Math 21a Supplement on Electricity and Magnetism
Vector fields on
R
3
play a central role in Maxwell’s theory of electricity and magnetism.
In
particular, the electric field in space is described in Maxwell’s theory by a vector valued function of
space and time,
E
(t, x).
This vector provides the direction and magnitude of the electric field at
any given point and at any given time.
Likewise, the magnetic field is also described in Maxwell’s
theory by a vector valued function of space and time,
B
(t, x).
(Why not the letter ‘M’ for magnetic
field?
I am not sure.
In any event, the traditional letter is ‘B’.)
Maxwell proposed a set of equations which he postulated constrain the possible vector
valued function pairs (
E
,
B
) which can arise as real world electric and magnetic fields.
These are
the famous Maxwell equations.
(Actually, various portions of these equations were written down
by others prior to Maxwell’s culminating contribution.)
These equations involve the operations of
curl and divergence in a fundamental way.
In any event, here are Maxwell’s equations in a
vacuum (no charged, polarizable or magnetically susceptible materials or particles present):
•
div
E
= 0,
•
div
B
= 0,
•
∂
∂
t
E
= curl
B
,
•
∂
∂
t
B
=  curl
E
.
(1)
Here, I have written these equations in units where various natural constants (such as the speed of
light) are equal to 1.
In a physics book, these equations generally appear with various natural
constants whose values depend on the particular choice of units of measurement.
This preview has intentionally blurred sections. Sign up to view the full version.
View Full Document
This is the end of the preview.
Sign up
to
access the rest of the document.
 Fall '11
 GregoryJ.Galloway
 Math, Multivariable Calculus, Magnetic Field, Electric charge

Click to edit the document details