An infinitely long wire, located on the $z$-axis, carries a current $I$ along the $+z$-direction and produces the magnetic field $\vec{B}$. The magnitude of the line integral $\int \vec{B} \cdot \overrightarrow{d l}$ along a straight line from the point $(-\sqrt{3} a, a, 0)$ to $(a, a, 0)$ is…

Question

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

Aicharya · Accepted Answer

$\begin{aligned} & \Rightarrow|\overrightarrow{\mathrm{d} \ell}|=\mathrm{rd} 	heta \ & \Rightarrow|\overrightarrow{\mathrm{B}}|=\frac{\mu_0 \mathrm{I}}{2 \pi \mathrm{r}} \ & \Rightarrow \int \overrightarrow{\mathrm{B}} \cdot \overrightarrow{\mathrm{d} \ell}=\int|\overrightarrow{\mathrm{B}}||\overrightarrow{\mathrm{d} \ell}| \cos 0^{\circ} \ & =\int\left(\frac{\mu_0 \mathrm{I}}{2 \pi \mathrm{r}}ight) 	imes(\operatorname{rd} 	heta) \ & =\int_{	heta_1}^{	heta_2} \frac{\mu_0 \mathrm{I}}{2 \pi} \mathrm{d} 	heta=\frac{\mu_0 \mathrm{I}}{2 \pi}\left[	heta_2-\left(-	heta_1ight)ight]\end{aligned}$ [ $	heta_1$ is anticlockwise hence taken negative] $\begin{aligned} & \Rightarrow 	an 	heta_1=\frac{\mathrm{a} \sqrt{3}}{\mathrm{a}}=\sqrt{3} \ & 	heta_1=\frac{\pi}{3} \ & \Rightarrow 	an 	heta_2=\frac{\mathrm{a}}{\mathrm{a}}=1 \ & 	heta_2=\frac{\pi}{4} \ & \Rightarrow \int \mathrm{B} \cdot \mathrm{d} \ell=\frac{\mu_0 \mathrm{I}}{2 \pi}\left[\frac{\pi}{3}+\frac{\pi}{4}ight] \ & =\frac{7 \mu_0 \mathrm{I}}{24} \ & \Rightarrow 	ext { Ans. Option (1) }\end{aligned}$

An infinitely long wire, located on the $z$-axis, carries a current $I$ along the $+z$-direction and…

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