A conducting circular loop is placed in a uniform magnetic field with its plane perpendicular to the field. The magnetic field is increasing at a constant rate of 0.4 T s ⁻¹ , while simultaneously the radius of the loop is shrinking at a constant rate of 5 mm s ⁻¹ . At the instant when the radius of the loop is 10 cm and the magnetic field is 1 T , the net induced emf in the loop is x mV . The value of x is _______.

Step-by-step solution

The magnetic flux through the loop is given by: = B A = B r^2 Since both the magnetic field B and the radius r are changing with time, we differentiate the flux with respect to time using the product rule: d dt = ( r^2 dB dt + 2rB dr dt ) From the given data: Instantaneous radius, r = 10 cm = 0.1 m Instantaneous magnetic field, B = 1 T Rate of change of magnetic field, dB dt = 0.4 T s ⁻¹ Rate of change of radius, dr dt = -5 mm s ⁻¹ = -0.005 m s ⁻¹ (negative because the radius is shrinking) Substituting the values i