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JEE Advanced 2024Paper 1
Exam
JEE Advanced
Year
2024
Paper
Paper 1
Subject
Physics
Chapter
Magnetic Effects of Current

JEE Advanced 2024 · Paper 1 · Physics · Magnetic Effects of Current

An infinitely long wire, located on the z -axis, carries a current...

An infinitely long wire, located on the zz-axis, carries a current II along the +z+z-direction and produces the magnetic field B⃗\vec{B}. The magnitude of the line integral ∫B⃗⋅dl→\int \vec{B} \cdot \overrightarrow{d l} along a straight line from the point (−3a,a,0)(-\sqrt{3} a, a, 0) to (a,a,0)(a, a, 0) is given by [ μ0\mu_0 is the magnetic permeability of free space.]

Options

  1. A7μ0I/247 \mu_0 I / 24
  2. B7μ0I/127 \mu_0 I / 12
  3. Cμ0I/8\mu_0 I / 8
  4. Dμ0I/6\mu_0 I / 6

Correct answer

A. 7μ0I/247 \mu_0 I / 24

Step-by-step solution

⇒∣dℓ→∣=rdθ⇒∣B→∣=μ0I2πr⇒∫B→⋅dℓ→=∫∣B→∣∣dℓ→∣cos⁡0∘=∫(μ0I2πr)×(rd⁡θ)=∫θ1θ2μ0I2πdθ=μ0I2π[θ2−(−θ1)]\begin{aligned} & \Rightarrow|\overrightarrow{\mathrm{d} \ell}|=\mathrm{rd} \theta \\ & \Rightarrow|\overrightarrow{\mathrm{B}}|=\frac{\mu_0 \mathrm{I}}{2 \pi \mathrm{r}} \\ & \Rightarrow \int \overrightarrow{\mathrm{B}} \cdot \overrightarrow{\mathrm{d} \ell}=\int|\overrightarrow{\mathrm{B}}||\overrightarrow{\mathrm{d} \ell}| \cos 0^{\circ} \\ & =\int\left(\frac{\mu_0 \mathrm{I}}{2 \pi \mathrm{r}}\right) \times(\operatorname{rd} \theta) \\ & =\int_{\theta_1}^{\theta_2} \frac{\mu_0 \mathrm{I}}{2 \pi} \mathrm{d} \theta=\frac{\mu_0 \mathrm{I}}{2 \pi}\left[\theta_2-\left(-\theta_1\right)\right]\end{aligned} [ θ1\theta_1 is anticlockwise hence taken negative] ⇒tan⁡θ1=a3a=3θ1=π3⇒tan⁡θ2=aa=1θ2=π4⇒∫B⋅dℓ=μ0I2π[π3+π4]=7μ0I24⇒ Ans. Option (1) \begin{aligned} & \Rightarrow \tan \theta_1=\frac{\mathrm{a} \sqrt{3}}{\mathrm{a}}=\sqrt{3} \\ & \theta_1=\frac{\pi}{3} \\ & \Rightarrow \tan \theta_2=\frac{\mathrm{a}}{\mathrm{a}}=1 \\ & \theta_2=\frac{\pi}{4} \\ & \Rightarrow \int \mathrm{B} \cdot \mathrm{d} \ell=\frac{\mu_0 \mathrm{I}}{2 \pi}\left[\frac{\pi}{3}+\frac{\pi}{4}\right] \\ & =\frac{7 \mu_0 \mathrm{I}}{24} \\ & \Rightarrow \text{ Ans. Option (1) }\end{aligned}
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