Question

A coaxial cable having radius "$a$" of inner wire and inner and outer radii " $b$ " and "$c$" respectively of the outer shell carries equal and opposite currents of magnitude $i$ on the inner and outer conductors as shown. What is the magnitude of the magnetic induction at point $P$ of the cable at a distance $r(b<r<$ c) from the axis?

• A. Zero
• B. $\frac{{\mu }_{0}ir}{2\pi a^2}$
• C. $\frac{{\mu }_{0}i}{2\pi r}$
• D. $\frac{{\mu }_{0}i}{2\pi r}\frac{c^2-r^2}{c^2-b^2}$

Answer 1

Answer: (D)

By Ampere's law
$B(2\pi r)={\mu }_0\left(i-i^{\prime }\right)$
Now, current density in the outer wire is constant
$\frac{i}{\pi \left(c^2-b^2\right)}=\frac{i^{\prime }}{\pi \left(r^2-b^2\right)}$
$i^{\prime }=i\left(\frac{r^2-b^2}{c^2-b^2}\right)$
$\therefore B=\frac{{\mu }_{0}i}{2\pi r}\left[1-\frac{r^2-b^2}{c^2-b^2}\right]$
$=\frac{{\mu }_{0}i}{2\pi r}\left(\frac{c^2-r^2}{c^2-b^2}\right)$