1. An electromagnetic velocity filter (sometimes called a Wien filter) is used in a variety of experiments and other applications, such as electron microscopes, in which we need a beam of charged particles all going at the same speed. Generally the souce of the particles will produce particles having a broad range of speeds. As the particles move away from the source they pass through the velocity filter. Only the particles going the “correct” speed are able to pass straight through it. All other particles are deflected and hit surfaces inside the filter.
The filter is implemented by having a region containing constant electric and magnetic fields. The electric field is usually achieved by having parallel plates and the magnetic field is achieved by having a solenoid. Suppose that the particles are positive charges and they enter the velocity filter moving along the x-axis in the positive x-direction. The electric field in this region will have a strength, E, and will point in the positive y-direction.
(a) Draw a free body diagram for a particle passing through the velocity filter at the “correct” speed.
(b) For the electric force on the particles to be canceled by the magnetic force which way must the magnetic field point?
(c) For an electric field strength of E and a desired speed of v, what magnetic field is needed (i.e. write B as a function of E and v)?
(d) Let’s see how this plays out in practice. Suppose we want to select for particle speeds of 2.9 × 108 m/s (about 97% of the speed of light). Our parallel plates are separated by a distance of 1.00 mm and can be charged to a voltage difference of 25 kV. What magnetic field strength will we need?
(e) Suppose we are using a solenoid with 1 mm thick wires so the coil density is N/l = 1000 wires/m. What current will be needed in the wires to achieve the field called for by this velocity filter? Does this seem like it will be easy to build this?
(f) If you send negative particles through the selector will you have to change the direction of the magnetic field? Explain.