Question 188: Electromagnetic Induction

Question

A circular conducting coil of radius 1 m is being heated by the change of magnetic field $$\overrightarrow B $$ passing perpendicular to the plane in which the coil is laid. The resistance of the coil is 2 $$\mu$$$$\Omega$$. The magnetic field is slowly switched off such that its magnitude changes in time as $$B = {4 \over \pi } 	imes {10^{ - 3}}T\left( {1 - {t \over {100}}} ight)$$The energy dissipated by the coil before the magnetic field is switched off completely is E = ___________ mJ.

Accepted Answer

$$\phi  = \overrightarrow B .\overrightarrow S $$
$$\phi  = {4 \over \pi } 	imes {10^{ - 3}}\left( {1 - {t \over {100}}} ight).\pi {R^2}$$
$$\phi  = 4 	imes {10^{ - 3}} 	imes {(1)^2}\left( {1 - {t \over {100}}} ight)$$
$$\varepsilon  = {{ - d\phi } \over {dt}}$$
$$\varepsilon  = {{ - d} \over {dt}}\left( {4 	imes {{10}^{ - 3}}\left( {1 - {t \over {100}}} ight)} ight)$$
$$\varepsilon  = 4 	imes {10^{ - 3}}\left( {{1 \over {100}}} ight) = 4 	imes {10^{ - 5}}V$$
When B = 0
$$1 - {t \over {100}} = 0$$
t = 100 sec
Heat $$ = {{{\varepsilon ^2}} \over R}t$$
Heat $$ = {{{{(4 	imes {{10}^{ - 5}})}^2}} \over {2 	imes {{10}^{ - 6}}}} 	imes 100$$ J
Heat $$ = {{16 	imes {{10}^{ - 10}} 	imes 100} \over {2 	imes {{10}^{ - 6}}}}$$ J
Heat = 0.08 J
Heat = 80 mJ

Electromagnetic Induction

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