Question

A cylindrical cavity of diameter 'a' exists inside a cylinder of diameter '2a' as shown in the figure. Both the cylinder and the cavity are infinitely long. A uniform current density $J$ flows along the length. If the magnitude of the magnetic field at the point P is given by $\frac{N}{12}{\mu }_{0}J$ , then the value of $N$ is :

• A. $5$
  
• B. $6$
  
• C. $7$
  
• D. $8$
  

Answer 1

The correct option is A $5$

The magnetic field for an infinitely long cylinder is given by,

$B_{in}=\frac{{\mu }_{0}Jr}{2}$

$B_{out}=\frac{{\mu }_{0}J{R}^2}{2r}$

$r=$ distance from the axis of the cylinder.

$R=$ Radius of the cylinder.

Assuming the bigger cylinder to carry a positive current density and the smaller cylinder carry a negative current density of magnitude J each.

$\therefore$ Magnetic field at point P = $B=B_1+B_2$

$B_1=\frac{{\mu }_{0}Ja}{2}$

$B_2=\frac{-{\mu }_{0}J{\left(\frac{a}{2}\right)}^2}{2\frac{3a}{2}}$

$\therefore B_2=\frac{-{\mu }_{0}Ja}{12}$

$\therefore B=\frac{{5\mu }_{0}Ja}{12}$.

$\therefore N=5$