According to the Lorentz force law, an electron moving in a constant magnetic field follows a path which satisfies , where is a constant vector on . Prove that this path is a helix.
We can choose to align the z axis with the magnetic field B. Then:
, ,
So:
Solving the third equation by integrating twice with respect to t: where k is a constant.
Now if you differentiate the second equation and then substitute for from the first then you will get:
which has the general solution:
So
Now using equation 1
So
Now using , , we can write:
This is the equation of a helix.