Electric and magnetic forces on a rotating charged particle beam

[jughead4466]

Particles having mass=m and charge = Q travel parallel to the z axis, forming a beam of radius = R and uniform charge density = p. To keep the beam focused, an external uniform magnetic field, B, parallel to the z axis is provided, and the beam is made to rotate with a constant, uniform angular velocity = w.

4. Use Gauss' Law to find the radial electric field in the beam on a cylinder of radius r<R

6a. Find the tangential velocity of a particle in the beam at r<R

8b. Find the total (electric and magnetic) force on a particle at r<R

 

[member 553083]

The radial electric field in the beam on a cylinder of radius r<R can be found using Gauss' Law: E_r = \frac{pQ}{2\pi r}.The tangential velocity of a particle in the beam at r<R is given by v_t = rw, where w is the angular velocity of the beam.The total force on a particle at r<R is equal to the vector sum of the electric and magnetic forces. The electric force is given by F_e = \frac{pQ^2}{2\pi r^2}, and the magnetic force is given by F_m = \frac{Qv_tB}{c}, where c is the speed of light. Thus, the total force is given by F = \frac{pQ^2}{2\pi r^2} + \frac{QrwB}{c}.