Question 130: Electromagnetic Induction

Question

A conducting metal circular-wire-loop of radius r is placed perpendicular to a
magnetic field which varies with time as 
B = B0e$${^{{{ - t} \over r}}}$$ , where B0 and $$	au $$ are constants, at time t = 0. If the resistance of the loop is R then the heat generated in the loop after a long time (t $$ 	o $$ $$\infty $$)  is :

Accepted Answer

Given,
B = B
0
e$$^{ - {t \over 	au }}$$
Area of the circular  loop, A = $$\pi $$ r
2
$$ 	herefore $$   Flux $$\phi $$ = BA = $$\pi $$ r
2
B
0
e$$^{ - {t \over 	au }}$$
Induced emf in the loop,
$$\varepsilon $$ = $$-$$ $${{d\phi } \over {dt}}$$ = $$\pi $$ r
2
B
0
$${1 \over 	au }$$e$$^{ - {t \over 	au }}$$
Heat generated
= $$\int\limits_0^ \propto  {{i^2}R\,dt} $$
= $$\int\limits_0^ \propto  {{{{\varepsilon ^2}} \over R}} \,dt$$
= $${1 \over R}{{{\pi ^2}{r^4}B_0^2} \over {{	au ^2}}}\int\limits_0^ \propto  {{e^{ - {{2t} \over 	au }}}} \,dt$$
= $${{{\pi ^2}{r^4}B_0^2} \over {{	au ^2}R}} 	imes {1 \over {\left( { - {2 \over 	au }} ight)}}\left[ {{e^{ - {{2t} \over 	au }}}} ight]_0^ \propto $$
= $${{ - {\pi ^2}{r^4}B_0^2} \over {2{	au ^2}R}} 	imes 	au \left( {0 - 1} ight)$$
= $${{{\pi ^2}{r^4}B_0^2} \over {2	au R}}$$

Answer

$${{{\pi ^2}{r^4}B_0^4} \over {2	au R}}$$

Answer

$${{{\pi ^2}{r^4}B_0^2} \over {2	au R}}$$

Answer

$${{{\pi ^2}{r^4}B_0^2R} \over 	au }$$

Answer

$${{{\pi ^2}{r^4}B_0^2} \over {	au R}}$$

Electromagnetic Induction

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