(a) Two earthed conducting half-planes meet along the -axis, with the field-filled…

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) Two earthed conducting half-planes meet along the

z

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

\beta = 5\pi/4

(an exterior, re-entrant edge). Following the wedge derivation, write the leading angular and radial dependence of the potential near the edge, and give the exponent

p

such that

E

and

\sigma

vary as

s^p

with distance

s

from the edge. State whether surface charge crowds towards or drains away from this edge. (b) A small body of polarisability

\alpha

sits in a static, curl-free field

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

whose magnitude varies with position, so that it acquires an induced dipole moment

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

. Starting from the force law

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

, show that

\mathbf{F}=\nabla\left(\tfrac12\alpha E^2\right)

, and hence that the potential energy of the induced dipole is

U=-\tfrac12\alpha E^2

. Apply this to a small conducting sphere of radius

2.0\,\mathrm{mm}

(polarisability

\alpha = 4\pi\varepsilon_0a^3

) placed where the field magnitude is

1.0\times10^5\,\mathrm{V\,m^{-1}}

, and give its energy.

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