Chapter 35
Gauss’s Law for the Magnetic Field, and, Ampere’s Law Revisited
309
35
Gauss’s Law for the Magnetic Field, and, Ampere’s Law
Revisited
Gauss’s Law for the Magnetic Field
Remember Gauss’s Law for the
electric
field?
It’s the one that, in conceptual terms, states that
the number of
electric
field lines poking outward through a closed surface is proportional to the
amount of
electric
charge inside the closed surface.
In equation form, we wrote it as:
We called the quantity on the left the electric flux
.
Well, there is a Gauss’s Law for the
magnetic
field as well.
In one sense, it is quite similar
because it involves a quantity called the magnetic flux which is expressed mathematically as
and represents the number of magnetic field lines poking outward through a closed
surface.
The big difference stems from the fact that there is no such thing as “magnetic charge.”
In other words, there is no such thing as a
magnetic monopole
.
In Gauss’s Law for the
electric
field we have electric charge (divided by
e
o
) on the right.
In Gauss’s Law for the magnetic field,
we have
0
on the right:
(351)
As far as calculating the magnetic field, this equation is of limited usefulness.
But, in
conjunction with Ampere’s Law in integral form (see below), it can come in handy for
calculating the magnetic field in cases involving a lot of symmetry.
Also, it can be used as a
check for cases in which the magnetic field has been determined by some other means.
Ampere’s Law
We’ve talked about Ampere’s Law quite a bit already.
It’s the one that says
a current causes a
magnetic field
.
Note that this one says nothing about anything changing.
It’s just a cause and
effect relation.
The integral form of Ampere’s Law is both broad and specific.
It reads:
(352)
where:
the circle on the integral sign, and,
l
harpoonrightnosp
d
, the differential length, together, tell you that
the integral (the infinite sum) is around an imaginary
closed
loop
.
B
harpoonrightnosp
is the magnetic field,
l
harpoonrightnosp
d
is an infinitesimal path element of the closed loop,
μ
o
is a universal constant called the magnetic permeability of free space, and
I
THROUGH
is the current passing through the region enclosed by the loop.
o
ENCLOSED
e
Q
=
⋅
A
d
E
harpoonrightnosp
harpoonrightnosp
A
d
E
harpoonrightnosp
harpoonrightnosp
⋅
=
Φ
E
THROUGH
o
I
μ
=
⋅
l
harpoonrightnosp
harpoonrightnosp
d
B
0
=
⋅
A
d
B
harpoonrightnosp
harpoonrightnosp
A
d
B
harpoonrightnosp
harpoonrightnosp
⋅
=
Φ
B
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Chapter 35
Gauss’s Law for the Magnetic Field, and, Ampere’s Law Revisited
310
What Ampere’s Law in integral form says is that, if you sum up the magneticfieldalongapath
segment times the length of the path segment for all the path segments making up an imaginary
closed loop, you get the current through the region enclosed by the loop, times a universal
constant.
The integral
on whatever closed path upon which it is carried out, is called the
circulation
of the magnetic field on that closed path.
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 Fall '08
 RABE
 Physics, Gauss' Law, Magnetic Field

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