MHTCET 16 May 2024 Shift 2 Paper

Concept:The magnetic field at the centre of a circular arc is directly proportional to the angle subtended by the arc at the centre.Explanation:For a circular arc of radius $R$ carrying current $I$, the magnetic field at the centre is given by$$B = \frac{\mu_0 I \theta}{4 \pi R}$$where $\theta$ is the angle subtended by the arc in radians.The given arc is $\frac{3}{4}$ of the full circumference of the circle.Since a full circle subtends an angle of $2 \pi$ radians, the angle subtended by this arc is$$\theta = \frac{3}{4} \times 2 \pi = \frac{3 \pi}{2}$$Substituting this value into the formula,$$B = \frac{\mu_0 I \left(\frac{3 \pi}{2}\right)}{4 \pi R} = \frac{3 \mu_0 I}{8 R}$$Now, determine the direction using the right-hand rule.When the current flows in the anticlockwise direction, curl the fingers of your right hand along the current path.Your thumb then points upward, which is out of the plane of the circle.Therefore, the magnetic field at the centre is directed upward.Answer:The magnetic field at the centre is $\frac{3 \mu_0 I}{8 R}$ in the upward direction.Hence, the correct option is Option D.