First slide

Magnetic field due to current element

Question

Current i is flowing in hexagonal coil of side a. The magnetic induction at the center of the coil will be

Moderate

A

$3 \sqrt{3} \frac{μ_{o} i}{π a}$
B

$\frac{μ_{o} i}{3 \sqrt{3} π a}$
C

$\frac{μ_{o} i}{\sqrt{3} π a}$
D

$\frac{\sqrt{3} μ_{o} i}{π a}$

Solution

$F i e l d d u e t o f i n i t e w i r e, B_{A B} = \frac{μ_{o} i}{4 π x} (S i n 30^{0} + S i n 30^{0}) = \frac{μ_{o} i}{4 π x} (\frac{1}{2} + \frac{1}{2}) = \frac{μ_{o} i}{4 π (\frac{\sqrt{3} a}{2})} = \frac{μ_{o} i}{\sqrt{3} (2 π a)} (∵ O A = a, △ A O B i s e q u i l a t e r a l) ∴ T o t a l m a g n e t i c i n d u c t i o n d u e t o a l l t h e s i d e s a t O, B = 6 B_{A B} = 6 \frac{μ_{o} i}{2 \sqrt{3} π a} = \sqrt{3} \frac{μ_{o} i}{π a}$

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