Charge passing through a magnetic field of uniform magnetic flux density

[3OPAH]

My reasoning:

The magnetic force on charge q is

F_m = qv x B

B does not change |v|. Therefore, |F_m| is constant at time t > 0 and F_m is always perpendicular to the direction of movement of charge q. F_m behaves as a centripetal force, and thus the charge moves along the circumference of a circle.

Here is my drawing depicting what I think is happening:

Now, knowing that an electric charge q either at rest or in motion, experience an electric force F_e in the presence of an electric field E, that is,


F_e = qE

Then if we have a charge q moving with velocity v in the presence of both an electric field E and a magnetic flux density B, the total force exerted on the charge is therefore

F = F_e + F_m = q(E + v x B)

which is the Lorentz force equation.

I am having trouble using what I have done so far and what I know about the magnetic force and electric force to compute the necessary electric force needed to make charge q move in a straight line. If the charge is moving in a circular path about the xy-plane under the influence of a magnetic field in the positive z direction, then the necessary electric field needed to counteract the circular movement of the charge and make it move in a straight line will be a combination of x and y coordinates, correct?

 

[mfb]

then the necessary electric field needed to counteract the circular movement of the charge and make it move in a straight line will be a combination of x and y coordinates, correct?

Correct.

What is the net force on the charge if it moves in a straight line at constant speed*?

*this is not required, but if you allow a variable speed things get really messy

 

[3OPAH]

Correct.

What is the net force on the charge if it moves in a straight line at constant speed*?

*this is not required, but if you allow a variable speed things get really messy

If a charge experiences no net force, then its velocity is constant; the charge is either at rest (if its velocity is zero), or it moves in a straight line with constant speed. So the net force is zero.

 

[mfb]

Right.

 

[3OPAH]

Right.

So we are given the initial velocity of v = ae_x + be_y. The magnitude of the velocity vector has to be the same, but opposite in direction. So the necessary electric field is (with the magnetic flux density in there as well):

E = -bB_0e_x + aB_0e_y

correct?

 

[mfb]

Correct.