Q. 12 3.6( 10 Votes )

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

Answer :

Given: side of square loop = 12 cm

In metre, the side of square loop = 0.12m

Area of the square loop = 0.12 × 0.12 = 144 × 10 - 4m2

Velocity of the loop = 8 cms-1

In metre/s, the velocity of the loop, v= 0.08m/s

Since gradient of electric field is along negative x - direction

In Tm - 1,

The rate of decrease of magnetic field in time is given as:

Resistance of the square loop = 4.50 mΩ =4.5 × 10 - 3

Due to change in the motion of the square loop in presence of non - uniform magnetic field, the decrease in the magnetic flux is given as:

Due to explicit time variation in magnetic field, the rate of change of flux is given by:

Substituting values in above, we get

The total emf induced in the square loop is :

e =1.44 × 10–5 V+ 11.52× 10-5 V

e = 12.96 × 10-5 V

The induced current can be written as:

i =e/R

i = 2.88 × 10 - 2A

The current direction will be in the way that there will be increase in the flux along positive z direction.

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