An Apparent Contradiction

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E Near the surface of any conductor in electrostatics

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What have we learned about conductors?

• There is no electric field inside a conductor• Net charge can only reside on the surface

of a conductor • Any external electric field lines are

perpendicular to the surface (there is no component of electric field that is tangent to the surface).

• The electric potential within a conductor is constant

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For any two points and inside the conductor 1r

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The conductor’s surface is an equipotential.

Equipotential Surfaces

An equipotential surface is a surface on which the electric potential V is the same at every point.

Because potential energy does not change as a test charge moves over an equipotential surface, the electric field can do no work on such a charge. So, electric field must be perpendicular to the surface at every point so that the electric force is always perpendicular to the displacement of a charge moving on the surface.

Field lines and equipotential surfaces are always mutually perpendicular.

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Method of images: What is a force on the point charge near a conducting plate?

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Equipotential surface

The force acting on the positive charge is exactly the same as it would be with the negative image charge instead of the plate.

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The point charge feels a force towards the plate with a magnitude:

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Method of images: A point charge near a conducting plane.

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Equipotential surface

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Equilibrium in electrostatic field: Earnshaw’s theorem

There are NO points of stable equilibrium in any electrostatic field!

How to prove it? Gauss’s Law will help!

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Imaginary surface surrounding P

If the equilibrium is to be a stable one, we require that if we move the charge away from P in any direction, there should be a restoring force directed opposite to the displacement. The electric field at all nearby points must be pointing inward – toward the point P. But that is in violation of Gauss’ law if there is no charge at P.

Thomson’s atom1899

If charges cannot be held stably, there cannot be matter made up of static point charges (electrons and protons) governed only by the laws of electrostatics. Such a static configuration would collapse!

Hw quiz

An infinitesimally thin, insulating, uniformly charged horizontal sheet has a small charged object “floating” above it. If the object has mass m and charge Q, find σ, the charge per unit area on the sheet. Assume the sheet is of infinite extent.

A sphere of radius A has a charge Q uniformly spread throughout its volume. Find the difference in the electric potential, in other words, the voltage difference, between the center and a point 2A from the center.

There is a conducting spherical shell, inner radius A and outer radius B. A charge Q1 is put at the center. If you now put charge -2Q1 on the shell, find the charge density everywhere.