By Peter W. Hawkes (Ed.)

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**Additional resources for Advances in Electronics and Electron Physics, Vol. 76**

**Sample text**

It is based on the potential representation in space in the form of so-called single (or double) layer potentials, that is, in the form of an integral over the electrode surface, the subintegral function being a known kernel and an unknown density function. Equipotential electrode surfaces are replaced by surface charges with a certain density distribution. Since the potential at the system boundary IS known, the density function may be found from the conditions that the integral is equal to the potential value at any point on the boundary (or, as it is generally phrased, from the condition of boundary collocation).

For most lenses, the analytical solution of integral equations is impossible, so they are replaced by a set of linear algebraicequations, whose order is equal to the number of collocation points. Such a set of equations can be obtained, for example, if the terminal electrodes are divided into a finite number of intervals so that the continuous charge distribution is replaced by the discrete one. A charge equal to the product of the mean charge density within the interval and its length or area is placed at the central point of each interval 18 L.

Equipotential electrode surfaces are replaced by surface charges with a certain density distribution. Since the potential at the system boundary IS known, the density function may be found from the conditions that the integral is equal to the potential value at any point on the boundary (or, as it is generally phrased, from the condition of boundary collocation). The first step in this method is to find the charge distribution on the electrodes. The second step is to determine the potential in the whole space.