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

Explanation

As per data given in the question, magnetic field due to current in solenoid S, `B = mu_(0) n I`
Magnetic flux linked with the smaller coil due to this field is
`phi = NBA`, where A = area of smaller coil `= pi b^(2)`
`:.` emf induced in the smaller coil, `e = - (d phi)/(dt) = - (d)/(dt) (NB pi b^(2)) = - N pi b^(2) (d)/(dt) (mu_(0) n I) = - N pi b^(2) mu_(0) n (dI)/(dt)`
As current I varies as a function of `(t^(2) + C)`, therefore,
`e = - N n pi mu_(0) b^(2) (d)/(dt) (mt^(2) + C) = - N n pi mu_(0) b^(2) (2 mt)`
`= - mu_(0) N n pi b^(2) (2 mt)`
Negative sign indicates that emf induced oppose the main current in the long solenoid.
From (i), we find that
Hence, the variation of induced emf `|e|` with time t is representedd by a straight line graph as shown in Fig.