= Solution
For a stationary closed circuit, <Faraday's law of induction> is
$$
\boxed{\mathcal E=\oint\mathbf E\cdot d\boldsymbol\ell=-\frac{d\Phi_B}{dt},\qquad
\nabla\times\mathbf E=-\frac{\partial\mathbf B}{\partial t}.}
$$
The loop orientation and magnetic-flux normal obey the right-hand convention. For a moving material circuit, the emf is instead $\oint(\mathbf E+\mathbf v\times\mathbf B)\cdot d\boldsymbol\ell$, 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>. <University Physics Volume 2 by Samuel J. Ling and others>[Chapter13, Faraday induction, Lenz direction and motional emf]{ref}
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