A coil of inductance L is divided into 6 equal parts. All these parts are connected in parallel. The…
A coil of inductance L is divided into 6 equal parts. All these parts are connected in parallel. The resultant inductance of this combination is
$\frac{L}{6}$
$\frac{L}{36}$
$\frac{L}{24}$
6L
Solution
The inductance of a coil is
$\begin{aligned}
& \mathrm{L}=\frac{\mathrm{M}_0 \mathrm{~N}^2 \mathrm{~A}}{1} \Rightarrow \mathrm{~L} \propto \frac{\mathrm{~N}^2}{1} \\
& \therefore \frac{\mathrm{~L}_2}{\mathrm{~L}_1}=\left(\frac{\mathrm{N}_2}{\mathrm{~N}_1}\right)^2\left(\frac{\mathrm{l}_1}{\mathrm{l}_2}\right)
\end{aligned}$
When the coil is divided into 6 equal parts, then
$\begin{aligned}
& N_2=\frac{N_1}{6}, 1_2=\frac{l_1}{6} \\
& \therefore \quad L_2=4\left(\frac{\frac{N_1}{6}}{N_1}\right)^2\left(\frac{\mathrm{l}_1}{\frac{l_1}{6}}\right)=\frac{L_1}{6}=\frac{L}{6}
\end{aligned}$ For parallel combination,
$\frac{1}{L_e}=\left(\frac{6}{L}\right) 6=\frac{36}{L} \Rightarrow L_e=\frac{L}{36}$