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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 :

143368_b85d6abf0379450fb095b84946f8b8c0.png


A
33μ0iπa
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B
μ0i33πa
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C
μ0i3πa
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D
3μ0iπa
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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 $$\Rightarrow12\dfrac { { \mu  }_{ 0 }I }{ 4\sqrt { 3 } \pi a } \left( \dfrac { 1 }{ 2 } -\left( \dfrac { -1 }{ 2 }  \right)  \right) \\$$
$$=\dfrac { \sqrt { 3 } { \mu  }_{ 0 }I }{ \pi a } $$

335303_143368_ans.png

Physics
NCERT
Standard XII

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