In the figure, there is a conducting wire having current i and which has a shape of closed half ellipse [x2a2+y2 b2=1] is kept in a uniform magnetic field B as shown. The magnitude of magnetic dipole moment of loop and torque acting on it are-

To find the magnitude of the magnetic dipole moment of the loop, we can use the formula:

μ = iA

where μ is the magnetic dipole moment, i is the current flowing through the loop, and A is the area enclosed by the loop.

In this case, the loop has the shape of a closed half ellipse, given by the equation x^2/a^2 + y^2/b^2 = 1. To find the area enclosed by the loop, we can integrate the equation over the appropriate limits.

Let's assume that the loop lies in the xy-plane, with the major axis along the x-axis and the minor axis along the y-axis. The limits of integration for y will be -b to b, and for x, we can solve the equation for x in terms of y to get the limits.

Solving the equation x^2/a^2 + y^2/b^2 = 1 for x, we get:

x = ± a * sqrt(1 - y^2/b^2)

So the limits of integration for x will be -a * sqrt(1 - y^2/b^2) to a * sqrt(1 - y^2/b^2).

Now, we can calculate the area enclosed by the loop by integrating the equation over the given limits:

A = ∫[from -b to b] ∫[from -a * sqrt(1 - y^2/b^2) to a * sqrt(1 - y^2/b^2)] dx dy

Once we have the value of A, we can calculate the magnitude of the magnetic dipole moment using the formula μ = iA.

To find the torque acting on the loop, we can use the formula:

τ = μ x B

where τ is the torque, μ is the magnetic dipole moment, and B is the uniform magnetic field.

By taking the cross product of μ and B, we can find the direction and magnitude of the torque acting on the loop.