(a) Starting from Laplace's equation in cylindrical coordinates, ^2V = 1 s s (s V s )…

PHYSICS · Subjective · Fermi University Physics · Electricity, Magnetism & Electromagnetism · Electrostatic fields and potential · Conductors, capacitance and boundary-value electrostatics

The Boundary-Value Problem, Ch Cylindrical geometry, edges, and what a field does to a dipole

Problem

(a) Starting from Laplace's equation in cylindrical coordinates,

\nabla^2V = \frac{1}{s}\frac{\partial}{\partial s}\left(s\frac{\partial V}{\partial s}\right) + \frac{1}{s^2}\frac{\partial^2V}{\partial\phi^2} + \frac{\partial^2V}{\partial z^2} = 0,

separate

V(s,\phi,z) = R(s)\Phi(\phi)Z(z)

and show that it splits into three ordinary differential equations linked by two separation constants,

k

(for

Z

) and

m

(for

\Phi

), and write down the equation obeyed by

R(s)

. (b) Two earthed conducting half-planes meet along the

z

axis, with the field region between them opening through an angle

\beta

. Using the

k=0

case of part (a), impose

V=0

on both faces and finiteness at the edge, and derive the leading power-law dependence

E,\sigma \propto s^{\pi/\beta - 1}

of the field and surface charge density near the edge. (c) A small body of polarisability

\alpha

sits in a static, non-uniform applied field

\mathbf{E}(\mathbf{r})

that is curl-free, so that it acquires an induced dipole moment

\mathbf{p}=\alpha\mathbf{E}

. Starting from the force law for a rigid dipole,

\mathbf{F} = (\mathbf{p}\cdot\nabla)\mathbf{E}

, derive the force on the induced dipole in terms of

\nabla(E^2)

, and hence show that its energy is

-\tfrac12\alpha E^2

Browse the problem library