Why Does a Charged Particle in a Magnetic Field Perform Circular Motion?

[KLscilevothma]

http://www.geocities.com/bridgestein/Bffield.jpg
(copy and paste the above address to your browser please)

In fact I know the answers of this problem, but I have a question.

Here's the question.
A charge +q moving with constant velocity v enters into a uniform magnetic field region as shown. The magnetic flux density is B.
a) What is the direction of the force F experienced by q?
b) What is the work done by F?
c) What kind of motion it is performing?
d) Express the radius of curvature r of the change in terms of B, v, m and q. Where m is the mass of the charge.

ans:
a) Direction of force is always perpendicular to the direction of q. The point charge will go upward.

From the part (d), I know it is performing circular motion and that the work done by F is 0 J!

d) r = mv/(Bq)

Why it is performing circular motion? Can't it be an elliptical motion?

 

[quantumdude]

Originally posted by KL Kam
Why it is performing circular motion? Can't it be an elliptical motion?

When you have a constant magnitude force that is always perpendicular to the velocity, uniform circular motion is the only possible result. To show this, you can use Newton's second law for the radial component of the force (that is the only nonzero component in this problem):

ΣF_r=qvxB=ma_r

where v=vi and B=-Bk. You would then solve the differential equation for the trajectory, using the initial conditions, to see that the path is necessarily circular.

 

[Claude Bile]

Also, if the motion were elliptical, the acceleration would not be perpendicular to the motion and the magnetic field would therefore do work, which is contradictory to your (correct) answer to b).

 

[KLscilevothma]

Thanks Tom and Claude Bile