A rectangular loop of sides $12\ \text{cm}$ and $5\ \text{cm}$, with its sides parallel to the $x$-axis and $y$-axis respectively, moves with a velocity of $5\ \text{cm s}^{-1}$ in the positive $x$ direction, in a space containing a variable magnetic field in the positive $z$ direction. The field has a gradient of $10^{-3}\ \text{T cm}^{-1}$ along the negative $x$ direction and it is decreasing with time at the rate of $10^{-3}\ \text{T s}^{-1}$. If the resistance of the loop is $6\ \text{m}\Omega$, the power dissipated by the loop as heat is ______ $\times10^{-9}\ \text{W}$.
Numerical value type. Enter your answer.
Answer: 216
Area of the loop: $A = 12\times5 = 60\ \text{cm}^2 = 6\times10^{-3}\ \text{m}^2$.
The field increases towards $-x$, so as the loop moves along $+x$ the field it sees decreases at
$$\left|\frac{dB}{dx}\right|v = 10^{-3}\ \text{T cm}^{-1}\times5\ \text{cm s}^{-1} = 5\times10^{-3}\ \text{T s}^{-1}$$
The field also decreases with time at $10^{-3}\ \text{T s}^{-1}$. Both effects reduce the flux, so
$$\varepsilon = A\left(5\times10^{-3} + 1\times10^{-3}\right) = 6\times10^{-3}\times6\times10^{-3} = 3.6\times10^{-5}\ \text{V}$$
$$P = \frac{\varepsilon^2}{R} = \frac{(3.6\times10^{-5})^2}{6\times10^{-3}} = 2.16\times10^{-7}\ \text{W} = 216\times10^{-9}\ \text{W}$$
Solution by Sreeraj P, M.Sc Physics