Question 191: Electromagnetic Induction

Question

Two concentric circular coils, C1 and C2 are
placed in the XY plane. C1 has 500 turns, and
a radius of 1 cm. C2 has 200 turns and radius
of 20 cm. C2 carries a time dependent current
I(t) = (5t2 – 2t + 3) A where t is in s. The emf
induced in C1 (in mV), at the instant t = 1 s is
$${4 \over x}$$. The value of x is ___ .

Accepted Answer

$${B_2} = {{{\mu _0}{I_2}{N_2}} \over {2{R_2}}}$$
Total flux $$\phi  = {N_1}{B_2}\pi {R_1}^2 = {N_1}{N_2}{{{\mu _0}I} \over {2{R_2}}}\pi {R_1}^2$$
$$ 	herefore $$ $$\phi  = {{500 	imes 200 	imes 4\pi  	imes {{10}^{ - 7}} 	imes (5{t^2} - 2t - 3)\pi {{({{10}^{ - 2}})}^2}} \over {2 	imes 20 	imes {{10}^{ - 2}}}}$$
= $${{{{10}^5} 	imes 4{\pi ^2} 	imes {{10}^{ - 7}}(5{t^2} - 2t + 3) 	imes {{10}^{ - 4}}} \over {40 	imes {{10}^{ - 2}}}}$$
$$ \Rightarrow $$ $$\phi  = (5{t^2} - 2t + 3) 	imes {10^{ - 4}}$$
We know, $$e = \left| {{{d\phi } \over {dt}}} ight| = (10t - 2) 	imes {10^{ - 4}}$$
At $$t = 1\sec $$
$$e = 8 	imes {10^{ - 4}} = 0.8\,mV = {{0.8} \over {10}} = {4 \over 5}$$
$$ 	herefore $$ $$x = 5$$

Electromagnetic Induction

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