Near a knife edge

PHYSICS · Multi-part · Fermi University Physics · Electricity, Magnetism & Electromagnetism · Conductors, capacitance and boundary-value electrostatics · Conductors, capacitance and boundary-value electrostatics

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

Problem

Near a knife edge, where the field-filled region opens through

\beta = 2\pi

, the field grows as

E \propto s^{-1/2}

as the distance

s

from the edge shrinks. Near a cube's edge, where

\beta = 3\pi/2

, it grows more gently, as

E \propto s^{-1/3}

. ((a)) Estimate the factor by which

s

must be reduced to double

E

near the knife edge. ((b)) Estimate the same factor near the cube edge. ((c)) State in one sentence what the comparison shows about how the sharpness of an edge governs the crowding of field lines close to it.

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