lower edge of a horizontal magnetic field, and then drop it so that it falls under gravity. An induced magnetic force slows the loop's fall until it leaves the field. Assume the loop has side length L, mass m, and resistance R. At the instant shown in the figure, the loop is partly in and partly out of the field and has an instantaneous velocity v. What is the induced emf in the loop at the instant shown? The emf you found in part a) drives a current in the loop. This current will cause the loop to experience a magnetic force. Find the magnitude and direction of the magnetic force at the instant shown. Write a force balance equation for the loop at the instant shown. The loop reaches "terminal velocity" when it no longer accelerates (that is, the acceleration of gravity has been cancelled out by the action of the magnetic force). Find an expression for this terminal velocity v_t, assuming it is reached before the loop fully leaves the field. How could you make a loop that would fall very slowly out of a field of given magnitude B? What would happen if you cut a slit in the loop so that it was no longer a complete square, then repeated the experiment?

Explanation / Answer

Here,

a) for the induced emf

induced emf in the loop = B * v * L

b)

current in the loop , I = induced emf/R

I = B * v * L/R

magnetic force on the loop = B*I*L

magnetic force on the loop = B^2 * L^2 * v/R

the direction of magnetic force is upwards

as the gravity is pulling the loop downwards

c)

for the force balance equation

m * g - B^2 * L^2 * v/R = m * a

a is the acceleration

d)

for the terminal velocity

m * g - B^2 * L^2 * v/R = m * a

m * g - B^2 * L^2 * v/R = 0

Vt = m * g *R/(B^2 * L^2)

the terminal velocity is m * g *R/(B^2 * L^2)

e)

for a very slow fall , the terminal velocity must be very small

hence , the mass of loop and resistance must be very small

f)

as there will no current in the loop

no magnetic force acts on the loop

and it will be under freefall

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