Electromagnetic induction > Mutual inductance. Inductance of conductors. Transformers > Problem 11.3.8

Statement

$11.3.8.$

Determine the inductance of the unit length of a two-wire line consisting of
two thin flat busbars with a width of d = 0.1 m, located at a distance of h = 5
mm from each other. The busbars flow modulo equal but oppositely directed
currents.

Solution

For two thin conducting flat plates of width d separated by a distance" $h (h \ll d)$, equal and opposite currents produce a practically uniform magnetic field in the space between them. Applying Ampère's law to a rectangular path that crosses one plate, we obtain $ H = I/d $ and $ B = \mu_0 I/d $

The magnetic flux per unit length crossing the area$ h \times 1 $between the plates is

$ \Phi' = B h = \mu_0 I h / d $

Therefore, the inductance per unit length is:

$L' = \frac{\Phi'}{I} = \frac{\mu_0 h}{d}$

Substituting the numerical values
$(h = 5\ \text{mm} = 0.005\ \text{m}, d = 0.1\ \text{m}, \mu_0 = 4\pi \times 10^{-7}\ \text{H/m})$

$L' = \frac{4\pi \times 10^{-7} \times 0.005}{0.1} = 2\pi \times 10^{-8}\ \text{H/m} \approx 6.28 \times 10^{-8}\ \text{H/m}$

$\boxed{L' \approx 6.3 \times 10^{-8}\ \text{H/m} \; (= 63\ \text{nH/m})}$

Answer

$\boxed{L' \approx 6.3 \times 10^{-8}\ \text{H/m} \; (= 63\ \text{nH/m})}$