In the figure below a beam of electrons whose kinetic energy emerges from a thin-foil “window” at the end of an accelerator tube. A metal plate at distance from this window is perpendicular to the direction of the emerging beam. I am going to show that the beam can be prevented from hitting the plate if a uniform magnetic field is applied.
I start by stating that the Kinetic Energy is given by the formula . Because the beam of particles exist within a magnetic field, they experience a , directed radially inward.. Because the applied magnetic field causes them to follow a circular path, and that magnetic force is . If we plug all of these values into the radius of a circle that is created when the beam of electrons move perpendicular to the applied uniform magnetic field I obtain the formula . If I solve the Kinetic Energy formula for and then plug it into the circle equation I obtain . Next, if I solve for after squaring both sides and state that the distance of the circle must be less than , I obtain the forumla . If I now take out of the fraction and multiply each side of the equation by its reciprocal, and then bring under the radical, I obtain the following solution which is the end that I seek, namely . After using the Right-Hand Rule, I find that the Magnetic Field is pointing out of the page.