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.

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.

$$\vec F=q\vec E+q\vec v\times\vec B$$

which is known as the Lorentz Force Law.
TO COME: Here's a field and a moving charge, find the force.