Question

# A current $$i$$ is flowing in a hexagonal coil of side $$a$$ as shown in figure. The magnetic induction at the centre of the coil will be :

A
33μ0iπa
B
μ0i33πa
C
μ0i3πa
D
3μ0iπa

Solution

## The correct option is D $$\dfrac {\sqrt 3\mu_0i}{\pi a}$$From figure, $$OA=r{\dfrac {\sqrt{3}}{2}}$$$${r^2}-{r^2}\bigg ({\dfrac {3}{4}}\bigg )={\dfrac {a^2}{4}}$$$${r=a}{\Rightarrow}OA={a}{\dfrac {\sqrt{3}}{2}}$$$$\Rightarrow$$ magnetic field due to a straight line wire at distance $$x=\dfrac { { \mu }_{ 0 }I }{ 4\pi x } \left( \sin { { \theta }_{ 2 }-\sin { { \theta }_{ 1 } } } \right) \\$$$$=\dfrac { { 2\mu }_{ 0 }I }{ 4\pi \left( a\sqrt { 3 } \right) } \left( { \sin { \left( 30^o \right) } -\sin { \left( -30^o \right) } } \right) \\$$Now due to 6 wires of hexagon $$\Rightarrow12\dfrac { { \mu }_{ 0 }I }{ 4\sqrt { 3 } \pi a } \left( \dfrac { 1 }{ 2 } -\left( \dfrac { -1 }{ 2 } \right) \right) \\$$$$=\dfrac { \sqrt { 3 } { \mu }_{ 0 }I }{ \pi a }$$PhysicsNCERTStandard XII

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