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# Motion of Charged Particles in Magnetic and Electric Fields

"A fact is a simple statement that everyone believes. It is innocent, unless found guilty. A hypothesis is a novel suggestion that no one wants to believe. It is guilty, until found effective"
Edward Teller

MAGNETIC FIELDS

- Return now to the case of a "point" charge moving with velocity v in a region of constant magnetic field (B). We previously stated (without proof) that such a particle would move in a circle or helix. Now let's prove it.

- Consider the positive charge (e) at the top of the figure at right. With the directions of v and B indicated the right-hand-rule leads to a downwards force as shown. This force changes the direction of v but does not change the magnitude of v, as shown.

- Quantitatively we find

> Assuming that this force causes circular motion, FB is the centripetal force so that
> which leads to a radius

> The period (time taken to perform one revolution) is given by
> The frequency, f, is

- Notice that this frequency is independent of the velocity. That is fast particles will move in large radius circles, slow particles will have smaller radii. This phenomenon is the basis for the cyclotron one of the earliest charged particle accelerators.

- If the velocity of the particle has a component along B this will be unchanged and the resultant trajectory will be helical.

MAGNETIC and ELECTRIC FIELDS

- There are many useful applications in which B and E fields are applied simultaneously. Perhaps the simplest application is the "crossed" B and E fields.

> For this configuration the electric and magnetic forces are in opposite directions. If a charged particle moves through this region of space with no deflection then

> so that

- Other applications include mass spectrometers and particle accelerators. (EXPAND ?)

I'm reading a book about anti-gravity. I can't put it down.

Dr. C. L. Davis
Physics Department
University of Louisville
email: c.l.davis@louisville.edu