198 Week 6: Moving Charges and Magnetic Force
- Thetorqueon a magnetic dipole in a uniform magnetic field is:
~τ=~m×B~ (405)Associated with this are its potential energy:U=−~m·B~ (406)and its force in anon-uniform magnetic field:F~=−∇~U=∇~(~m·B~) (407)Magnetic dipoles align with the field due to the torque, and then follow the field back to where
it is stronger, just as do electric dipoles. Students have experienced this with toy magnets and
refrigerator magnets from when they were very small – this is why bar magnets attract one
another.
You should be able to compute the magnetic moment of simple currentloops, although we’ll
get more practice at this in the next chapter/week.6.1: Magnetic Force versus Magnetic Field
In our discussions of the electrostatic force, we were able to start with a fundamental experimental
result – Coulomb’s Law – and proceed to systematically deduce nearlyall of electrostatics including
the more fundamentalexpressionof Coulomb’s Law: Gauss’s Law for the Electric Field. Coulomb’s
Lawalonetold usbothhow to create an electric fieldandwhat the force was in terms of the field.
Life is not quite so simple for the magnetostatic field (where the “static” aspect refers to the
field itself, not to the charges moving in or acting as sources of the field). In this and the next
chapter we will learn that moving charges in a magnetic field experience a force according to a basic
experimental rule (given a field) and moving charges in turn act as sources for a magnetic field (as
one can experimentally verify by measuring forces). However, theoriginalexperiments, conducted
by Ampere, that demonstrated both together involvedcurrentsand notmoving elementary charges.
We, on the other hand, are interested in developing a “microscopic”description of fields that
works for elementary point charges like electrons and quarks and that can be suitably coarse-grain
averaged into continuous distributions of charge and current (using the methods explored in the first
part of the course). This suggests that we start with either force acting onorfield produced by
moving point charges and work our wayupto Ampere’s experimental results with current balances,
instead of trying to work our way backwards.
For better or worse we will therefore begin with the force exertedby a magnetic field that we can
think of as beingdefinedby this force law, without (yet) worrying about where the field comes from.
In the next chapter (next week), we will explore in great detail thesources of that field. Do not
hestitate, however, to skip forward and backward between the two chapters as you study, as knowing
at least thesummaryof the next chapter will help you with this one, just as you will certainly need
to not instantly forget this chapter to move on and learn the next one. Together they ultimately
produce asingleview of the magnetic force between two moving charges and how it becomes the
magnetic force between two currents.
6.2: Magnetic Force on a Moving Point Charge
With that said, let us proceed directly to the basic relation thatexperimentallydescribes the force
exerted by a magnetic field on a charged particle. Note well that thisforce law can be more or less