In a certain region, a hypothetical magnetic field is described by B = B₀ e_r , where e_r represents the unit vector in the radial direction. A circular loop of radius a carrying a current i is positioned such that its plane is parallel to the x - y plane, with its centre located at (0, 0, d) . Determine the magnitude of the magnetic force exerted on the loop.

Step-by-step solution

The position vector of any point on the circular loop is given by: r = a i + a j + d k The distance of any point on the loop from the origin is: r = | r | = a^2 + d^2 The unit vector in the radial direction is e_r = r r . Thus, the magnetic field at the loop is: B = B₀ e_r = B₀ a^2 + d^2 (a i + a j + d k ) The current element vector d l along the loop is: d l = d r = (-a i + a j ) d The magnetic force on this small current element is d F = i(d l B ) . Calculating the cross product: d l B = B₀ d a^2 + d^2 vmatrix i