A rectangular loop of sides 12 ~cm and 5 ~cm, with its sides para… - Vidyadip

Question

A rectangular loop of sides $12 \mathrm{~cm}$ and $5 \mathrm{~cm}$, with its sides parallel to the $x$-axis and $y$-axis respectively moves with a velocity of $5 \mathrm{~cm} / \mathrm{s}$ in the positive $\mathrm{x}$ axis direction, in a space containing a variable magnetic field in the positive $z$ direction. The field has a gradient of $10^{-3} \mathrm{~T} / \mathrm{cm}$ along the negative $\mathrm{x}$ direction and it is decreasing with time at the rate of $10^{-5} \mathrm{~T} / \mathrm{s}$. If the resistance of the loop is $6 \mathrm{~m} \Omega$, the power dissipated by the loop as heat is______ $\times 10^{-9} \mathrm{~W}$.

✓

Answer

$\mathrm{B}_0$ is the magnetic field at origin

$\frac{d B}{d x}=-\frac{10^{-3}}{10^{-2}}$

$\int_{B_0}^B d B=-\int_0^x 10^{-1} d x$

$B-B_0=-10^{-1} x$

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

Motional emf in $\mathrm{AB}=0$

Motional emf in $\mathrm{CD}=0$

Motional emf in $\mathrm{AD}=\varepsilon_1=\mathrm{B}_0 / \mathrm{v}$

Magnetic field on $\operatorname{rod} B C B$

$=\left(\mathrm{B}_0-\frac{\left(-12 \times 10^{-2}\right)}{10}\right)$

Motional emf in $\mathrm{BC}=\varepsilon_2=\left(\mathrm{B}_0+\frac{12 \times 10^{-2}}{10}\right) \ell \times \mathrm{v}$

$\varepsilon_{\text {eq }}=\varepsilon_2-\varepsilon_1=300 \times 10^{-7} \mathrm{~V}$

For time variation

$\left(\varepsilon_{\text {eq }}\right)^{\prime}=\mathrm{A} \frac{\mathrm{dB}}{\mathrm{dt}}=60 \times 10^{-7} \mathrm{~V}$

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

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

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