Page 199 - Engineering Electromagnetics, 8th Edition

CHAPTER 7 The Steady Magnetic Field 181

Figure 7.1 The law of Biot-Savart
expresses the magnetic field intensity dH 2
produced by a differential current element
I 1 dL 1 . The direction of dH 2 is into the
page.

case of a cylindrical conductor of circular cross section as the radius approaches zero.
We assume a current I flowing in a differential vector length of the filament dL. The
1
law of Biot-Savart then states that at any point P the magnitude of the magnetic
field intensity produced by the differential element is proportional to the product of
the current, the magnitude of the differential length, and the sine of the angle lying
between the filament and a line connecting the filament to the point P at which
the field is desired; also, the magnitude of the magnetic field intensity is inversely
proportional to the square of the distance from the differential element to the point P.
The direction of the magnetic field intensity is normal to the plane containing the
differential filament and the line drawn from the filament to the point P.Of the two
possible normals, that one to be chosen is the one which is in the direction of progress
of a right-handed screw turned from dL through the smaller angle to the line from the
filament to P. Using rationalized mks units, the constant of proportionality is 1/4π.
The Biot-Savart law, just described in some 150 words, may be written concisely
using vector notation as
IdL × R
IdL × a R
dH = = (1)
4πR 2 4πR 3
The units of the magnetic field intensity H are evidently amperes per meter (A/m).
The geometry is illustrated in Figure 7.1. Subscripts may be used to indicate the point
to which each of the quantities in (1) refers. If we locate the current element at point 1
and describe the point P at which the field is to be determined as point 2, then

I 1 dL 1 × a R12
dH 2 = (2)
2
4πR 12

1 Biot and Savart were colleagues of Amp`ere, and all three were professors of physics at the Coll`ege de
France at one time or another. The Biot-Savart law was proposed in 1820.