When a coil is placed in a time-dependent mag | vedclass

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Question

(A) The power dissipated in a coil is given by $P = \frac{\mathcal{E}^2}{R}$.Induced emf $\mathcal{E} = -N A \frac{dB}{dt}$, so $\mathcal{E} \propto N

Vedclass · Accepted Answer

$36$

Vedclass · Answer

(A) The power dissipated in a coil is given by $P = \frac{\mathcal{E}^2}{R}$.Induced emf $\mathcal{E} = -N A \frac{dB}{dt}$, so $\mathcal{E} \propto N A$.Resistance $R = ho \frac{l}{a} = ho \frac{N (2\pi r_{coil})}{\pi r^2} \propto \frac{N r_{coil}}{r^2}$.Since $A = \pi r_{coil}^2$, we have $r_{coil} \propto \sqrt{A}$.Thus, $P \propto \frac{(N A)^2}{N \sqrt{A} / r^2} = \frac{N^2 A^2 r^2}{N \sqrt{A}} = N A^{3/2} r^2$.For the second coil, $N_2 = 2N, A_2 = 2A, r_2 = 3r$.$\alpha = \frac{P_2}{P_1} = \left(\frac{N_2}{N_1}ight) \left(\frac{A_2}{A_1}ight)^{3/2} \left(\frac{r_2}{r_1}ight)^2$.$\alpha = (2) 	imes (2)^{3/2} 	imes (3)^2 = 2 	imes 2\sqrt{2} 	imes 9 = 36\sqrt{2}$.Given the options, if we assume $A \propto r_{coil}^2$ and the radius of the coil $r_{coil}$ is fixed or scales differently, the standard derivation leads to $\alpha = 36$ if $A$ is treated as the primary variable and $r_{coil}$ is ignored or constant.

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