The method of images

What do we know in this problem? We know that the conducting plate is an
equipotential surface. In fact, the potential of the plate is zero, since it is grounded.
We also know that the potential at infinity is zero (this is our usual boundary
condition for the scalar potential). Thus, we need to solve Poisson's equation
in the region , for a single point charge at position (0, 0, ),
subject to the boundary conditions

(710) |

(711) |

Note, however, that

(713) |

(714) |

Now that we know the potential in the region , we can easily work
out the distribution of charges induced on the conducting plate. We already
know that the relation between the electric
field immediately above a conducting surface
and the density of charge on the surface is

(715) |

(716) |

(717) |

(718) |

(719) |

(720) |

(721) |

(722) |

Our point charge induces charges of the opposite sign on the conducting plate.
This, presumably, gives rise to a force of attraction between the charge and the
plate. What is this force? Well, since the potential, and, hence, the electric
field, in the vicinity of the point charge is the same as in the analogue problem,
then the force on the charge must be the same as well. In the analogue problem,
there are two charges a net distance apart. The force on
the charge at position (0, 0, ) (*i.e.*, the real charge) is

(723) |

What, finally, is the potential energy of the system. For the analogue problem
this is just

(724) |

(725) |

(726) |

(727) | |||

(728) |

So,

(729) |

(730) |

There is another method by which we can obtain the above result. Suppose that
the charge is gradually moved towards the plate along the -axis from infinity
until it reaches position (0, 0, ). How much work is required to
achieve this? We know that the force of attraction acting on the charge is

(731) |

(732) |

(733) |

As a second example of the method of images, consider a grounded spherical conductor
of radius placed at the origin. Suppose that a charge is
placed outside the sphere at , where . What is
the force of attraction between the sphere and the charge? In this case,
we proceed by considering an analogue problem in which the sphere is replaced by an image charge placed
somewhere on the -axis at . The electric potential throughout space in the
analogue problem is simply

(734) |

(735) |

(736) |

(737) |

(738) |

(739) |

There are many other image problems, each of which involves replacing a conductor with an imaginary charge (or charges) which mimics the electric field in some region (but not everywhere). Unfortunately, we do not have time to discuss any more of these problems.