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A point charge

q

is brought to a position a distance

d

away from an infinite...

Feb 15, 2024

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 the surface-charge density induced on the plane, and find the force between the plane and the charge by using Coulomb’s law for the force between the charge.

Key Concept

Method of Images and Surface Charge Density

Explanation

The method of images involves replacing the conducting plane with a virtual charge (image charge) of equal magnitude but opposite sign, located symmetrically on the other side of the plane at a distance

d

. The surface charge density

\sigma

on the plane can be found using the principle that the electric field just outside a conductor is perpendicular to the surface and has a magnitude of

\frac{\sigma}{\epsilon_0}

. For a point charge

q

at a distance

d

from the plane, the surface charge density at a point

r

from the projection of

q

onto the plane is given by

\sigma(r) = \frac{-q}{2\pi\epsilon_0(d^2+r^2)^{3/2}}

Key Concept

Coulomb's Law and Force Calculation

Explanation

The force between the point charge

q

and its image charge

-q

can be calculated using Coulomb's law as if they were two point charges in free space. The force is attractive and given by

F = \frac{q^2}{4\pi\epsilon_0(2d)^2}

. This is the magnitude of the force that the plane exerts on the point charge

q