Electromagnetic induction > Mutual inductance. Inductance of conductors. Transformers > Problem 11.3.12

Statement

$11.3.12.$
All dimensions of the conductor were increased by k times. How many times
will the inductance of the conductor change?

Solution

For this problem we can use scaling arguments. The inductance of a system of conductors with linear media is proportional to the characteristic length.

If each linear dimension is multiplied by k, the total inductance is also multiplied by k. This follows from dimensional analysis:
inductance has units of

$\text{H} = \text{Wb/A} = \text{T·m}^2/\text{A}$

and since the magnetic field of a current configuration scales inversely with distance, the magnetic flux ends up being proportional to the current and to the length, so that L is proportional to length.

Therefore, when all linear dimensions (radii, distances between conductors, and total length of the system) are scaled by a factor k, the inductance increases by a factor of k:

$\boxed{L_{\text{new}} = k\, L_{\text{original}}}$.

Answer

$\boxed{L_{\text{new}} = k\, L_{\text{original}}}$.