Method of image charges

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The method of image charges (also known as the method of images and method of mirror charges) is a basic problem-solving tool in electrostatics. The name originates from the replacement of certain elements in the original layout with imaginary charges, which replicates the boundary conditions of the problem.

1 Uniqueness theorem
- 1.1 Extension
2 See also
3 References

[edit] Uniqueness theorem

Any uniqueness theorem says that an object satisfying some set of given conditions or properties is the only such object that exists; it is uniquely determined by the specified conditions.

To illustrate, draw a closed loop to define a region inside and a surface on the line. If:

$\nabla^2V=f$ within the region and v = g on the surface

then $V (x, y, z)$ is unique (given constant f and g).

In this case of image charges we can use the uniqueness theorem to say that provided the boundary conditions laid out by the problem are satisfied, any element can be replaced and then that this is the only alternative arrangement.

Initial system.

The system and its image.

The simplest example of a use of this method is that in 2-dimensional space of a point charge, with charge +q, located at (0, a) above an 'infinite' grounded (ie: V = 0) conducting plate, lying along the x-axis. Deriving any results from this setup, such as the charge distribution on the plate, or the force felt by the point charge, is not trivial.

In order to simplify this problem, we may replace the plate of equipotential with a charge, located at (0, −a) and with charge −q. This arrangement will produce the same electric field at any point for which y > 0 (ie: above the conducting plate), and satisfies the boundary condition, that the potential along the plate must be zero. This new setup is depicted below.

This situation is equivalent to the original setup, and so calculating the force on the real charge is now trivial, by use of Coulomb's law. Finding the charge density on the plate is less obvious, but still easily attainable by using Gauss' law. In order to use Gauss' law, we now extend this case to three dimensions, so that we may construct 2-dimensional Gaussian surfaces (ie: the plate now lies on the xz-plane).

The potential, in cylindrical coordinates, at any point in space due to two point charges is simply:

$V\left(r,z\right) = \frac{1}{4 \pi \epsilon_0} \left( \frac{q}{\sqrt{r^2 + \left(z-a \right)^2}} + \frac{-q}{\sqrt{r^2 + \left(z+a \right)^2}} \right) \,$

And because of the uniqueness theorem, this turns out to be the only solution to this problem.

The surface charge on the grounded plane is given by

$\sigma = -\epsilon_0 \frac{\partial V}{\partial z} \Bigg|_{z=0} \,$

Which, after simplifying, ends up being:

$\sigma \left(r \right) = \frac{-q d}{2 \pi \left(r^2 + a^2\right)^{3/2} } \,$

In addition, the total charge induced on the conducting plane involves integrating this, so:

$Q_t \,$	$= \int_{0}^{2\pi}\int_{0}^{\infty} \sigma\left(r\right)\, r dr d\theta \,$
	$= \frac{-qd}{2\pi} \int_{0}^{2\pi}d\theta \int_{0}^{\infty}\frac{r\,dr}{\left(r^2 + a^2\right)^{3/2}}$
	$= -q \,$

So the total charge induced on the plane turns out to be simply -q.

[edit] Extension

The system and its image for the extended case of two charges (k is a constant).

This method can be extended to two or more charges, replacing the plate with the 'image charge' of each real charge. As the total electrostatic potential is equal to the scalar sum of the potentials, at any point on the xz plane the potential of any real charge will cancel with that of its image. Hence the potential anywhere on the plane will be zero, and the boundary condition satisfied.

The diagram to the right depicts a specific case of this extension, in which there are two (real) charges present, each a distance a above the plate.

[edit] See also

[edit] References

Griffiths, David J. (1998). Introduction to Electrodynamics (3rd ed.). Prentice Hall. ISBN 0-13-805326-X.

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Categories: Electromagnetism | Electrostatics | Introductory physics

Method of image charges

From Wikipedia, the free encyclopedia

Contents

[edit] Uniqueness theorem

[edit] Extension

[edit] See also

[edit] References

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