phy1020.DVI

HereAis an area vector whose magnitude is equal to the area of the surface, and whose direction is perpen-
dicular to the surface. Magnetic flux is measured in units ofwebers(Wb), where 1 WbD1Tm^2. The weber
is named after the German physicist Wilhelm Eduard Weber.

31.6 Gauss’s Law for Magnetism

We’ve already seen one of the four Maxwell’s equations, Gauss’s law. Another of Maxwell’s equations is the
analogous equation for magnetism. It has no proper name, but we may call itGauss’s law for magnetism.It
states:

ˆBD0: (31.5)

In other words, the magnetic flux through any closed surface is always equal to zero. This is due to the fact
that there are no magnetic monopoles, so magnetic field lines never terminate.

31.7 Biot-Savart Law

As mentioned in the previous chapter, magnetic fields are produced by electric currents. TheBiot-Savart
law^1 gives the magnetic fieldBproduced by an electric currentIrunning through a short length of wire
l, wherelis a vector whose length is equal to the length of the wire, and which points in the direction of
the conventional current. The Biot-Savart law states:

B.r/D

 0

4

IlOr

r^2

(31.6)

Hereris a vector pointing from the current elementIlto the field point (the point at which the magnetic
field is being observed). Note the presence of the vector cross product operatorin this equation; the cross
product is described in Appendix O.
The Biot-Savart law is a magnetic counterpart of Coulomb’s law: just as Coulomb’s law gives the electric
field due to a point chargeq, the Biot-Savart law gives the magnetic field due to a currentIflowing through
a short wirel.
Comparing Eq. (31.2) with Eq. (31.6), we can find the pole strengthqdue to a currentIthrough a short
wire of lengthl:

qDIl: (31.7)

31.8 Magnetic Field due to a Long Wire

By making use of the calculus, one may use of the Biot-Savart law (Eq. (31.6)) to find the magnetic fieldB
due to a very long wire carrying an electric currentI, at a perpendicular distancerfrom the wire. The result
is:

B.r/D

 0

2

I

r

: (31.8)

Thedirectionof the magnetic field is given by the right-hand rule: if you point the thumb of your right hand
in the direction of the conventional currentI, then the fingers of your right hand curl in the direction of the
magnetic field lines.

(^1) PronouncedBEE-oh sav-AR, and named for the French physicists Jean-Baptiste Biot and F ́elix Savart.