Electricity & Magnetism I — formula sheet
Every boxed result from the 20 derivations of this unit on one page, for revision. Each title links to the full derivation, with its assumptions and where the formula breaks down. Constants: physical constants table.
Electric Field from Coulomb's Law
Every element of source charge \(dq'=\rho\,d^3r'\) contributes a little Coulomb field pointing radially away from itself (for \(dq'>0\)), falling off as the inverse square of the distance; the total field is the vector superposition of all these contributions. The field is a property of the sources alone — the test charge has been divided out. A charge \(q\) dropped at \(\vec{r}\) then feels \(\vec{F}=q\vec{E}\).
Gauss's Law (Integral Form) from Coulomb
The net field lines threading out through any closed surface count only the charge inside it, in units of \( 1/\varepsilon_0 \). Deforming the surface, or moving external charges around, leaves the flux untouched — external charges and the shape of \( S \) cancel out exactly.
Differential Gauss's Law via the Divergence Theorem
At each point of space the divergence of the electric field — the net outward flux per unit volume in the limit of a vanishing region — equals the local charge density divided by \(\varepsilon_0\). Positive charge is a source from which field lines emanate; negative charge is a sink into which they converge; where \(\rho=0\) the field is divergence-free (though not necessarily zero). The field's sources sit exactly where the charge is.
Irrotationality of the Electrostatic Field
The three statements are equivalent ways of saying the electrostatic field is conservative. Because each Coulomb contribution is the gradient of an inverse-distance potential, and the curl annihilates any gradient, the total field has no circulation. The work done moving a test charge between two points is path-independent and defines a potential difference.
Scalar Potential, Poisson and Laplace Equations
The electric field is the negative slope of a scalar potential landscape. The curvature of that landscape, measured by the Laplacian, is fixed pointwise by the local charge density: positive charge makes \(\varphi\) locally peaked (\(\nabla^2\varphi<0\)), and where there is no charge the potential can have no interior maxima or minima — it is the smooth interpolation set entirely by the boundaries.
Uniqueness Theorem for Boundary-Value Problems
The energy-like quantity \( \int_V |\nabla W|^2\,dV \) — proportional to the field energy stored in the difference solution — is forced to zero by the shared source and shared boundary data. A non-negative quantity that sums to zero must be zero pointwise, so the two candidate solutions cannot differ in any physically observable way. Guessing a valid solution therefore is solving the problem.
Electrostatic Energy and Field Energy Density
The total electrostatic energy equals the volume integral of a strictly non-negative density \(\tfrac12\varepsilon_0 E^2\). Because every point where the field is nonzero carries energy — even far from any charge — energy is best regarded as stored in the field, not on the charges. Where the field is strong (near sharp conductor edges, between close capacitor plates) energy piles up quadratically. Note that \(\tfrac12\rho\varphi\) can be locally negative, yet its integral equals that of the manifestly non-negative \(\tfrac12\varepsilon_0 E^2\).
Fields at Conductor Surfaces
A conductor in equilibrium expels the field from its interior and pushes every excess charge onto its surface. Outside, the field emerges at exactly a right angle with strength set purely by the local surface charge density, and the whole conductor sits at one voltage. The single factor \(\sigma/\varepsilon_0\) — not \(\sigma/2\varepsilon_0\) — already contains the field the local sheet makes plus the field of all other charges, which cancel inside and add outside.
Multipole Expansion of the Potential
Far from any bounded static source the potential is an ordered sum: a Coulomb tail set by the net charge \(Q\); if \(Q=0\), a dipole field set by \(\vec p=\int\vec r'\rho\,d^3r'\); if \(\vec p=0\) too, a quadrupole field set by the tensor \(Q_{ij}\); and so on. The leading non-vanishing moment controls the far field and, uniquely, is independent of where the origin is placed.
Dielectrics, Bound Charge and the D Field
Where the polarization "piles up" (nonzero divergence), positive charge has been pushed out faster than it arrived, leaving a net negative bound charge — hence the minus sign. At a surface, \(\vec{P}\cdot\hat{n}\) counts the dipole heads left stranded on the boundary. The displacement \(\vec{D}\) is the specific combination of the true field \(\vec{E}\) and the material response \(\vec{P}\) whose flux through any closed surface counts only the free charge you placed — the bound charge cancels internally.
Continuity Equation and Charge Conservation
At each point, the divergence of the current density (the net outflow of charge per unit volume) is exactly balanced by the local rate of decrease of charge density. Where charge is piling up (\(\partial_t\rho>0\)) the current lines must be converging (\(\nabla\cdot\mathbf{J}<0\)); charge never simply appears — every local increase is fed by current arriving from neighbouring points.
Biot–Savart Law from the Current Force Law
Each steady current element sources a field that (i) falls off as \(1/s^2\), the same inverse-square as Coulomb; (ii) is transverse — the cross product forces \(d\vec B\) perpendicular to both the current direction \(d\vec l'\) and the line of sight \(\hat s\), so the field circulates around the wire rather than pointing along it; and (iii) scales with \(\mu_0/4\pi\), the constant carrying the experimental magnitude of the current force law. Its size also carries a factor \(\sin\theta\), the angle between \(d\vec l'\) and \(\hat s\). The element form has physical meaning only summed around a closed circuit.
Absence of Magnetic Monopoles
The magnetic field of any steady current has zero divergence at every point. By the divergence theorem this is equivalent to the statement that the net magnetic flux through any closed surface is zero: whatever flux enters a closed surface must leave it. Field lines have no sources or sinks — they close on themselves. There is no magnetic charge for \( \mathbf{B} \) to diverge from, i.e. no magnetic monopole.
Ampère's Circuital Law from Biot–Savart
The magnetostatic field circulates around current: its curl at a point is set entirely by the local current density there, and the net circulation around any loop counts only the current piercing that loop, weighted by \(\mu_0\). Regions with no current carry a curl-free (but not necessarily zero) field.
Magnetic Vector Potential and Gauge Freedom
The field \(\mathbf{A}\) is a "reservoir" whose curl is the physical \(\mathbf{B}\); only the gauge-invariant curl is measurable, and any two \(\mathbf{A}\)'s differing by a gradient describe the same magnetism. Fixing the redundancy with \(\nabla\cdot\mathbf{A}=0\) makes each Cartesian component of \(\mathbf{A}\) obey exactly the electrostatic Poisson equation with \(\mu_0 J_i\) playing the role of \(\rho/\varepsilon_0\). The immediate solution with the Coulomb Green's function is \(\displaystyle \mathbf{A}(\mathbf{r})=\frac{\mu_0}{4\pi}\int\frac{\mathbf{J}(\mathbf{r}')}{|\mathbf{r}-\mathbf{r}'|}\,dV'\).
Magnetic Dipole: Field, Torque and Energy
Lorentz Force and Cyclotron Motion
A charge feels an electric force along \( \vec E \) that does work, plus a magnetic force perpendicular to both \( \vec v \) and \( \vec B \) that does none. In a uniform \( \vec B \) the path is a helix: uniform glide along the field at \( v_\parallel \), circular gyration across it at frequency \( \omega_c \) (independent of speed) and radius \( r_c \) (set by the transverse momentum \( mv_\perp \)). A perpendicular \( \vec E \) adds a uniform \( \vec E\times\vec B \) drift, identical for every charge regardless of sign or mass, so it drives no net current.
Magnetized Matter, Bound Currents and H
Magnetization that curls in the interior (\(\nabla\times\vec{M}\neq0\)) is indistinguishable from a real volume current; magnetization that terminates at a surface (\(\vec{M}\times\hat{n}\neq0\)) acts as a surface sheet of current, exactly like the wound surface of a solenoid. The field \(\vec{H}\) is engineered to subtract the bound part, so its curl "sees" only the current we put in by hand.
Ohm's Law, Resistance and Kirchhoff's Rules for DC Circuits
Ohm's law relates voltage, current, and resistance in a linear, proportional way. Resistance depends on the material (\(\rho\)) and geometry (\(L/A\)). Power dissipation is quadratic in current or inversely proportional to resistance for a fixed voltage. Kirchhoff's rules provide the conservation laws (charge and energy) that allow solving for currents and voltages in complex networks.
The RC Circuit: Charging, Discharging and the Time Constant
The charge, current, and capacitor voltage all evolve exponentially with time constant \(\tau = RC\). The current starts at \(\mathcal{E}/R\) and decays to zero, while the capacitor voltage rises from zero to \(\mathcal{E}\). The time constant \(\tau\) is the time required for the charge to reach \(63.2\%\) of its final value.