 Solution

For a stationary closed circuit, Faraday's law of induction isThe loop orientation and magnetic-flux normal obey the right-hand convention. For a moving material circuit, the emf is instead $\oint (\mathbf{E}+\mathbf{v}\times \mathbf{B})\cdot d\mathbf{\ell }$, with the same negative derivative of the flux through the moving circuit. The additional magnetic force term accounts for motional emf.

Lenz's law expresses the minus sign physically: an induced current opposes the change in flux that produces it. It does not always oppose the field itself. Increasing positive flux drives a current producing negative flux; decreasing it reverses that response. A straight conducting rod of length l moving across a perpendicular uniform field has emf magnitude Blv. A resistive return circuit then develops a magnetic braking force, and the supplied mechanical power equals Joule heating in the ideal steady limit. This connects induction with conservation of energy. ^[ref]