Theory
The equation for the change in radial velocity for a particle as it passes between any pair of cylinders in the lens is:
z is the axis passing through the middle of the lens. r is the direction normal to z. If the lens is constructed with cylindrical plates, the field is symmetrical around z. is the magnitude of the electric field in the radial direction for a particle at a particular radial distance and distance across the gap, the rest mass of the particle passing through the field, is the velocity of the particle and q is the charge of the particle. The integral occurs over the gap between the plates. This is also the interval where the lensing occurs.
The pair of plates is also called an electrostatic immersion lens, thus an einzel lens can be described as two or more electrostatic immersion lenses. Solving the equation above twice to find the change in radial velocity for each pair of plates can be used to calculate the focal length of the lens.
Read more about this topic: Einzel Lens
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