The method of images

Infinite conducting plane

Problem:

Two parallel metallic plates (infinite extension) are kept at potentials Φ = 0 and are located at z = 0 and z = L.  A point charge q is at z0.  What is the electrostatic potential between the plates?

Solution:

Problem:

A charge q is released from rest a distance d from an infinite conducting plane.  How long will it take for the charge to strike the plane?

Solution:

Problem:

A particle of mass m and charge e is suspended on a string of length L.  At a distance d (d > L) under the point of suspension there is an infinite plane conductor.  Ignore gravity.  Compute the frequency of the pendulum if the amplitude is sufficiently small such that Hooke's law is valid.  (Ignore radiation losses).

Solution:


Conducting spheres

Problem:

(a)  A point charge q is located a distance d away from the center of a grounded conducting sphere of radius a.  Does the electrostatic force between the charge and the sphere pull these two objects together or push them apart.  Why?
(b)  Now suppose the conducting sphere is not grounded and carries no net charge.  Does the electrostatic force pull the two objects together or push them apart?  Why?
Is the magnitude of the force greater or less than the force between the charge and the grounded sphere?  Why?

Solution:

Problem:

Consider a point charge q located a distance d from the center of a grounded, conducting sphere of radius a < d.  Use spherical coordinates and locate the charge on the z-axis.
(a)  Find the potential outside the sphere.
(b)  Derive an expression for the induced surface charge density on the sphere.
(c)  Expand the potential in a power series in 1/r for r >> d and keep the first two terms.  What is the significance of these two terms?  Find the electric field to the same order

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Solution:

Problem:

Inside a thin-walled metal sphere with radius R = 20 cm there is a metal sphere with the radius   r = 10 cm.  The two spheres have a common center.  The inner sphere is connected with a very long wire to the Earth via an opening in the outer sphere.  A charge Q = 10-8 C is placed onto the outer sphere.
(a)  Find an expression for the electrostatic potential everywhere.  Neglect the potential due to the wire. (Hint:  Think method of images.)
(b)  Find the potential of the outer sphere.  Give a numerical answer.
(c)  Find the capacitance of the system.  Give a numerical answer.

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Solution:

Problem:

Compute the force of attraction between a neutral metallic sphere of radius a and a point charge q positioned a distance r from the center of the sphere, where r > a.
How does the force of attraction behave as a function of r if r >> a?

Solution:

Problem:

A conductor at potential V = 0 has the shape of an infinite plane with a hemispherical bulge of radius R.  A charge q is placed above the center of the bulge, a distance d from the plane (which means a distance d - R from the top of the bulge).  What is the electrostatic force on the charge?

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Solution:


Conducting cylinders

Problem:

The ultimate goal of this problem is to determine the capacitance of a straight telephone wire suspended at a fixed height h above at ground surface.  The length of the wire is L, the radius of the wire is a, and the ground potential is equal to zero.
(a)  As a first step, determine the capacitance of a system consisting of two parallel straight wires. The length of each wire is L, the radius of each wire is a, and the distance between the wires is 2h.   Assume that L >> h >> a.  You can assume that the charges on the upper and lower wires are q and -q, respectively.
(b)  Using the result of part (a) and symmetry arguments, find the capacitance of the telephone wire described in the beginning of the problem.

Solution: