One of the most useful laws in magnetism
Highly applicable when there is a significant amount of symmetry.
Most common shapes are circles and rectangles, which are usually oriented such that the line integral only calls the B field “function” at a single input parameter (see APhO 2013 T1 last part).
$$ \textrm{Point charge: }\mathbf F= q(\mathbf E+\mathbf v \times \mathbf B) $$
$$ \textrm{Wire: }\mathbf F= \int I \, \mathrm{d} \mathbf s \times \mathbf B $$
When a conductor moves through an EM field, it gains a current density according to Ohm’s law. This can be analysed through Drude’s model of conduction (assuming charge carrier density is very high, such that drift velocity is negligible). At steady state, there is charge buildup which causes an electric field to be created, affecting current density. There is also a magnetic field created but this is usually ignored due to the complexity of self-induction. Due to these effects, the boundary conditions are usually solved using some uniqueness theorem shenanigans.
The “first-order” Lorentz force usually creates zero net force on the conductor due to the zero charge density (but not always. There can be a force on the charge buildup). However, there is a net force due to the current density created, because only the electrons flow in the current.
For a wire loop moving through a time-dependent magnetic field, the emf across the loop is always given as
$$ \mathcal E = -\frac{\mathrm d \Phi}{\mathrm d t} $$
Kind of like electric potential but for magnetism
$$ \renewcommand\vec\mathbf \vec B = \nabla \times \vec A \qquad \vec A = \frac{\mu_0}{4\pi}\int\frac{\vec J \, \mathrm{d}V}{r} $$
Subsituting the above equation yields the Biot-Savart law:
$$ \renewcommand\vec\mathbf \vec B= \frac{\mu_0}{4 \pi} \int\frac{I \mathrm{d} \vec s \times \hat{r}}{r^2} $$
$$ \mathbf m = NI \mathbf A $$