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Let us consider two solenoids A and B, made from same magnetic material of relative permeability $\mu_r$ and equal area of cross-section. Length of A is twice that of B and the number of turns per unit length in A is half that of B. The ratio of self inductances of the two solenoids, $L_A:L_B$ is
Detailed Solution
Self inductance $L=\mu_0\mu_r n^2 A l$, so $\dfrac{L_A}{L_B}=\left(\dfrac{n_A}{n_B}\right)^2\times\dfrac{l_A}{l_B}=\left(\dfrac{1}{2}\right)^2\times2=\dfrac{1}{2}$. So $L_A:L_B=1:2$.