Question

Please write your handwriting easy to understand :)

2. A beam of particles of charge g, mass m, and velocity v is emitted from a point source, roughly parallel with a magnetic field B (say along the z axis), but with a small angular dispersion. Show that the effect of the field is to focus the beam to a point at a distance z 2Tmv/Iq|B from the source.

Answer 1

Assume that the particle makes a small angle φ wrt the B field. The force on the particle will be F = qv x B, which in the small angle approximation becomes F = qvBφ. this force is in the direction perpendicular to the initial velocity and the B field. If you think about it this means the particle will spiral. The force will cause a torque given by T = Fr = FZQ-qvB2фл2. Setting torque equal to the moment of inertia I mr-2-mz2φ^2 times the angular acceleration a gives us: qvB2φ^2-amz-29"2 or a sqvB / mz (Note the angle φ cancels itself out). Now if you think about it when the particle goes through a half of a spiral it will be back in line with the axis parallel to the B field passing through the point source hole. Using the kinematic equation: θ = (12)at-2 and noting that t-z/v and a half spiral means we take θ-π we have: π-( 1 /2)(qvB/m2(z"2 / v"2) or π-qB2 / 2mv. Solve for z to get