A n infinitely long wire, located on the 𝑧-axis, carries a current 𝐼 along the +𝑧-direction and produces the magnetic field $\stackrel{\to <!--\; \to \; -->}{B}$. The magnitude of the line integral $\int <!--\; \int \; -->\stackrel{\to <!--\; \to \; -->}{B}\cdot <!--\; \cdot \; -->\stackrel{\to <!--\; \to \; -->}{dl}$ along a straight line from the point (−√3𝑎, 𝑎, 0) to (𝑎, 𝑎, 0) is given by

[${\mathrm{\mu <!--\; \mu \; -->}}_0$ is the magnetic permeability of free space.]

The correct option is:
7𝜇₀𝐼/24

Step-by-Step Explanation:

1. Understand the Geometry and Magnetic Field:
We have an infinitely long wire along the $z$-axis carrying a current $I$ in the $+z$-direction.
The magnetic field $\overrightarrow{B}$ produced by this wire at any point in the $xy$-plane (since $z=0$ for the path) is given in cylindrical coordinates by:
$\overrightarrow{B}=\frac{{\mu }_{0}I}{2\pi r}\hat{\varphi }$
where $r=\sqrt{x^2+y^2}$ is the perpendicular distance from the $z$-axis, and $\hat{\varphi }=-sin\varphi \hat{i}+cos\varphi \hat{j}$ is the azimuthal unit vector.

2. Express the Line Integral:
We need to calculate the line integral $\int \overrightarrow{B}\cdot \overrightarrow{dl}$ along a straight line path from the point $(-\sqrt{3}a,a,0)$ to $(a,a,0)$.
Along this path:
- The $y$-coordinate is constant: $y=a$, which means $dy=0$.
- The $z$-coordinate is constant: $z=0$, which means $dz=0$.
- The path differential vector is $\overrightarrow{dl}=dx\hat{i}$.
In Cartesian coordinates, the magnetic field is:
$\overrightarrow{B}=\frac{{\mu }_{0}I}{2\pi (x^2+y^2)}(-y\hat{i}+x\hat{j})$
Taking the dot product:
$\overrightarrow{B}\cdot \overrightarrow{dl}=-\frac{{\mu }_{0}Iy}{2\pi (x^2+y^2)}dx$

3. Integrate with Respect to $x$:
Substitute $y=a$ into the integrand:
${\int }_{x_i}^{x_f}\overrightarrow{B}\cdot \overrightarrow{dl}=-\frac{{\mu }_{0}Ia}{2\pi }$
Using the standard integration formula $\int \frac{dx}{x^2+a^2}=\frac{1}{a}{tan}^{-1}\left(\frac{x}{a}\right)$:
${\int }_{-\sqrt{3}a}^a\overrightarrow{B}\cdot \overrightarrow{dl}=-\frac{{\mu }_{0}Ia}{2\pi }$
$=-\frac{{\mu }_{0}I}{2\pi }$

4. Evaluate the Trigonometric Terms:
- ${tan}^{-1}\left(1\right)=\frac{\pi }{4}$
- ${tan}^{-1}(-\sqrt{3})=-\frac{\pi }{3}$
Substituting these values back in:
$\int \overrightarrow{B}\cdot \overrightarrow{dl}=-\frac{{\mu }_{0}I}{2\pi }$

5. Find the Magnitude:
The question asks for the magnitude of the line integral:
$|\int \overrightarrow{B}\cdot \overrightarrow{dl}|=\frac{7{\mu }_{0}I}{24}$