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 infinitity long. A uniform current density `j` flows along the length . If the magnitude of the magnetic field at the point `P` is given by `(N)/(12) mu_(0)aJ`, then the value of `N` is

Answer

`(5)`

Explanation

Due to current in complete cylinder, the magnetic field induction at P is
`B_1=(mu_0)/(4pi)(2I)/(a)=(mu_0)/(4pi)2/a(pia^2J)=(mu_0aJ)/(2)`
current through the cylindrical cavity of radius
`a//2` is `I^'=(I)/(pia^2)xxpi(a//2)^2=I/4`
Due to current in cylindrical cavity, the magnetic field induction at P is
`B_2=(mu_02)/(4pi)((I//4))/((a+a//2))=(mu_0)/(4pi)(2I)/(4xx(3a//2))`
`=(mu_02)/(4pi)((pia^2J))/(6a)=(mu_0aJ)/(12)`
Net magnetic field induction at P is
`B=B_1-B_2=(mu_0aJ)/(2)-(mu_0aJ)/(12)`
`=(mu_0aJ)/(2)[1-1/6]=(5mu_0aJ)/(12)`
As per question, `=(5mu_0aJ)/(12)=(Nmu_0aJ)/(12)`
`:. N=5`