Vacuum permeability doubled

Problem

Imagine that the magnetic permeability of the vacuum were doubled. Discuss how the following would change (a) the magnetic field of a straight wire for the same current, (b) the force between two parallel wires for the same geometry, (c) the speed of light $c = 1/\sqrt{\mu_0\varepsilon_0}$.

Solution

(a) The field of a straight wire is $B=\dfrac{\mu_0 i}{2\pi r}\propto\mu_0$: doubling $\mu_0$ doubles the field.

(b) The force per unit length between two parallel wires is $\dfrac{F}{\ell}=\dfrac{\mu_0 I_1 I_2}{2\pi d}\propto\mu_0$: doubling $\mu_0$ doubles the force.

(c) The speed of light is $c=\dfrac{1}{\sqrt{\mu_0\varepsilon_0}}\propto\dfrac{1}{\sqrt{\mu_0}}$: doubling $\mu_0$, $c$ decreases by a factor $1/\sqrt2$.

$\ev{\text{(a) }B\times2;\ \text{(b) }F/\ell\times2;\ \text{(c) }c\times\tfrac{1}{\sqrt2}}$

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Topics: Magnetismo Concepts: Campo magnetico Skills: Analisi di casi limite e fantafisica