Question

Self inductance of unit length of two co-axial cylinders, the inner one of radius a and outer one of radius b, carrying same current but in opposite direction is

• A. $\frac{{\mu }_0}{2\pi }\frac{b}{a}$
• B. $\frac{{\mu }_0}{2}$
• C. $\frac{{\mu }_0}{2\pi }ln\left(\frac{b}{a}\right)$
• D. $\frac{{\mu }_0}{\pi }ln\left(\frac{b}{a}\right)$

Answer 1

The correct option is C $\frac{{\mu }_0}{2\pi }ln\left(\frac{b}{a}\right)$


B=$\frac{{\mu }_0}{2\pi }\frac{i}{r}$ (By Amperes Law)

$d\Phi =BdA=\frac{{\mu }_0}{2\pi }\frac{i}{r}$ [For unit length]

$\Phi =\int d\Phi ={\int }_a^b\frac{{\mu }_0}{2\pi }\frac{i}{r}dr$

$\Phi =\frac{{\mu }_0}{2\pi }iln\left(\frac{b}{a}\right)$ By $\Phi =Li$

$L=\frac{{\mu }_0}{2\pi }ln\left(\frac{b}{a}\right)$