Function Conjunction → Functions composed of Physical Expressions

Functions for a point charge

The electric scalar potential at due to a point charge at is:

The magnetic vector potential at due to a point charge which had a velocity at is:

Functions for an ordered pair of point charges

A charge subject to an electric scalar potential at due to a point charge at has an electric potential energy of:

A charge subject to a magnetic vector potential at due to a point charge which had a velocity at has a potential momentum of:

Lorentz Force for

The Lorentz Force between charges can be derived from the scalar potential and the vector potential .

A charge which has a velocity of at will experience a Lorentz force due to a point charge at of:

The electric field is:

The magnetic field is:

The Lorentz Force can be expressed directly in terms of the potentials:

Where:

• = negative the gradient of the scalar potential .
• = negative the partial derivative of the magnetic vector potential with respect to time .
• = the cross product of the velocity of the charge and the curl of the magnetic vector potential due to charge .

To restate from a previous section, the magnetic vector potential of a charge experienced by a charge is:

The partial derivative of this with respect to time is:

Where:

• = the velocity of at the retarded time according to an observer at time sharing the same inertial frame as .
• = the acceleration of at the retarded time according to an observer at time sharing the same inertial frame as .
• = the proximity of the position of at time to the position of at the retarded time (i.e. the inverse of the norm of the vector difference between the two position vectors and ).
• = the partial derivative of this proximity with respect to time .