A pair of straight parallel thin wires, such as a lamp cord, each of radius r, are a distance 𝓁 apart and carry current to a circuit some distance away. Ignoring the field within each wire, show that the inductance per unit length is (μ₀/π) ln[(𝓁 - r) /r].

Verified step by step guidance

1. Identify the relevant physical principles: The inductance per unit length of parallel wires can be calculated using the magnetic field around the wires and the mutual inductance concept. The formula for the inductance per unit length (L') of two parallel wires is given by L' = \(\frac{\mu_0}{\pi}\) ln\(\frac{d}{r}\), where d is the center-to-center distance between the wires and r is the radius of each wire.
2. Calculate the effective separation distance: Since the wires have a radius r and are separated by a distance 𝓁 from center to center, the effective separation distance for the magnetic field interactions is (𝓁 - r). This is because the closest approach between the surfaces of the wires is what influences the magnetic field interactions.
3. Substitute the effective separation distance into the inductance formula: Replace d in the inductance formula with (𝓁 - r) to account for the actual distance between the surfaces of the wires. The formula becomes L' = \(\frac{\mu_0}{\pi}\) ln\(\frac{𝓁 - r}{r}\).
4. Understand the logarithmic term: The term ln\(\frac{𝓁 - r}{r}\) represents the natural logarithm of the ratio of the effective separation distance to the radius of the wires. This ratio indicates how much farther apart the wires are compared to their radius, which affects the magnetic coupling between them.
5. Interpret the result: The expression (μ₀/π) ln[(𝓁 - r) /r] shows that the inductance per unit length depends logarithmically on the ratio of the separation distance minus the radius to the radius itself. This reflects how the geometry and spacing of the wires influence the magnetic fields and thus the inductance.