= Solution
Neglect <fringing field> near the edges of the <parallel-plate capacitor>. Put the plates at $z=0,d$, with prescribed <electric potentials> $V_A,V_B$. <Translational symmetry parallel to a plane> makes the <electrostatic potential> independent of the coordinates parallel to the plates. Thus <Laplace's equation> reduces to $d^2V/dz^2=0$. Its <general solution> is $V=A+Bz$, and the two <boundary conditions> determine both constants:
$$
\boxed{V(z)=V_A+\frac{V_B-V_A}{d}z,\qquad \mathbf E=\frac{V_A-V_B}{d}\hat{\mathbf z}.}
$$
The <electric field> is uniform and points toward decreasing <electric potential>. No plate <voltages> are specified, so a numerical <electric potential> cannot be fixed; choosing one plate as the reference only fixes the arbitrary additive constant.
Back to article page
