An infinitely long wire, located on the $z$-axis, carries a current $I$ along the +z-direction and produces the magnetic field $\overrightarrow{\mathrm{B}}$. The magnitude of the line integral $\int \overrightarrow{\mathrm{B}} . \overrightarrow{d l}$ along a straight line from the point $(-\sqrt{3} \mathrm{a}, \mathrm{a}, 0)$ to $(a, a, 0)$ is given by [ $\mu_{0}$ is the magnetic permeability of free space.]

JEE Advanced · 2024 · 26 May 2024 · Paper 1 · Magnetic Effects of Current

An infinitely long wire, located on the

z

-axis, carries a current

I

along the +z-direction and produces the magnetic field

B

. The magnitude of the line integral

\int B . d l

along a straight line from the point

(- 3 a, a, 0)

(a, a, 0)

is given by
[

μ_{0}

is the magnetic permeability of free space.]

Answer: (A)

The magnetic field at distance

r

from the wire is

∣ B ∣ = \frac{μ _{0} I}{2 π r}

. For a path element

∣ d ℓ ∣ = rd θ

, the line integral becomes

\int B \cdot d ℓ = \int \frac{μ _{0} I}{2 π} d θ

. From

(- 3 a, a, 0)

tan θ_{1} = 3

giving

θ_{1} = \frac{π}{3}

. From

(a, a, 0)

tan θ_{2} = 1

giving

θ_{2} = \frac{π}{4}

. The integral is

\frac{μ _{0} I}{2 π} [\frac{π}{3} + \frac{π}{4}] = \frac{7 μ _{0} I}{24}

JEE Advanced · 2024 · 26 May 2024 · Paper 1 · Single correct option