A rectangular loop of sides 12 cm and 5 cm , with its sides parallel to the x -axis and y -axis respectively…

Let $B_0$ is the magnetic field at origin. Then,

$\frac{dB}{dx}=-\frac{{10}^{-3}}{{10}^{-2}}$

$\Rightarrow {\int }_{B_0}^{B}dB=-{\int }_0^x{10}^{-1}dx$

$\Rightarrow B-B_0=-{10}^{-1}\mathrm{x}$

$\Rightarrow B=\left(B_0-\frac{x}{10}\right)$

Motional emf in $AB=0$

Motional emf in $CD=0$

Motional emf in $AD={\epsilon }_1=B_0\ell v$

Magnetic field on rod $BC$ 

$=\left(B_0-\frac{\left(-12\times {10}^{-2}\right)}{10}\right)$

Motional emf in  $BC={\epsilon }_2=\left(B_0+\frac{12\times {10}^{-2}}{10}\right)\ell \times v$

Net equivalent emf ${\mathrm{\epsilon }}_{\mathrm{eq}}={\mathrm{\epsilon }}_2-{\mathrm{\epsilon }}_1=300\times {10}^{-7} \mathrm{V}$

For time variation

${\left({\epsilon }_{eq}\right)}^{\text{'}}=A\frac{dB}{dt}=60\times {10}^{-7} V$

${\left({\epsilon }_{\mathrm{eq}}\right)}_{\text{net }}={\epsilon }_{\mathrm{eq}}+{\left({\epsilon }_{\mathrm{eq}}\right)}^{\text{'}}=360\times {10}^{-7} \mathrm{V}$

Power $r=\frac{{\left({\epsilon }_{\text{eq }}\right)}_{\text{net }}^2}{R}=216\times {10}^{-9} \mathrm{W}$