Talk-Polywell.org

This is a seemingly dumb question.

What is the right formulation of the Lorenz force? I have seen it two ways:

Lorenz Force = charge*[E field + (velocity x Magnetic field)]

I have also seen it written...

Lorenz Force = charge*[E field x Magnetic field)]

So here is the question. Say I have an electron sitting here. No velocity, no force, it is just sitting. I put a uniform magnetic field on it.

1. It should turn, right? All electrons have an intrinsic spin. This spin makes them behave like mini-magnetics. Its mini-magnetic poles should align with the big poles of the field, correct? North pointing to south, and south pointing to North. It should turn.

2. Will it revolve around the magnetic field line? There is no initial velocity, so it should sit there right? I mean the Lorenz force reduces too: Charge*Magnetic field. Are you sure it will not move?

3. Now let us turn to a positively charged ion. Say I have one of those sitting here. No motion, no field. I put a uniform magnetic field on it. The ion has no intrinsic magnetic field, right? So it should not move. Right? The Lorenz force is = charge*[Magnetic Field]. The positive ion should not do anything, turn OR move.

4. Now let’s say I kick the electron with a velocity perpendicular to the magnetic field line. That means the electron is going to revolve around in a plane with no backward or forward motion. The Lorenz force should be perpendicular to both the magnetic field direction AND the velocity field direction. In fact the Lorenz force should point towards the magnetic field line. The electron should revolve around the magnetic field line. Clockwise or counterclockwise should ONLY matter on which direction my initial kick was in. Remember, I am kicking the electron only in a plane perpendicular to the magnetic field line.

5. Now let’s say I kick that ion. I kick it in a plane perpendicular to the magnetic field line. The ion should revolve the same way the electron did. Specifically, it will revolve counterclockwise or clockwise based SOLELY on the direction I kick it.

Got all that? So with no velocity, nothing moves, electron or ion. With no velocity, the electron turns, but remains in place. You move either particle, and they revolve. And the revolving depends on which way you kick it. If that is true EXPLAIN HOW ELECTRONS ARE SUPPOSED TO ROTATE CLOCKWISE AND ELECTRONS ROTATE COUNTERCLOCKWISE UNDER A Uniform MAGNETIC FIELD. Namely shown in this Wikipedia article:

http://en.wikipedia.org/wiki/Guiding_center

This is very confusing. Since we know that the rotation shown cannot happen unless there is some initial velocity. But then if there is some initial velocity, then the direction of rotation should depend of the velocity, not wither the particle is positive or negatively charged.

Lets see if I can answer this for you.

The Lorentz force equation is:

Lorenz Force = charge*[E field + (velocity x Magnetic field)]

I don't know what the other formula would be. It looks somewhat like the E-cross-B drift velocity, v=(E x B)/B^2, which is derived using the Lorentz force and the equation of motion.

1) - No, it will not "turn" - it's a point particle so it can't have a direction at all. To think of an electron as a tiny bar magnet can be misleading: The spin is a quantum mechanical property that is either parallel or anti parallel to a magnetic field. This is independent of the actual direction of the magnetic field. The spin does not affect the motion in a homogeneous magnetic field. (See for example wikipedia: Stern-Gerlach experiment)

2) - v=0 and E=0 => F=0. It will stay put.

3) - Same as 2.

4) and 5)

Now let’s say I kick the electron with a velocity perpendicular to the magnetic field line. That means the electron is going to revolve around in a plane with no backward or forward motion. The Lorenz force should be perpendicular to both the magnetic field direction AND the velocity field direction.

- Correct, but I'm not sure what you mean by velocity field...

In fact the Lorenz force should point towards the magnetic field line. The electron should revolve around the magnetic field line.

- Point towards the field line? The field line only shows direction, it's not an actual line. Every point in space has a corresponding field line (all equal in a homogeneous field).

Clockwise or counterclockwise should ONLY matter on which direction my initial kick was in.

- No, it's decided by the cross product between the velocity and the B-field (in combination with the sign of the charge). Example:

A homogeneous magnetic field pointing "up" (along z-axis), particle velocity straight forward (along y-axis): v x B points 90 degrees to the right (along x-axis, perpendicular to the yz-plane). So a particle with positive charge will turn right and a particle with negative charge will turn left. If we keep aligning the y-axis with the velocity the result will be the same, independent on the actual direction of the velocity. So a particle will move in a circle in the plane perpendicular to B (xy-plane). Positive particles will rotate CW and negative paticles CCW if one looks into the B-field (not along), that is "down" in the example. This is exactly what's shown at the top of the picture on wikipedia.