A. F. Peterson:
Notes on Electromagnetic Fields & Waves
11/04
Fields & Waves Note #17
Maxwell’s Equations
Objectives:
Present Ampere’s Law as it was extended by Maxwell to
incorporate a displacement current term.
Summarize the entire set of
Maxwell’s equations and related equations.
Show that Maxwell’s equations
admit waves as solutions and discuss the characteristics of electromagnetic
waves.
James Clerk Maxwell
After Michael Faraday discovered in 1831 that a change in magnetic flux caused an
electrical effect, the equations describing electricity and magnetism remained basically the
same for about 30 years.
Then, in the 1860s, James Clerk Maxwell extended Ampere’s
Circuital Law to include a new term that took into account a timevarying electric flux and
the associated magnetic effect.
Maxwell organized the existing knowledge base into a
collection of about 20 different equations describing electricity and magnetism.
Using these
equations, he predicted the existence of electromagnetic waves, and also postulated that light
was of an electromagnetic nature.
Because of Maxwell’s discoveries, today the primary
equations of electromagnetic fields are named after him.
In 1888, after Maxwell’s death, Hertz demonstrated that electromagnetic waves could be
launched and received using simple antennas.
Around 1900, Oliver Heaviside simplified
Maxwell’s equations into the 4 primary equations we still work with today.
Heaviside also
developed the vector notation for divergence and curl that we use today, more than 100 years
later!
In this Note, we review these developments and walk through a mathematical exercise that
may be similar to the path James Clerk Maxwell followed to obtain electromagnetic waves.
Ampere’s Law as modified by Maxwell
The integral form of Ampere’s Circuital Law, as modified by James Clerk Maxwell in the
1860s, is given by
mmf
=
∑
=
∑
+
∑
ÚÚ
ÚÚ
Ú
H
d
J
dS
d
dt
D
dS
S
S
C
l
(17.1)
where
S
is an open surface that terminates on the closed contour
C
, and the righthand screw
convention (Note #13) is used to link the direction of
dS
with the orientation of
C
.
The
abbreviation “mmf” was originally used to denote “magnetomotive force,” which actually
isn’t a force at all.
In fact, the units of (17.1) are Amperes.
We observe that if the fields are
static, the new term on the right side drops out, leaving Ampere’s Law as it was presented in
Note #13.
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A. F. Peterson:
Notes on Electromagnetic Fields & Waves
11/04
The term on the right side of (17.1),
I
d
dt
D
dS
S
displacement
=
∑
ÚÚ
(17.2)
was introduced by Maxwell and given the name “displacement” current.
Observe that the
units of electric flux are Coulombs, and the time derivative converts the units to Amperes, so
the term is equivalent to a current.
Maxwell reasoned that in certain parts of an electrical
system, a time varying electric field played the role of a current.
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 Spring '08
 CITRIN
 Electromagnet, James Clerk Maxwell, A. F. Peterson

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