The magnetic field at the centre of a wire loop formed by two semicircular wires of radii \(\mathrm{R}_{1}=2 \pi \mathrm{~m}\) and \(R_{2}=4 \pi \mathrm{~m}\) carrying current \(\mathrm{I}=4 \mathrm{~A}\) as per figure given below is \(\alpha \times 10^{-7} \mathrm{~T}\). The value of \(\alpha\) is \_\_\_\_. (Centre O is common for all segments)

JEE Main · 2024 · 27 Jan 2024 · Shift 2 · Magnetic Effects of Current

The magnetic field at the centre of a wire loop formed by two semicircular wires of radii

R_{1} = 2 π m

and

R_{2} = 4 π m

carrying current

I = 4 A

as per figure given below is

α \times 1 0^{- 7} T

. The value of

α

is
____. (Centre O is common for all segments)

Answer: 3

The magnetic field at the center due to a semicircular arc of radius

R

carrying current

I

is half that of a full loop:

B = \frac{μ _{0} I}{4 R}

. Both semicircles here produce field in the same direction at the common center O, so their contributions add:

B = \frac{μ _{0} I}{4} (\frac{1}{R _{1}} + \frac{1}{R _{2}}) = \frac{( 4 π \times 1 0 ^{- 7} ) ( 4 )}{4} (\frac{1}{2 π} + \frac{1}{4 π}) = 4 π \times 1 0^{- 7} \times \frac{3}{4 π} = 3 \times 1 0^{- 7} T .

α = 3

JEE Main · 2024 · 27 Jan 2024 · Shift 2 · Integer answer