Question

Problem 1a: Velocity Selector: Show that with the right ratio of electric to magnetic field strength a particle of velocity v will proceed through both fields in a straight line at constant speed (hint: you will need an equation containing v. Also: what does the straight line at constant speed give you?). Assume that the angle of the velocity vector relative to the magnetic field vector is 90 degrees.

b: Show mathematically that the charge magnitude and sign do not matter.

c: Draw and label the electric field vector, the electric force vector, the magnetic field vector, the velocity vector and the magnetic force vector. Hint: start with the two force vectors. They have to add to zero. Then use the vector nature of the Eq = F(E) equation and the right hand rule to get the other vectors.) Assume that the particle is negatively charged. Use into and out of the page vector notation where necessary.

d. Explain in terms of what happens with the force vectors when the charge sign changes to allow a particle of either charge sign pass through the velocity selector at constant velocity v. In other words, explain physically why the particle charge sign makes no difference.

e. Explain in terms of what happens with the force vectors when the charge magnitude changes. In other words, explain physically why the charge magnitude makes no difference in the velocity selector.

Answer 1

consider a charge q in space
let its motion be along x axis depicted by v = |v| i
let magnetic field be perpendicular to the x axis along y axis, B = |B| j
let the electric field be E = Exi + Eyj + Ezk
a. from force balance
   q v x B = q E
   |v||B| k = Exi + Eyj + Ezk

   for both the electric nd the magnetic forces to balance
   Ex = 0
   Ey = 0
   Ez = |v||B|
   hence
   for mutually perpendicular electric magnetic fields and a ahcrged particle moving mutually perpendicular to both will move with no net force if E/B = |v|

b. from the above derivation we can see that the chare sign and magnitude dont matter
c. the following diagram shows the x,y,z axis

d. as the net force vector is 0, the force vector undergoes no change when sign of charge is changed as net force is still 0 and the particle with velocity v can pass through undeflected

e. when charge magnitude changes the net force still remains the same and hence the particles with constant velcity still pass through undelflected