A close and uniformly wound coil is in the form of a spiral of inner radius a and outer radius b as shown in the figure. The coil is having N turns and is placed in a time varying magnetic field $B = B_{0} t .$ The emf induced in the coil is

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$\frac{π B_{0} N}{3 {(b - a)}^{2}} \times (b^{3} - a^{3})$

$\frac{π B_{0} N}{3} \times (b^{2} + a^{2} + a b)$

$\frac{π B_{0} N}{3}$

$\frac{π B_{0} N}{2} \times (b^{2} - a^{2})$

answer is B.

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Detailed Solution

As the coil is wound uniformly, the number of turns per unit radial distance is $n = \frac{N}{b - a}$

Consider a section of coil of radius r and thickness dr the magnetic flux linked with the element is $Δ ϕ = π r^{2} \times B_{0} t \times n d r = \frac{π B_{0} t \times N}{b - a} r^{2} d r$

emf induced in this section is $d e = - \frac{d (Δ ϕ)}{d t} = \frac{π B_{0} N}{b - a} r^{2} d r$

$e = \int d e = - \int \frac{π B_{0} N}{b - a} r^{2} d r = - - \frac{π B_{0} N}{b - a} \frac{(b^{3} - a^{3})}{3}, | e | = \frac{π B_{0} N}{3} (b^{2} + a^{2} + a b)$