Title: Part IB / Electromagnetism
Tags: Fundamentals

Q: Definition. All objects possess a property called electric charge Q s.t. there exists a force between 2 bodies at rest given by Coulomb's Law [...]
A: \[\mathbf{F} = \frac{Q_1 Q_2 (\mathbf{x_1 - x_2})}{4 \pi \epsilon_0 |\mathbf{x_1 - x_2} |^2}\]
Back Extra: Q measured in Coulombs
Opposites attract, like charges repel

Definition. Charge density given by \(\rho(t, \mathbf{x})\) and current density given by  \(\mathbf{J}(t, \mathbf{x}) \). Charge in volume dV centred at \(\mathbf{x}\) given by  \(\rho(t,\mathbf{x})dV \), charge crossing surface element \(d\mathbf{S}\) in time \(dt\) given by  \(\mathbf{J} \cdot d\mathbf{S} dt \).

Q: Definition. A test body - object with small charge and size - experiences a force given by Lorentz Force Law [...]
A: \[\mathbf{F} = q(\mathbf{E}(t, \mathbf{x}) + \mathbf{v} \times \mathbf{B}(t, \mathbf{x}))\]
Back Extra: E : electric field(vector field)
B : magnetic field (pseudovector field)

Q: Proposition. Conservation of Charge. If V is a time independent volume with boundary S, then the charge in V is \(Q(t) = \int_V \rho(t, \mathbf{x}) dV\) and the charge crossing S in time dt is \(I dt\). Charge conservation then tells us that [...]
A: \[- \frac{dQ}{dt} dt = I dt \implies \frac{dQ}{dt} = - \int_S \mathbf{J} \cdot dS\]
Back Extra: We can use the divergence theorem to get \[\frac{\partial \rho}{\partial t} + \nabla \cdot \mathbf{J} = 0\]