= Electrostatic boundary conditions at a conductor
= Electrostatic boundary condition
{synonym}
The <electrostatic boundary conditions at a conductor> relate the one-sided total <electric fields> at a smooth metal–<vacuum> interface. Choose a <unit normal> $\mathbf n$ \b[from metal into the adjacent vacuum], let $\mathbf E_{\rm metal}$ and $\mathbf E_{\rm vac}$ be the two limiting <electric fields>, and let $\sigma$ be the signed <surface charge density>. A thin <Gaussian pillbox> with cap <area> $A$ has outward cap normals $+\mathbf n$ on the <vacuum> side and $-\mathbf n$ on the metal side. As its height tends to zero, its side <electric flux> tends to zero for bounded local <electric fields>, leaving
$$
A\mathbf E_{\rm vac}\cdot\mathbf n-A\mathbf E_{\rm metal}\cdot\mathbf n=\frac{\sigma A}{\epsilon_0},\qquad \mathbf n\cdot(\mathbf E_{\rm vac}-\mathbf E_{\rm metal})=\frac{\sigma}{\epsilon_0}.
$$
The minus sign comes from the second outward cap normal, not from the sign of the <electric charge>. A shrinking rectangular loop across the interface and the zero closed <line integral> in <electrostatics> give continuity of the <tangential component>:
$$
\mathbf n\times(\mathbf E_{\rm vac}-\mathbf E_{\rm metal})=\mathbf0.
$$
For a <conductor in electrostatic equilibrium>, $\mathbf E_{\rm metal}=\mathbf0$, hence
$$
\boxed{\mathbf E_{\rm vac}=\frac{\sigma}{\epsilon_0}\mathbf n,\qquad \Phi|_S=V_c,\qquad \sigma=-\epsilon_0\left.\frac{\partial\Phi}{\partial n}\right|_{\rm vac}.}
$$
Thus positive <surface charge density> gives an <electric field> away from metal and negative <surface charge density> gives an <electric field> toward metal. The derivative uses the same chosen <unit normal>, from the <vacuum> side. The ideal surface layer has no unique <electric field> value on the layer itself; these are limiting values. Ordinary surface <electric charge> preserves continuity of <electric potential>; an ideal <electric dipole> sheet would require a different model. https://farside.ph.utexas.edu/teaching/em/lectures/node70.html[Richard Fitzpatrick, Boundary conditions for E and D: pillbox and loop conditions]{ref}.
Here “exterior” means just outside the \b[metal], including the <vacuum> within a <cavity in a conductor>. On a spherical inner wall of radius $a$, its <unit normal> from metal into the <cavity in a conductor> is $-\widehat{\mathbf r}$. A central positive <point charge> $q$ gives $\mathbf E_{\rm vac}=q\widehat{\mathbf r}/(4\pi\epsilon_0a^2)$ there, so
$$
\sigma_{\rm inner}=-\frac{q}{4\pi a^2},\qquad \int_{S_{\rm inner}}\sigma_{\rm inner}\,dA=-q.
$$
The outward normal of the <cavity in a conductor> as a volume is instead $+\widehat{\mathbf r}$; confusing these two <unit normals> reverses the inferred sign. For a stipulated spherical enclosure with inner radius $a$, outer radius $b$, no external sources and total isolated metal <electric charge> $Q_c$, <spherical symmetry> and <Gauss's law> give the analytical model
$$
E_r(r)=\begin{cases}
q/(4\pi\epsilon_0r^2),&0<r<a,\\
0,&a<r<b,\\
(Q_c+q)/(4\pi\epsilon_0r^2),&r>b.
\end{cases}
$$
Its outer <surface charge density> is $(Q_c+q)/(4\pi b^2)$. Grounding to $V_c=0$ with zero <electric potential> at infinity instead gives $Q_c=-q$ and zero exterior <electric field>, while preserving the same nonzero <electric field> inside the charged <cavity in a conductor>.
\Image[/electrostatic-conductor-equilibrium-pillbox-cavity.png]
{title=Signed <Gaussian pillbox> accounting and an analytical charged <cavity in a conductor>}
{description=Original schematic and analytical model, not measured data. Left: the two cap normals give the normal <electric field> jump at a positive metal–<vacuum> surface. Right: a central positive <point charge> in a spherical enclosure with $b=2a$; solid blue is an isolated initially neutral enclosure, and dashed orange is a grounded enclosure. Both have zero <electric field> in the metal and the same nonzero <electric field> in the <cavity in a conductor>. The jumps show one-sided limits, with the point-source singularity excluded.}
As a numerical sign check, stipulate $q=1\,\mathrm{nC}$, $a=0.10\,\mathrm m$, $b=0.20\,\mathrm m$ and $1/(4\pi\epsilon_0)\simeq8.99\times10^9\,\mathrm{N\,m^2/C^2}$. The inner <electric field> is approximately $899\,\mathrm{N/C}$ along $+\widehat{\mathbf r}$, but $\mathbf E_{\rm vac}\cdot\mathbf n=-899\,\mathrm{N/C}$ and $\sigma_{\rm inner}\simeq-7.96\,\mathrm{nC/m^2}$. An isolated neutral enclosure has $+1\,\mathrm{nC}$ on the outer wall, approximately $+1.99\,\mathrm{nC/m^2}$ and $225\,\mathrm{N/C}$ just outside. Multiplying each <surface charge density> by its spherical <surface area> recovers the signed wall <electric charges>.
The local result uses the total <electric field>, including all induced and imposed sources. The factor $\sigma/\epsilon_0$ follows from the zero metal-side <electric field>; a symmetric isolated sheet in <vacuum> instead has magnitude $|\sigma|/(2\epsilon_0)$ on each side. https://www.feynmanlectures.caltech.edu/II_05.html#Ch5-S9[Feynman, Leighton and Sands, II.5.9: why a conductor differs from an isolated charge sheet]{ref}. At sharp corners, limiting <electric fields> can be singular and the smooth local pillbox argument needs care. In material outside the metal, use the normal free-charge jump of the <electric displacement field> and the material's <permittivity>, as in <electrostatic interface boundary conditions>. https://farside.ph.utexas.edu/teaching/em/lectures/node70.html[Richard Fitzpatrick, Boundary conditions for E and D: free and bound surface charge]{ref}.
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