Gravitational field at the centre of a semicircle formed by a thin wire AB of mass m and length l is

The correct option is 2πGm/l² along y axis.

Let us derive the gravitational field at the center of the semicircle step-by-step.
Consider a thin wire of mass _m and length _l bent into a semicircle. Let the center of the semicircle be at the origin (0,0) of a Cartesian coordinate system, and let the semicircle lie in the upper half-plane (y > 0) symmetric about the y-axis. The ends of the wire lie on the x-axis.

The length of the semicircular wire of radius _R is:
$l=\pi R$
Therefore, the radius _R of the semicircle is:
$R=\frac{l}{\pi }$

The mass per unit length (linear mass density _λ) of the wire is:
$\lambda =\frac{m}{l}$

Let us consider a small element of the wire subtending an angle $d\theta$ at the center, located at an angle $\theta$ with respect to the positive x-axis.
The length of this small element is $ds=Rd\theta$, and its mass $dm$ is:
$dm=\lambda ds=\lambda Rd\theta$

The gravitational field $dE$ at the center due to this mass element $dm$ is directed towards the element (along the radial direction at angle $\theta$):
$dE=\frac{Gdm}{R^2}=\frac{G\left(\lambda Rd\theta \right)}{R^2}=\frac{G\lambda }{R}$

Due to the symmetry of the semicircle about the y-axis, the horizontal components of the gravitational field ($d{E}_x=dEcos\theta$) from opposite halves of the semicircle cancel each other out. Thus, the net gravitational field along the x-axis is zero:
$E_x=0$

The vertical components ($d{E}_y=dEsin\theta$) reinforce each other and point in the positive y-direction (towards the wire since gravity is attractive).
Integrating $d{E}_y$ from $\theta =0$ to $\theta =\pi$:
$E_y={\int }_0^{\pi }dEsin\theta ={\int }_0^{\pi }\frac{G\lambda }{R}$

$E_y=\frac{G\lambda }{R}$

Substitute the values of $\lambda =\frac{m}{l}$ and $R=\frac{l}{\pi }$ into the equation:
$E_y=\frac{2G\left(\frac{m}{l}\right)}{\left(\frac{l}{\pi }\right)}=\frac{2\pi Gm}{l^2}$

Therefore, the net gravitational field at the center of the semicircle is 2πGm/l² along y axis.