To solve this problem it is necessary to apply the concepts related to inductance on an inductor, which is mathematically described as:
[tex]L = \mu_0 n^2 A l[/tex]
Where,
[tex]\mu_0[/tex]= Permeability constant (Described as 'n' at the problem equation)
l = Length
A = Cross-sectional Area
n = No of turn per unit length
The number of turns N is given by
[tex]N = \frac{l}{2a}[/tex]
The number of turns per unit length n is
[tex]n = \frac{N}{l} = \frac{1}{2a}[/tex]
The relationship of the cable lengths starts from assuming that the length 'a' is less than the length 'r', and therefore the length of the wire d would be related by:
[tex]d = N(2\pi r)[/tex]
[tex]d = \frac{l}{2a}2\pi r[/tex]
[tex]d = \frac{\pi r}{a}l[/tex]
Solving to obtain l,
[tex]l = \frac{ad}{\pi r}[/tex]
Substituting at the first equation,
[tex]L = \mu_0 n^2 A l[/tex]
[tex]L = \mu_0 (\frac{1}{2a})^2(\pi r^2)(\frac{ad}{\pi r})[/tex]
[tex]L = \mu_0 (\frac{rd}{4a})[/tex]
