Notes 5
Fall 2004.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
Recall 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 time derivative
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
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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Fall 2004.doc
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,
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 Fall '10
 Czarnecki
 Magnetic Field

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