Lorentz Force Law

When we think about the effects of a magnetic field, we're likely to think about compass needles, or the attraction of ferrous materials. Before we can talk about those, though, we need to start with something more basic, and a little surprising.
The magnetic field \(\vec B\) exerts a force on a moving charge \(q\), equal to $$\vec F=q\vec v\times\vec B$$ where \(\vec v\) is the velocity of the charge.
Yes, a magnetic field exerts a force on an electric force.

The cross-product appears a lot in the next two chapters, and you can review Section 1.5 if you need a refresher.

If I place a positive charge next to the N pole of a bar magnet, and let go, what force does the charge feel?
The charge isn't moving, so \(\vec v=0\) and the force is zero. The magnetic field only pushes moving charges.


If a positive charge moves to the right, in a magnetic field that points to the right, what is the force on the charge?
The force on the charge is $$\vec F=q\vec v\times\vec B$$ but \(\vec v\) and \(\vec B\) point in the same direction, and so their cross-product is zero. Thus the charge feels no force.

OK, fine then…



If a positive charge moves to the right in a magnetic field that points downward, what is the force on the charge?
The cross product of \(\vec v\) and \(\vec B\) must be perpendicular to both, and can only point into or out of the page. Using the right-hand rule from Section 1.5, point the fingers of your right hand in the direction of \(\vec v\) (downward), and rotate your hand so the second vector \(\vec B\) shoots out of your palm. Your thumb should be pointing out of the page.


What if a negative charge moves downward in a magnetic field that points right? What is the force on that?
The cross-product \(\vec v\times\vec B\) still points downward, but the force on the charge is \(\vec F=q(\vec v\times\vec B\)\). When \(q<0\), this reverses the direction of the force, and so the negative charge will feel a force pointing into the page. Obviously, this can trip you up if you're not careful! I recommend a two-step process: find \(\vec v\times\vec B\) first with the right-hand rule, and then twist your wrist 180 degrees if the charge is negative. (I call this the "electron twist".)


In a magnetic field that points to the right, in what direction does a positive charge have to move to feel a force to the right? To the left?
The force on the charge is \(\vec F=q\vec v\times\vec B\), and so the force is always perpendicular to both \(\vec v\) and \(\vec B\). Thus if the magnetic field points to the right, it will never push a charge to the right, or the left.

While the electric field tells you the direction a charge will be pushed, while the magnetic field tells you the direction a charge cannot be pushed.


If a charge is in the presence of electric and magnetic fields, it feels a force
$$\vec F=q\vec E+q\vec v\times\vec B$$
which is known as the Lorentz Force Law.
TO COME: Here's a field and a moving charge, find the force.