A tightly-wound, long solenoid is kept with its axis parallel to a large metal sheet carrying a surface current. The surface current through a width dl of the sheet is K , dl and the number of turns per unit length of the solenoid is n . The magnetic field near the centre of the solenoid is found to be zero. Determine the current in the solenoid.

Step-by-step solution

The magnetic field produced by a large metal sheet carrying a surface current density K is given by B_ sheet = ₀ K 2 . The magnetic field inside a long solenoid with n turns per unit length and carrying a current I is given by B_ solenoid = ₀ n I . Since the net magnetic field near the centre of the solenoid is zero, the magnetic field due to the solenoid must be equal and opposite to the magnetic field due to the metal sheet. Equating the magnitudes of the two magnetic fields: ₀ n I = ₀ K 2 I = K 2n