Page 345 - Intro to Tensor Calculus

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Figure 2.6-6. Magnetic field around wire.

Biot-Savart Law
The Biot-Savart law for magnetostatics describes the magnetic field at a point P due to a steady line
current moving along a curve C and is

Z ~
~
B(P)= µ 0 I × b e r ds (2.6.59)
4π r 2
C
with units [N/amp · m] and where the integration is in the direction of the current flow. In the Biot-Savart
~
2
law we have the constant µ 0 =4π × 10 −7 N/amp which is called the permeability of free space, I = I b e t is
the current flowing in the direction of the unit tangent vector b e t to the curve C, b e r is a unit vector directed
from a point on the curve C toward the point P and r is the distance from a point on the curve to the
~
general point P. Note that for a steady current to exist along the curve the magnitude of I must be the
same everywhere along the curve. Hence, this term can be brought out in front of the integral. For surface
~
~
currents K and volume currents J the Biot-Savart law is written
ZZ ~
~
B(P)= µ 0 K × b e r dσ
4π S r 2
ZZZ ~
~
and B(P)= µ 0 J × b e r dτ.
4π V r 2
EXAMPLE 2.6-5.
~
~
Calculate the magnetic field B a distance h perpendicular to a wire carrying a constant current I.
Solution: The magnetic field circles around the wire. For the geometry of the figure 2.6-6, the magnetic
field points out of the page. We can write

~
I × b e r = I b e t × b e r = Iˆ e sin α
where ˆ e is a unit vector tangent to the circle of radius h which encircles the wire and cuts the wire perpen-
dicularly.