Lenz's Law and the Alternating-Current Generator

Faraday's law gives the emf from changing flux linkage. Lenz's law gives its direction. Push a north pole toward a closed resistive coil: its near face becomes north, opposing the approach. Viewed from the magnet, the conventional current is counterclockwise. Curl your right fingers with the current; your thumb points toward the coil's north face.

With a resistive load, moving the magnet requires work against the induced magnetic force. That work supplies electrical energy and heat. If induction helped the motion while supplying electrical energy, the system could gain energy without an input. The minus sign in Faraday's law expresses this opposition.

For an ideal AC generator, a planar coil rotates at constant angular speed omega about an axis in its plane and perpendicular to uniform B. N is the turn count and A is the area. With its normal along B at t = 0, linkage is N B A cos(omega t). The derivative is the graph slope: d cos(omega t)/dt = -omega sin(omega t). Faraday's minus sign gives emf N B A omega sin(omega t). One revolution gives one cycle, so f = omega / (2 pi).

For steady rotation with the axis in the coil plane and perpendicular to uniform B, peak emf is N B A omega. Increasing any one factor while holding the others fixed raises the peak. N counts the coil's turns, not its revolutions per second. Increasing N raises the peak but does not change the frequency when the rotation rate is fixed.

Now withdraw the north pole from the same closed resistive coil. The magnetic flux decreases. The near face becomes south and attracts the departing north pole, opposing its withdrawal. Viewed from the magnet, the conventional current is clockwise. Induction opposes the change in flux, not always the existing field.

Keep the rotation axis in the coil plane and perpendicular to uniform B. If the constant angular speed doubles between two steady runs while N, A and B stay fixed, f = omega / (2 pi) doubles and peak emf N B A omega doubles. The cycles take half as long. These relations describe induced emf; the load's properties also affect current.