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 nfinitely long. A uniform current density j flows along the length. If the magnitude of the magnetic filed at the point P is given by $\frac{N}{12}{\mu }_{0}aj$, then the value of N is

Answer 1

For $r\le R,B=\frac{{\mu }_{0}I}{2\pi R^2}.r=\frac{{\mu }_{0}j\pi R^2}{2\pi R^2}.r=\frac{{\mu }_{0}jr}{2}$

For $r>R,B=\frac{{\mu }_{0}I}{2\pi r}=\frac{{\mu }_{0}j\pi R^2}{2\pi r}=\frac{{\mu }_{0}j{R}^2}{2r}$


Magnetic field due to the cylinder at P is
$B_1=\frac{{\mu }_{0}ja}{2}$

Magnetic field due to the cavity at P is

$B_2\frac{{\mu }_{0}j{\left(\frac{a}{2}\right)}^2}{2{\left(\frac{3a}{2}\right)}^2}=\frac{{\mu }_{0}ja}{12}$

Net magnetic field at point P is

$B=B_1-B_2=\frac{{\mu }_{0}ja}{2}(1-\frac{1}{6})=\frac{5{\mu }_{0}ja}{12}=\frac{N{\mu }_{0}ja}{12}$

Hence, N=5