A regular polygon of 6 sides is formed by bending a wire of length \(4 \pi\) meter. If an electric current of \(4 \pi \sqrt{3}\) A is flowing through the sides of the polygon, the magnetic field at the centre of the polygon would be \(\mathrm{x} \times 10^{-7} \mathrm{~T}\). The value of x is ______.

JEE Main · 2024 · 01 Feb 2024 · Shift 1 · Magnetic Effects of Current

A regular polygon of 6 sides is formed by bending a wire of length

4 π

meter. If an electric current of

4 π 3

A is flowing through the sides of the polygon, the magnetic field at the centre of the polygon would be

x \times 1 0^{- 7} T

. The value of x is ______.

Answer: 72

A regular hexagon of perimeter

4 π

m has side

a = \frac{4 π}{6} = \frac{2 π}{3}

m. Each side is at distance

r = \frac{a}{2 tan 3 0 ^{\circ}} = \frac{a 3}{2} = \frac{π}{3}

m from the centre and subtends

\pm 3 0^{\circ}

there, so the field at the centre is

B = 6 \times \frac{μ _{0} I}{4 π r} (sin 3 0^{\circ} + sin 3 0^{\circ}) = \frac{6 μ _{0} I}{4 π r} .

With

\frac{μ _{0}}{4 π} = 1 0^{- 7}

I = 4 π 3

A and

r = \frac{π}{3}

B = 6 \times 1 0^{- 7} \times \frac{4 π 3}{π / 3} = 6 \times 1 0^{- 7} \times 12 = 72 \times 1 0^{- 7} T,

x = 72

JEE Main · 2024 · 01 Feb 2024 · Shift 1 · Integer answer