Question

A closely packed straight solenoid has a self inductance of L. What would be its self inductance of the solenoif is cut into two equal pieces and the pieces are put in parallel to each other.

Answer 1

Dear student

The self inductance of a solenoid is given by
$L=\frac{{\mu }_o{N}^{2}A}{l}$
where N is the number of turns , A is the area of cross section of solenoid and l is length of the solenoid.
When a solenoid is cut into two equal parts then the number of turns will be N'=N/2 and length l'=l/2 so the self inductance of new solenoid will be
$$\begin{gathered} L\text{'}=\frac{{\mu }_o{{\displaystyle \frac{N}{4}}}^{2}A}{{\displaystyle \frac{l}{2}}}=\frac{1}{2}\frac{{\mu }_o{N}^{2}A}{{\displaystyle l}}=\frac{L}{2} \\ Sonewselfinduc\tan ceisL/2.Ifthesetwosolenoidareconnectedinparallelthen \\ resul\tan tselfinduc\tan cewillbe \\ \frac{1}{L_{net}}=\frac{2}{L}+\frac{2}{L}=\frac{4}{L} \\ {L}_{net}=\frac{L}{4} \end{gathered}$$