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

Solution

## The correct option is A $$5$$The magnetic field for an infinitely long cylinder is given by, $${ B }_{ in }\space =\space \dfrac { { \mu }_{ 0 }Jr }{ 2 }$$$${ B }_{ out }\space =\space \dfrac { { \mu }_{ 0 }J{R}^{2} }{ 2 r }$$$$r\space = \space$$ distance from the axis of the cylinder.$$R \space = \space$$ 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  =   $$B \space = \space {B}_{1} \space + \space {B}_{2}$$$${ B }_{ 1 }\space =\space \dfrac { { \mu }_{ 0 }Ja }{ 2 }$$$${ B }_{ 2 }\space =\space \dfrac { -{ \mu }_{ 0 }J{ (\dfrac { a }{ 2 } ) }^{ 2 } }{ 2\dfrac { 3a }{ 2 } }$$$$\therefore { B }_{ 2 }\space =\space \dfrac {- { \mu }_{ 0 }Ja }{ 12 }$$$$\therefore { B }\quad =\quad \dfrac { { 5\mu }_{ 0 }Ja }{ 12 }$$.$$\therefore N \space = \space 5$$PhysicsNCERTStandard XII

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