Question

Two very long co-axial cylinders of radius R and 2R carry current I in opposite directions. Then energy stored in cylindrical region of radius 4R of unit length is

• A. $\frac{3}{8}\frac{{\mu }_0{i}^2}{\pi }$
  
• B. $\frac{{\mu }_0{i}^2}{4\pi }ln\left(2\right)$
  
• C. $\frac{{\mu }_0{i}^2}{4\pi }ln\left(4\right)$
  
• D. zero

Answer 1

The correct option is B $\frac{{\mu }_0{i}^2}{4\pi }ln\left(2\right)$


$B=\frac{{\mu }_0}{2\pi }\frac{i}{r}$
Energy density $U^{\prime }=\frac{B^2}{2{\mu }_0}$
dU =U'(dV)=$\frac{B^2}{2{\mu }_0}\times 2\pi rdr$
$U=\int dU={\int }_R^{2R}\frac{B^2}{2{\mu }_0}2\pi rdr=\frac{{\mu }_0{i}^2}{4\pi }ln\left(2\right)$