EM.1 · Electrostatic superposition, flux and conductors

A charged conductor can have zero field inside its material while its potential remains nonzero.

Prerequisites: 2, 47.

• Add Coulomb forces as vectors using the actual geometry
• Compute flux from symmetry or solid angle 立体角 with a stated orientation
• Find conductor fields and potentials using induced surface charges

For a point charge Q at position r0, E(r)=Q(r−r0)/(4πε0|r−r0|³). Superpose vectors, not field magnitudes. For two positive charges attracting an electron, forces in the same direction add; opposite directions subtract; perpendicular components combine by Pythagoras. Define the observation point and the direction of each force first. Potential is the scalar sum V=ΣQ/(4πε0r) with zero at infinity. A zero potential does not generally mean a zero field, since E=−∇V.

Gauss’s law is the closed-surface integral ∮E·dA=Q_enclosed/ε0. It determines a local field simply only when symmetry makes the normal field uniform or makes other flux contributions zero. A charge near an infinite plane sends half its total flux through that plane in magnitude: the plane subtends solid angle 2π out of 4π. The result is independent of distance and lateral position, but its sign depends on the chosen plane normal and charge sign. An open plane does not enclose a charge; using closed-surface wording for it is incorrect.

In electrostatic equilibrium 静电平衡, the electric field inside conducting material is zero and the conductor has constant potential. Place a charge Q at the centre of a conducting spherical shell with inner radius a, outer radius b and shell net charge q. A Gaussian surface inside the material requires inner-surface charge −Q, so the outer surface has q+Q. Spherical symmetry makes the external field that of total Q+q at the centre. The material’s potential, zero at infinity, is (Q+q)/(4πε0b), independent of the material observation radius.

In the cavity of that centred-charge shell, the field is Q/(4πε0r²) radially for 0<r<a. Potential includes both the point-charge contribution and constant shell contributions; continuity holds across each surface even though the normal field jumps at a surface charge. Keep cavity, conductor material and exterior separate. An off-centre charge still induces total inner charge −Q, but its cavity field is no longer the simple centred radial field. Zero interior conductor field follows equilibrium, not an assumption that every surface charge distribution is uniform.

A centred point charge Q lies in a conducting spherical shell of net charge q. Electrostatic equilibrium makes the field zero inside the metal. A Gaussian surface in the metal encloses zero net charge, so $Q_{inner}=-Q$. Charge conservation gives $Q_{outer}=q+Q$. With outer radius b and zero potential at infinity,

Do not confuse the open-plane half-flux result with enclosed charge. A conductor’s zero field fixes a constant potential, not necessarily zero potential.

solid angle: Angular area subtended by a surface, measured in steradians.

induced surface charge 感应表面电荷: Charge rearranged on a conductor to satisfy electrostatic equilibrium.