2 Engineering Example 4
2.1 Magnetic flux
Introduction
The magnetic flux through a surface is given by where is the surface under consideration, is the magnetic field and is the vector normal to the surface.
Problem in words
The field generated by an infinitely long vertical wire on the -axis is given by
Find the flux through a rectangular region (with sides parallel to the axes) on the plane .
Mathematical statement of problem
Find the integral over the surface, , . (see Figure 11 which shows part of the plane for which the flux is found and a single magnetic field line. The strength of the field is inversely proportional to the distance from the axis.)
Figure 11:
Mathematical analysis
On , so and the flux is given by the double integral
Interpretation
The magnetic flux increases in direct proportion to the extent of the side parallel to the axis (i.e. along the -direction) but logarithmically with respect to the extent of the side perpendicular to the axis (i.e. along the -axis).
Example 25
If , evaluate where is the part of the plane bounded by , .
Solution
Here and hence and
The integral
Similarly
.
Thus
Key Point 5
- An integral of the form evaluates to a scalar.
- An integral of the form evaluates to a vector.
The vector function involved may be the gradient of a scalar or the curl of a vector.
Example 26
Integrate where and is the area between and for and . (See Figure 12.)
Figure 12:
Solution
Here and . Thus and
For integrals of the form , non-Cartesian coordinates e.g. cylindrical polar or spherical polar coordinates may be used. Once again, it is necessary to include any scale factors along with the unit normal.
Example 27
Using cylindrical polar coordinates, (see HELM booklet 28.3), find the integral for and being the complete surface (including ends) of the cylinder , . (See Figure 13.)
Figure 13:
Solution
The integral
must be evaluated separately for the curved surface and the ends.
For the curved surface,
(with the
coming from
the scale factor for
and the fact that
on the curved surface.) Thus,
and
On the bottom,
so
and the contribution to the integral is zero.
On the top,
and
and
and
So