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Physics 54
Sources of the Magnetic Field
Caution: Cape does not enable user to fy.
— 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 Fnding concerns the BFeld set up by inFnitesimal
segment of a currentcarrying wire.
In the situation shown, we are interested in the BFeld at Feld point
P
due to
an inFnitesimal bit of wire. The current is
I
, in the direction speciFed
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,
0
=
4
×
10
−
7
in SI
units (exactly, by deFnition).
The choice of this constant makes the
ampere
the deFning quantity for
SI electromagnetic units.
The direction of the BFeld comes from the vector product. In the case shown it is out of
the page. If the Feld point were at angle
below the horizontal, the direction would be
into the page. One can see that the Feld lines of this BFeld form circles of radius
r
sin
about a point on the horizontal line. This is an example of a general property:
Lines of the B±eld always form closed curves.
The magnitude, in the case shown, is
dB
=
0
4
I
sin
r
2
.
We see that the Feld strength falls off like
1 /
r
2
, as in the case of the EFeld of a point
charge.
d
l
r
r
sin
P
PHY 54
1
Magnetostatics 2
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View Full DocumentOf course there are never isolated inFnitesimal bits of wire; one must integrate over all
the current segments in the system in order to Fnd the total BFeld 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 EFeld of a
point charge. In principle, the BFeld 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 Fnite
piece of wire. But circuits of rectangular shape are made of straight pieces, and the B
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This note was uploaded on 10/19/2011 for the course PHYSICS 54L taught by Professor Thomas during the Summer '09 term at Duke.
 Summer '09
 Thomas
 Physics, Current

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