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Question

A square loop of side 12 cm with its sides parallel to X and Y axes ismoved with a velocity of 8 cm s–1 in the positive x-direction in anenvironment containing a magnetic field in the positive z-direction.The field is neither uniform in space nor constant in time. It has agradient of 10–3 T cm–1along the negative x-direction (that is it increasesby 10 – 3 T cm–1as one moves in the negative x-direction), and it isdecreasing in time at the rate of 10–3 T s–1. Determine the direction andmagnitude of the induced current in the loop if its resistance is 4.50 mW.

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Solution

Given: The side of the square loop is

The area of the loop is given as,

Where, the side of the square loop is

Since,

By substituting the given values in the above formula, we get

The change in flux is given as,

Where, the area of the square loop is

By substituting the given values in the above formula, we get

The rate of change of the flux due to time variation is given as,

Where, the rate of decrease of magnetic field is

By substituting the given values in the above expression, we get

The induced emf in the loop is given as,

By substituting the given values in the above expression, we get

The induced current in the loop is given as,

Where, the resistance of the loop is

By substituting the given values in the above formula, we get

Since, the flux is decreasing, then according to Lenz’s law the induced current should oppose the cause of the change. The induced current is induced in positive *z* direction.

Thus, the magnitude of the induced current is *z* direction.

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