Vector Properties of the Magnetic Field

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 4Π 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 = 2Πr . Thus, because the field is constant on the path, the line integral is simply:

lineintegral

B·ds = Bl = (2Πr) =

This equation, called Ampere's Law, is quite convenient. We have generated an equation for the line integral of the magnetic field, independent of the position relative to the source. In fact, this equation is valid for any closed loop around the wire, not just a circular one (see problems).

@@Equation @@ can be generalized for any number of wires carrying any number of currents in any direction. We won't go through the derivation, but will simply state the general equation.

B·ds = × total current enclosed by path

Note that the path need not be circular or perpendicular to the wires. The figure below shows a configuration of a closed path around a number of wires:

Figure %: A closed path enclosing 4 wires

The line integral around the circle in the figure is equal to (I 1 + I 2 - I 3 - I 4) . Notice that the two wires pointing downwards are subtracted, since their field points in the opposite direction from the curve.

This equation, similar to the surface integral equation for electric fields, is powerful and allows us to greatly simplify many physical situations.