Magnetic Lorentz Force: Moving charges in magnetic fields experience $\vec{F} = q(\vec{v}\times\vec{B})$. For current-carrying conductors of length $L$: $\vec{F} = I(\vec{L}\times\vec{B})$.
Biot-Savart & Solenoids: Magnetic fields generated by currents satisfy $d\vec{B} = \frac{\mu_0 I}{4\pi}\frac{d\vec{\ell}\times\hat{r}}{r^2}$. Inside an ideal solenoid, field strength is $B = \mu_0 n I$.
Faraday's & Lenz's Laws: Induced electromotive force equals the negative time rate of magnetic flux change: $\mathcal{E} = -\frac{d\Phi_B}{dt} = -\frac{d}{dt}(BA\cos\theta)$.
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