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 `mOmega`.

Explanation

The magnetic field in loop varies with position .x. of loop and also with time simultaneously. The rate of change of flux due to variation of .B. with time is `(dphi)/(dt) = A xx (dB)/(dt)`
The rate of change of flux due to variation B with position .x. is
`(dphi)/(dt) = A xx (dB)/(dt) = A (dB)/(dx) xx (dx)/(dt) = A (dB)/(dx) xx v `
Since both cause decrease in flux, the two effects will add up
` therefore ` The net emf induced
` e = (dphi)/(dt) = A (dB)/(dt) + A(dB)/(dx) xx v = A [ (dB)/(dt) + v (dB)/(dx) ]`
` = 144 xx 10^(-4) [ 10^(-3) + 8 xx 10^(-3)]`
` = 144 xx 9 xx 10^(-7) = 129.6 xx 10^(-6) V`