Lenz's Law and Faraday's Law

The minus sign in $e = -\frac{d\Phi}{dt}$ is not a mere convention -- it is Faraday's law's built-in encoding of Lenz's law. To see this, fix the loop's area vector $\vec{A}$ (perpendicular to the loop's plane) so that it points parallel to $\vec{B}$ (i.e. the angle between them is zero), and suppose $\vec{B}$ is increasing with time. Then $\Phi = \vec{B}\cdot\vec{A} = |B||A|$, and since $|A|$ is fixed and positive while $\frac{dB}{dt} > 0$, the quantity $\frac{d\Phi}{dt} = |A|\frac{dB}{dt}$ is positive, so the right-hand side $-\frac{d\Phi}{dt}$ of the equation is negative. …