Method of Images Applied to a Point Charge and Infinite Plane Conductor

Question:

A point charge q is brought to a position a distance d away from an infinite plane conductor held at zero potential. Using the method of images, find:
(a) The surface-charge density induced on the plane, and plot it.
(b) The force between the plane and the charge by using Coulomb’s law for the force between the charge and its image.
(c) The total force acting on the plane by integrating σ2/2ε0 over the whole plane.
(d) The work necessary to remove the charge q from its position to infinity.
(e) The potential energy between the charge q and its image [compare the answer to part d and discuss].
(f) Find the answer to part d in electron volts for an electron originally one angstrom from the surface.



Solution:

(a) By assuming a cartesian coordinate system with x and y axes on conductor plane and z-axis perpendicular to the conductor plane where the point charge and its image are on the z-axis, any arbitrary point p in space with position vector r , is at distances r1 and r2 from the point charge and its image respectively, as it is represented in figure (1).

Figure 1: Point charge q and its image -q at distance d from an infinite plane conductor
held at zero potential.

Electric field at point p is given by,

…………………………………………………………………………………………………………………………………………………………………………………………………………………

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