Question : A current i is flowing in a hexagonal coil of side l. The magnetic induction at the centre of the coil will be
a) √3µ₀i/πl
b) µ₀i/√3πl
c) µ₀i/3√3πl
d) 3√3µ₀i/πl
Doubt by Sonali
Ans : a) is the correct option.
Solution :
Total Magnetic field at the centre
= 6×Magnetic field due to a single segment of wire at the centre
Total Magnetic field at the centre
= 6×Magnetic field due to a single segment of wire at the centre
All side of hexagon are equal to 'l' so this must be a regular hexagon.
Here, central angle = 360°/6 = 60°
AM = BM = l/2
In Rt. ΔOMA
tan30° = P/B = AM/OA
tan30° = P/B = AM/OA
1/√3 = (l/2)/a
1/√3 = l/2a
a = √3l/2
1/√3 = l/2a
a = √3l/2
Magnetic field due to a small segment of straight current carrying wire
B = µ₀I(sinΦ1 + sinΦ2)/4πa
B = µ₀i(sin30° + sin30°)/4π(√3l/2)
B = µ₀i(1/2 + 1/2)/2√3lπ
B = µ₀i/2√3lπ
B = µ₀i(1/2 + 1/2)/2√3lπ
B = µ₀i/2√3lπ
Total Magnetic field at the centre due to all the segments
= 6×µ₀i/2√3lπ
=3µ₀i/√3lπ
= √3µ₀i/lπ
= √3µ₀i/πl
Hence, a) would be the correct option.
= 6×µ₀i/2√3lπ
=3µ₀i/√3lπ
= √3µ₀i/lπ
= √3µ₀i/πl
Hence, a) would be the correct option.
