References Longitudinal electrodynamic forces Summary

Appendix

Notation

The following notation is used throughout the report:

• Vectors are denoted by boldface fonts, as and . Unit vectors have a hat on top of them, as .
• Scalar quantities are written with normal italic fonts, as I and . The same goes for the absolute length of a vector, i.e.
• r (which appears e.g. in the discussion on Maxwell stresses) means the radial distance from the center, in cylindrical coordinates. is the unit vector in the radial direction.
• Not to be confused with the above, is the vector from point (current element) 1 to point (current element) 2; is the length of this vector; and is the unit vector in its direction.
• is the distance from the origin. (It is used in the discussion on surface charges.)

A derivation of Grassmann's law

Biot-Savart's law, well known from magnetostatics, gives the magnetic field from a circuit:

is the magnetic field; the electric current; is an infinitesimal section of the conductor; the distance from to the point where the magnetic field is to be measured,; and the unit vector from to that point.

In differential form it becomes:

The magnetic force ( ) that moving charge experiences in a magnetic field is given by the Lorentz force law:

Here q is the charge that is moving; its velocity; and the magnetic induction. With in vacuum (and air), and noting that we have:

which is Grassmann's law for the force between two current elements.