Chapter 32
The Lorentz Force
When we place an electric charge in an electric field, or a magnetic pole in a magnetic field, the resulting
motion is pretty simple: the charge or pole simply accelerates along the direction of the field. But some more
interesting physics goes on when we place anelectriccharge in amagneticfield.
Suppose we have an electric chargeqmoving with velocityvin a magnetic fieldB. Then it turns out that
the charge will experience a forceFgiven by
FDqvB: (32.1)Note once again the presence of the cross product operator. This means that the force acting on chargeqis
perpendicular to both its direction of motionvand to the magnetic fieldB.
Note also that since the force is always perpendicular to the direction of motion, the work done by a
magnetic field on an electric charge is always zero.
If both an electric fieldEanda magnetic fieldBare both present, then the net force on the chargeqis
found by combining Eqs. (16.1) and (32.1), and is called theLorentz force:^1
FDq.ECvB/: (32.2)32.1 Plasmas
A plasma is essentially an ionized gas. We can gain some understanding of the behavior of plasmas by
examining the motion of charged particles in the presence of electric and magnetic fields.
Suppose, for example, that we have a (negatively charged) electron moving with velocityvperpendicular
to a magnetic fieldB, and that there is no electric field present. Then there will be Lorentz force acting on the
electron that will eventually cause it to move perpendicular to its original direction. By that time, the Lorentz
force will be in the direction opposite the direction of the direction of motion of the electron, and so on. The
net motion will be that the electron will move in a circle. The direction of motion of a negative charge in a
magnetic field will be given by still another right-hand rule: if you point the thumb of your right hand in the
direction ofB, then the fingers of your right hand will curl in the direction of motion of the electron. (If the
magnetic field points into the page, for example, then the electron will move clockwise.)
By similar reasoning, a positively charge (such as a proton) initially moving perpendicular toBwill move
in a circle given by aleft-hand rule: point the thumb of yourlefthand in the direction ofB, and the fingers
of your left will curl in the direction of motion of the positive charge. For example, if the magnetic fieldB
points into the page, then a proton will move counterclockwise.
(^1) Hypothetically, if magnetic monopoles exist, then the forceFon a magnetic monopoleqin an electric fieldEand a magnetic field
Bwould be given by a similar expression:FDqŒB.v=c^2 /E.