A loop ABCDA , carrying current $\mathrm{I}=12 \mathrm{~A}$, is placed in a plane, consists of two semi-circular segments of radius $\mathrm{R}_{1}=6 \pi \mathrm{~m}$ and $\mathrm{R}_{2}=4 \pi \mathrm{~m}$. The magnitude of the resultant magnetic field at center O is $\mathrm{k} \times 10^{-7} \mathrm{~T}$. The value of k is ______。 (Given $\mu_{0}=4 \pi \times 10^{-7} \mathrm{Tm} \mathrm{A}^{-1}$ )

JEE Main · 2025 · 03 Apr 2025 · Shift 1 · Magnetic Effects of Current

A loop ABCDA , carrying current

I = 12 A

, is placed in a plane, consists of two semi-circular segments of radius

R_{1} = 6 π m

and

R_{2} = 4 π m

. The magnitude of the resultant magnetic field at center O is

k \times 1 0^{- 7} T

. The value of k is ______。
(Given

μ_{0} = 4 π \times 1 0^{- 7} Tm A^{- 1}

)

Answer: 1

The two straight radial segments

A B

and

C D

lie along lines through the centre, so they contribute no field at

O

. Each semicircular arc of radius

R

carrying current

I

produces a field of magnitude

\frac{μ _{0} I}{4 R}

at the centre, and the inner and outer arcs carry current in opposite senses about

O

, so their fields subtract:

B_{O} = \frac{μ _{0} I}{4} (\frac{1}{R _{2}} - \frac{1}{R _{1}}) = \frac{4 π \times 1 0 ^{- 7} \times 12}{4} (\frac{1}{4 π} - \frac{1}{6 π}) .

The bracket equals

\frac{1}{12 π}

, so

B_{O} = 12 π \times 1 0^{- 7} \times \frac{1}{12 π} = 1 \times 1 0^{- 7} T .

The value of

k

1

JEE Main · 2025 · 03 Apr 2025 · Shift 1 · Integer answer