Two pure inductor coils of self inductance L each are connected in series, the net inductance is

When two coils are connected in series, the sane current i is induced in both the coils and total magnetic flux linked with them will be equal to the sum of the flux linked with them individually, i.e.,
$\mathrm{\Phi }={\mathrm{L}}_1\mathrm{i}+{\mathrm{L}}_2\mathrm{i} .......\left(1\right)$
If equivalent inductance be L' then
$\begin{array}{l}\mathrm{\Phi }={\mathrm{L}}^{\mathrm{\prime }}\mathrm{i}\\ \mathrm{Hence} {\mathrm{L}}^{\mathrm{\prime }}\mathrm{i}={\mathrm{L}}_1^{\mathrm{i}}+{\mathrm{L}}_2^{\mathrm{i}}\\ \mathrm{or} {\mathrm{L}}^{\mathrm{\prime }}={\mathrm{L}}_1+{\mathrm{L}}_2\\ \mathrm{Here} {\mathrm{L}}_1={\mathrm{L}}_2=\mathrm{L}\\ \therefore {\mathrm{L}}^{\mathrm{\prime }}=\mathrm{L}+\mathrm{L}=2\mathrm{L}\end{array}$