Physics 54
Sources of the Magnetic Field
Caution: Cape does not enable user to fly.
— Label on Batman Costume
The BiotSavart law
Soon after they learned of Oersted's discovery that a current produces magnetic effects,
Biot and Savart undertook careful experiments to determine the details. In modern
notation and terminology, their finding concerns the Bfield set up by infinitesimal
segment of a currentcarrying wire.
In the situation shown, we are interested in the Bfield at field point
P
due to
an infinitesimal bit of wire. The current is
I
, in the direction specified
by the vector
d
l
. The displacement of
P
relative to the bit of wire is
r
, which makes angle
θ
with the direction of
d
l
.
The experimental answer found by Biot and Savart is a
fundamental law:
BiotSavart law
d
B
=
μ
0
4
π
I
d
l
×
r
r
3
In this formula we have introduced another universal constant,
μ
0
=
4
π
×
10
−
7
in SI
units (exactly, by definition).
The choice of this constant makes the
ampere
the defining quantity for
SI electromagnetic units.
The direction of the Bfield comes from the vector product. In the case shown it is out of
the page. If the field point were at angle
θ
below the horizontal, the direction would be
into the page. One can see that the field lines of this Bfield form circles of radius
r
sin
θ
about a point on the horizontal line. This is an example of a general property:
Lines of the Bfield always form closed curves.
The magnitude, in the case shown, is
dB
=
μ
0
4
π
I
sin
θ
r
2
.
We see that the field strength falls off like
1/
r
2
, as in the case of the Efield of a point
charge.
d
l
r
r
sin
θ
θ
P
PHY 54
1
Magnetostatics 2
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Of course there are never isolated infinitesimal bits of wire; one must integrate over all
the current segments in the system in order to find the total Bfield at
P
. In general such
an integration is complicated; we will do it only in a few very simple cases.
Nevertheless, the BiotSavart law gives us the magnetic equivalent of the Efield of a
point charge. In principle, the Bfield of any set of currents could be calculated using it.
Two examples
The simplest case geometrically is that of a straight wire carrying current
I
. This is also
unphysical by itself, because steady currents cannot start and stop at the ends of a finite
piece of wire. But circuits of rectangular shape are made of straight pieces, and the B
field of such a circuit can be obtained by adding their separate contributions.
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 Summer '09
 Thomas
 Physics, Current, Magnetic Field, Ampere, BiotSavart law, Bfield

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