Predict the direction of induced current in metal rings 1 and 2 when current I in the wire is steadily decreasing.

onsider a long solenoid of length l and radius r with r << 1 and having n turns per unit length. If a current I flow through the coil, then the magnetic field inside the coil is almost constant and is given by

$\mathrm{B}=\mathrm{\mu }0\mathrm{nl}$

Magnetic flux linked with each turn = BA =${\mathrm{\mu }}_0\mathrm{nIA}$

...where $[\mathrm{A}={\mathrm{\pi r}}^2= \mathrm{cross}-\mathrm{sectional} \mathrm{area} \mathrm{of} \mathrm{the} \mathrm{solenoid}$

$$\begin{gathered} \therefore \mathrm{Magnetic} \mathrm{flux} \mathrm{linked} \mathrm{with} \mathrm{the} \mathrm{entire} \mathrm{solenoid} \mathrm{is} \\ \mathrm{\varphi }=\mathrm{Flux} \mathrm{linked} \mathrm{with} \mathrm{each} \mathrm{turn} \times \mathrm{Total} \\ \mathrm{number} \mathrm{of} \mathrm{turns} \\ ={\mathrm{\mu }}_0\mathrm{nIA}\times \mathrm{nl}={\mathrm{\mu }}_0{\mathrm{n}}^2\mathrm{IAl} \\ \mathrm{But} \mathrm{\varphi }=\mathrm{LI} \\ \therefore \mathrm{Self}-\mathrm{inductance} \mathrm{of} \mathrm{the} \mathrm{long} \mathrm{solenoid} \mathrm{is} \\ \mathrm{L}={\mathrm{\mu }}_0{\mathrm{n}}^2\mathrm{lA} \\ \mathrm{If} \mathrm{N} \mathrm{is} \mathrm{the} \mathrm{total} \mathrm{number} \mathrm{of} \mathrm{turns} \mathrm{in} \mathrm{the} \mathrm{solenoid} \\ \mathrm{then} \mathrm{n}=\frac{\mathrm{N}}{\mathrm{l}} \\ \therefore \mathrm{L}=\frac{{\mathrm{\mu }}_{0\mathrm{l}}{\mathrm{N}}^2\mathrm{A}}{\mathrm{l}} \end{gathered}$$