Ncert Exercises & Solutions

A long solenoid S has n turns per meter, with a diameter a. At the centre of this coil, we place a smaller coil of N turns and diameter b (where b<a). If the current in the solenoid increases linearly, with time, what is the induced emf appearing in the smaller coil. Plot graph showing the nature of variation in emf, if current varies as a function of ${mt}^{2} + C$ .

Hint: The current is induced due to the change in magnetic flux.

Step 1: Magnetic field due to a solenoid S, B=

μ_{0} n i

where signs are usual.

Magnetic flux in smaller coil,

ϕ

=NBA, where A=

π b^{2}

Step 2: Applying Faraday's law of EMl, we have;

e = \frac{- d ϕ}{d t} = \frac{- d}{d t} (N B A)

= - N π b^{2} \frac{d (B)}{d t}

w h e r e, B = μ_{0} N i

= - N n π μ_{0} b^{2} \frac{d}{d t} (m t^{2} + C) = - μ_{0} N n π b^{2} 2 m t

Since the current varies as a function of

m t^{2} + C .

e = - μ_{0} N n π b^{2} 2 m t

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