Notes 5 Spring 2005.doc
Development of Maxwell’s equations
:
Faraday’s law of induction
Recall Stokes’ theorem:
( )
∫
∫
⋅
=
⋅
×
∇
l
d
A
S
d
A
s
G
G
G
G
G
Since Stokes’ theorem is valid for any vector field, we can substitute the electric field vector,
E
G
, for
A
G
to get
( )
∫
∫
⋅
=
⋅
×
∇
l
d
S
d
s
G
G
G
G
G
E
E
and note that the expression for Faraday induction,
dt
d
l
d
Φ
−
=
⋅
∫
G
E
, describes how a changing
magnetic flux induces a voltage (emf) about a closed path.
Substituting this result into the boxed
equation above yields
()
dt
d
S
d
s
Φ
−
=
⋅
×
∇
∫
G
G
G
E
Provided we are working with an area that is constant in time, we can associate the timederivative
operator with only the magnetic field while using the definition of magnetic flux,
∫
⋅
=
Φ
s
S
d
G
G
B
, to
obtain
∫
∫
∫
⋅
⎟
⎠
⎞
⎜
⎝
⎛
−
=
⋅
−
=
⋅
×
∇
s
s
s
S
d
dt
d
S
d
dt
d
S
d
G
G
G
G
G
G
G
B
B
E
then we equate the integrands to obtain
dt
d
B
E
G
G
G
−
=
×
∇
This is what we will refer to as Maxwell’s first equation
.
Since it was derived directly from Faraday’s
law of induction it is often also referred to as Faraday’s law of induction
.
It shows that whenever a
timevarying magnetic field is present there is also (necessarily present) a nonzero circulation or curl
of the electric field, i.e., an electric field is present.
1
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Ampere’s law
Similar to what we did with the electric field, we can substitute
H
G
into Stokes’ Theorem to obtain
()
∫∫
⋅
×
∇
=
⋅
s
S
d
l
d
G
G
G
G
G
H
H
We see from unit analysis that the left side of this equation has units of Amperes and so the right hand
side must also have units of Amperes.
This implies that
H
G
G
×
∇
has units of
2
m
A
which are the units
of a current density.
In fact it turns out that
H
G
G
×
∇
is the sum of two current densities,
and
c
J
G
d
J
G
, the
conduction and displacement current densities, respectively.
So we write
d
c
J
J
G
G
G
G
+
=
×
∇
H
The displacement current density,
, results from a timevarying electric flux density, so it is
common to write
d
J
G
t
J
c
∂
∂
+
=
×
∇
D
H
G
G
G
G
This is Maxwell’s second equation
, Ampere’s law
, which shows that a current density results in a
circulation or curl of a magnetic field around that current density.
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 Fall '10
 Czarnecki
 Magnetic Field, Electric charge

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