Week 9

Basic Facts

Gauss's Law for the Magnetic Field: All magnetic fields have their origin in moving charges (currents) and so all magnetic field lines form closed loops. Since magnetic field lines neither start nor end the net magnetic flux from a region of space is 0.

Ampere's Law: The line integral of the magnetic field round a closed loop is proportional to the amound of current that flows through a surface bounded by that loop. Thus we have

where I the total current flowing through a surface bounded by the curve L.
Ampere's law plays the same role in magnetostatics that Gauss's law plays in electrostatics..

Curl: the curl of a vector function A is defined as
.
It is also written as and, in cartesian coordinates, is evaluated as

Differential form of Ampere's Law: If we apply Stoke's Theorem (see below) to the Integral form of Ampere's Law (see Week 7) then we find where J is the current density at the point of interest.

Useful Facts

A magnetic dipole m produces a magnetic field B(r) at position r that has the form

Stoke's Theorem: For any vector function A which is continuous in a region of space of area S that is bounded by a closed curve G Stoke's theorem states that