Question

A square loop of side 12 cm with its sides parallel to X and Y axes is moved with a velocity of $8cm{s}^{-1}$ 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 ${10}^{-3}Tc{m}^{-1}$ along the negative x-direction (that is it increases by ${10}^{-3}Tc{m}^{-1}$ 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 m$\Omega$.

Answer 1

$\frac{dB}{dx}={10}^{-1}T{m}^{-1}$

$\frac{dB}{dt}={10}^{-3}T{s}^{-1}$

$\varphi =flux;A=area$

also, $\frac{d\varphi }{dt}=A\times \frac{dB}{dx}\times V=11.52\times {10}^{-5}$

Rate of change of flux to explicit time variation is $\frac{d{\varphi }^{{}^{\prime }}}{dt}=A\times \frac{dB}{dt}=1.44\times {10}^{-5}$

$e=11.52\times {10}^{-5}+1.44\times {10}^{-5}$

$i=e/R=2.8\times {10}^{-2}A$