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Question

Two wires of same length are shaped into square and a circle. if they carry same current ratio of magnetic moment is


A
2:π
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B
π:3
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C
π:4
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D
1:π
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Solution

The correct option is A $$\displaystyle \pi :4$$
Let length of wire be $$l$$. For a square loop, side of square $$\displaystyle \frac { l }{ 4 } $$. 
Hence its magnetic moment
$$\displaystyle { M }_{ sq } = iA=i{ \left[ \frac { l }{ 4 }  \right]  }^{ 2 }=\frac { i{ l }^{ 2 } }{ 16 } $$
Radius of a circular loop $$\displaystyle r=\frac { l }{ 2\pi  } $$.
Hence, its magnetic moment
$$\displaystyle { M }_{ cir }=iA=i\pi { r }^{ 2 }$$
$$\displaystyle M_{cir}=i\pi \left[ \frac { { l }^{ 2 } }{ 4{ \pi  }^{ 2 } }  \right] =\frac { i{ l }^{ 2 } }{ 4\pi  } $$
$$\displaystyle \Rightarrow \quad \frac { { M }_{ sq } }{ { M }_{ cir } } =\frac { \pi  }{ 4 } $$

Physics

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