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For planar charge rotating rigidly about its centre, $\mathbf{v}=\mathbf{\Omega }\times \mathbf{r}$. The magnetic contribution at the centre is $dB={\mu }_0\Omega dq/(4\pi r)$, along the rotation axis. Integrate over the actual charge distribution; charges on a rotating polygon do not follow the sides of a stationary wire circuit.

Charge invariance means that the total electric charge of a specified collection of particles is unchanged by a Lorentz transformation. It differs from conservation of electric charge, which compares the same isolated system at different times. For a convective charge distribution with rest density ${\rho }_0$, the four-current is $J^{\mu }={\rho }_0{U}^{\mu }$, where $U^{\mu }$ is its four-velocity; its measured charge density increases under a boost while the simultaneous spatial volume contracts, leaving total charge unchanged.

Christian Møller and Léon Rosenfeld used the electric quadrupole moment of the deuteron to constrain nuclear forces. An isotropic charge distribution has no preferred axis and therefore zero quadrupole moment. A nonzero value requires an internal structure or quantum state that produces an anisotropic charge pattern. Their 1939 work connected this measured electrical property with orientation-dependent forces between the proton and neutron, illustrating how a small static observable can test a microscopic interaction model.^[ref]

The Coulomb force is the electrostatic force between electric charges. For point charges in a uniform isotropic medium of permittivity $ϵ$, the force on charge $q_2$ from $q_1$ is ${\mathbf{F}}_{2\leftarrow 1}=q_1{q}_2{\hat{\mathbf{r}}}_{12}/(4\pi ϵr^2)$, where ${\hat{\mathbf{r}}}_{12}$ points from the first charge to the second.

Equal charge signs give repulsion and opposite signs give attraction. The inverse-square dependence allows superposition to construct fields of extended charge distributions. Nearby conductors or spatially varying dielectrics can redistribute charge and alter the field, so the simple uniform-medium pair formula does not automatically solve those boundary problems.

For a localized charge distribution, the far electric potential is an inverse-radius expansion whose first terms are the total charge monopole, the electric dipole moment, and the electric quadrupole moment.

An electric quadrupole is a charge distribution whose total charge and electric dipole moment vanish while its second charge moment does not. Its potential decays as $r^{-3}$.

The trace-free electric quadrupole moment is a symmetric tensor measuring the second-order term in an electric multipole expansion. For a localized charge distribution its contribution to the far-field electrostatic potential is $Q_{ij}{\hat{r}}_i{\hat{r}}_j/(8\pi {ϵ}_0{r}^3)$, with summation over indices. The moment may be nonzero even when the total electric charge or electric dipole moment is nonzero; it should not be confused with an electric quadrupole whose lower moments vanish. Changing the origin generally changes the tensor when lower moments are nonzero.

Léon Rosenfeld and Christian Møller related the observed electric quadrupole moment of the deuteron to orientation-dependent nuclear forces. A purely spherical charge distribution gives zero quadrupole moment, because averaging $3z^2-r^2$ over directions cancels. A measured nonzero value therefore constrains the internal quantum state and the angular dependence of the interaction. Their 1939 work used this small electrical observable to test descriptions of the force between a proton and a neutron.^[ref]

Conductors with net charges of the same sign can attract because their internal charge distributions respond to one another. A point-charge calculation misses this electrostatic induction contribution. Exact conducting-sphere results are given in Electrostatics of Two Charged Conducting Spheres by John Lekner.^[ref]

A line charge is a charge distribution idealized as lying on a curve, described by charge per unit length. An infinite straight line with constant density $\mu$ produces the radial electric field $E_r=\mu /(2\pi {\epsilon }_{0}r)$ by Gauss's law, and a logarithmic electrostatic potential.

A circular charge distribution rotating with angular speed $\omega$ can be decomposed into current loops. A hoop of line density $\eta$ and radius $R$ has $m=\pi \eta \omega R^3\hat{n}$; a disc of surface density $\sigma$ has $m=\pi \sigma \omega R^4\hat{n}/4$.

A uniformly charged solid sphere of total charge $Q$, radius $R$ and angular velocity $\mathbf{\omega }$ has magnetic dipole momentFor a uniformly charged thin spherical shell the coefficient is $1/3$ instead of $1/5$. At large distances the magnetic field is a magnetic dipole field, $\mathbf{B}={\mu }_0[3(\mathbf{m}\cdot \hat{\mathbf{r}})\hat{\mathbf{r}}-\mathbf{m}]/(4\pi r^3)$. Its angularly averaged magnitude and root mean square magnitude both scale as $r^{-3}$; its angularly averaged vector is zero. Steady rotation of this axisymmetric charge distribution produces magnetostatics, not radiation.

The space ${\mathbb{R}}^3$ is simply connected, and the electric field has zero curl. Hence it is a conservative vector field. More explicitly, Stokes theorem gives zero circulation around any closed loop, since such a loop bounds a spanning surface in ${\mathbb{R}}^3$. Therefore the line integral between two fixed endpoints is independent of the chosen path.

Fix a reference point ${\mathbf{r}}_0$ and define the electric potentialTaking a short straight terminal segment in each coordinate direction shows ${\partial }_j\phi =-E_j$. ConsequentlyThe electric potential is determined up to an additive constant; the later condition at infinity fixes that constant for the finite charge distribution in part (b).

Set $r=R$ in the enclosed electric charge formula:Outside the entire charge distribution, Gauss's law gives $E_r(r)=Q/(4\pi {ϵ}_0{r}^2)$. Integrating $E_r=-{\phi }^{\prime }$ with $\phi (\infty )=0$ givesThus the exterior electric potential and electric field are exactly those of a point charge $Q$ placed at the center. This is the electrostatic shell theorem: outside a spherically symmetric charge distribution, only its total electric charge affects the electric field, not how that electric charge is distributed radially. If positive and negative electric charges cancel to give $Q=0$, both exterior quantities vanish.

Robert Hofstadter investigated nuclear charge distributions through energetic electron scattering. His 1956 review compares $400\mathrm{MeV}$ electrons scattered from helium gas with predictions for point charges and several extended charge profiles. Beyond about $70^{\circ }$, the elastic scattering was over a hundred times smaller than the point-charge prediction. The extracted nuclear charge form factor distinguishes candidate spatial profiles, connecting measured angular counts with a model of nuclear size and shape.^[ref]

A physicist.^[ref]

A charge distribution invariant under rotations about one center. For its spherically symmetric potential, Gauss's law gives the radial electric field $\mathbf{E}=Q(r){\mathbf{e}}_r/(4\pi {ϵ}_0{r}^2)$, where $Q(r)$ is the enclosed electric charge.