We are making progress. Our aim, the description of the electrons motion, is not far beyond reach. In Sec.1.3 of Chap.1 (Equation of Motion) we derived the general equation of motion (1c.05). With the help of this equation we can solve our problem.

Again let us refer to the Lorentz force. For convenience we bring this formula once more; especially for and we have:


Inserting this expression of the Lorentz force into Newton's equation of motion (1c.05) we get:

For brevity we introduce:

Thus we have the vector equation

or the coordinate equations:
(2a.01)
(2a.02)
(2a.03)

To make things easy we start with the last equation. We make the ansatz:

It is easily verified:


The meaning of the parameters C₁ and C₂ can easily be recognized. When we insert t = 0 into the solution we have

Thus
(2a.04)

Now let us try to solve the other two equations. The problem we have is the equations are coupled; they are a system of differential equations. That means we have to consider both simultaneously to find a suitable solution. Supposing the property of the electrons motion is circular we make following ansatz:

Verification:

Inserting the four results in our two differential equations (2a.01) (2a.02) we receive:

We have found two functions describing the motion in the x-y-plane:
(2a.05/06)
This solution (2a.04) (2a.05/06) represent an especially simple motion of the electron in a magnetic field.

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Chapter Two