JEE Advanced 2024 — Paper 1 — PHY-I solution

JEE Advanced2024Paper 1PHY-I

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 $\int B \cdot d l$ along a straight line from the point $(- 3 a, a, 0)$ to $(a, a, 0)$ is given by

[ $μ_{0}$ is the magnetic permeability of free space.]

A
$7 μ_{0} I /24$
B
$7 μ_{0} I /12$
C
$μ_{0} I /8$
D
$μ_{0} I /6$

Solution

The magnetic field of the long wire at any point in the $x$ - $y$ plane is azimuthal. For a path between two points, $\int B \cdot d l$ depends only on the change in azimuthal angle $Δ ϕ$ measured about the wire:

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

The point $(- 3 a, a, 0)$ has azimuthal angle $π - π /6 = 5 π /6$ , and $(a, a, 0)$ has azimuthal angle $π /4$ . The magnitude of the change is

$∣Δ ϕ ∣ = \frac{5 π}{6} - \frac{π}{4} = \frac{10 π - 3 π}{12} = \frac{7 π}{12}$

Therefore

$\int B \cdot d l = \frac{μ _{0} I}{2 π} \cdot \frac{7 π}{12} = \frac{7 μ _{0} I}{24}$

$∴$ Option (A) is correct.

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