Vector Properties of the Magnetic Field
Using vector calculus, we can generate some properties of any magnetic field, independent of the
particular source of the field.
Line Integrals of Magnetic Fields
Recall that while studying electric fields we established that the surface integral through any
closed surface in the field was equal to
4
Π
times the total charge enclosed by the surface. We
wish to develop a similar property for magnetic fields. For magnetic fields, however, we do not
use a closed surface, but a closed loop. Consider a closed circular loop of radius
r
about a
straight wire carrying a current
I
, as shown below.
A closed path around a straight wire
What is the line integral around this closed loop? We have chosen a path with constant radius, so
the magnetic field at every point on the path is the same:
B
=
. In addition, the total length of
the path is simply the circumference of the circle:
l
= 2
Πr
. Thus, because the field is constant on
the path, the line integral is simply:
lineintegral
B
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 Fall '10
 DavidJudd
 Physics, Vector Space, Electric Fields, Magnetic Field, Line integral

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