= Solution
Specifying potential on the complete boundary gives a <Dirichlet boundary condition> and a unique harmonic solution. Specifying its normal derivative gives a <Neumann boundary condition>, with the zero-net-flux compatibility condition and an undetermined additive constant. Mixed conditions on complementary portions also give uniqueness when a potential reference is fixed. In an exterior domain, conditions at infinity are needed. These statements follow by applying <Green's first identity> to the difference of two solutions. A metal in <electrostatic equilibrium> is an equipotential: $\Phi=V_0$, tangential <electric field> zero, and normal field $E_n=\sigma/\epsilon_0$ just outside a vacuum interface.
For the <conducting-wedge potential expansion>, put $u=\Phi-V_0$ in the empty sector $0<\theta<\alpha$. For $u=r^\nu\Psi(\theta)$, Laplace's equation gives $\Psi''+\nu^2\Psi=0$. Vanishing values on both faces require $\Psi\propto\sin(n\pi\theta/\alpha)$, hence
$$
\boxed{\Phi=V_0+\sum_{n=1}^{\infty}\left(a_nr^{n\pi/\alpha}+b_nr^{-n\pi/\alpha}\right)\sin\frac{n\pi\theta}{\alpha}.}
$$
Bounded potential at the corner sets $b_n=0$. The coefficients still require outer boundary data; the metal potential alone cannot select them. For a generic nonzero $a_1$, put $p=\pi/\alpha$. Differentiation gives
$$
\boxed{E_r\simeq-pa_1r^{p-1}\sin p\theta,\qquad E_\theta\simeq-pa_1r^{p-1}\cos p\theta.}
$$
If $a_1=0$, the first nonzero higher harmonic replaces this leading term. For $\alpha=\pi/2$, $u\simeq2a_1xy$, so $\mathbf E=-2a_1(y,x)$ and field lines satisfy \b[$x^2-y^2=\mathrm{constant}$], meeting the metal faces normally and tending to zero field at the corner. For $\alpha=3\pi/2$, the <electrostatic corner-field exponent> is $-1/3$. A real tip's rounding cuts off this divergence; breakdown or <field emission> can limit the attainable field.
\Image[/past-exam-of-the-physics-b-course-of-the-university-of-cambridge/2014/paper-1-wedge-field.png]
{title=Original right-angle field-line calculation and the two corner-field power laws}
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