# 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:

Using the product rule, the partial derivative of this with respect to time can be found. For example, the derivative of a product of two variables and with respect to time is:

.

Therefore, the partial derivative of the magnetic vector potential at due to with respect to time is:

 The partial derivative, with respect to time , of the proximity of the position of at time to the position of at the retarded time . According to an observer at time : the velocity a charge had at the retarded time , when it emitted a light signal which has now reached at position and time
 The proximity of the position of at time to the position of at the retarded time . According to an observer at time : the acceleration a charge had at the retarded time , when it emitted a light signal which has now reached at position and time