Question

A long cylindrical wire of radius $a$ carries a current $i$ distributed uniformly over its cross section. If the magnetic fields at distances $r<a$ and $R>a$ from the axis have equal magnitude, then:

• A. $a=\frac{r+R}{2}$
• B. $a=\sqrt{Rr}$
• C. $a=\frac{Rr}{R+r}$
• D. $a=\frac{R^2}{r}$

Answer 1

The correct option is B $a=\sqrt{Rr}$

Apply Ampere's law:

$B_{outside}=\frac{{\mu }_{o}i}{2\pi r}$
$B_{inside}=\frac{{\mu }_{o}ir}{2\pi R^2}$
Since $B_{atR>a}=B_{atr<a}$
$\frac{{\mu }_i}{2\pi R}=\frac{{\mu }_{o}i}{2\pi a^2}$
hence, $a=\sqrt{Rr}$