m/The field of an infinite U.
n/The geometry of the Biot-
Savart law. The small arrows
show the result of the Biot-Savart
law at various positions relative
to the current segment d`. The
Biot-Savart law involves a cross
product, and the right-hand
rule for this cross product is
demonstrated for one case.
o/Example 12.
which is known as the Biot-Savart law. (It rhymes with “leo bazaar.”
Both t’s are silent.) The distances d`andrare now defined as
vectors, d`andr, which point, respectively, in the direction of the
current in the end-piece and the direction from the end-piece to
the point of interest. The new equation looks different, but it is
consistent with the old one. The vector cross product d`×rhas a
magnituderd`sinφ, which cancels one ofr’s in the denominator
and makes the d`×r/r^3 into a vector with magnitude d`sinφ/r^2.The field at the center of a circular loop example 11
Previously we had to do quite a bit of work (examples 9 and 10),
to calculate the field at the center of a circular loop of current of
radiusa. It’s much easier now. Dividing the loop into many short
segments, each d`is perpendicular to thervector that goes from
it to the center of the circle, and everyrvector has magnitudea.
Therefore every cross product d`×rhas the same magnitude,
ad`, as well as the same direction along the axis perpendicular
to the loop. The field isB=∫
k Iad`
c^2 a^3=k I
c^2 a^2∫
d`=
k I
c^2 a^2(2πa)=
2 πk I
c^2 aOut-of-the-plane field of a circular loop example 12
.What is the magnetic field of a circular loop of current at a point
on the axis perpendicular to the loop, lying a distancezfrom the
loop’s center?
.Again, let’s writeafor the loop’s radius. Thervector now has
magnitude√
a^2 +z^2 , but it is still perpendicular to the d`vector.
By symmetry, the only nonvanishing component of the field is
along thezaxis,Bz=∫
|dB|cosα=∫
k I rd`
c^2 r^3a
r
=
k Ia
c^2 r^3∫
d`=
2 πk Ia^2
c^2 (a^2 +z^2 )^3 /^2.
698 Chapter 11 Electromagnetism