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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 :

28861_9f887f71bc6441d49ffdfda71da688da.png


A
5
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B
6
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C
7
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D
8
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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  =   $$ B \space = \space {B}_{1} \space + \space {B}_{2} $$
$$ { B }_{ 1 }\space =\space \dfrac { { \mu  }_{ 0 }Ja }{ 2 } $$
$$ { B }_{ 2 }\space =\space \dfrac { -{ \mu  }_{ 0 }J{ (\dfrac { a }{ 2 } ) }^{ 2 } }{ 2\dfrac { 3a }{ 2 }  } $$
$$ \therefore { B }_{ 2 }\space =\space \dfrac {- { \mu  }_{ 0 }Ja }{ 12 } $$
$$ \therefore { B }\quad =\quad \dfrac { { 5\mu  }_{ 0 }Ja }{ 12 } $$.
$$ \therefore N \space = \space 5 $$

Physics
NCERT
Standard XII

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