Potentials and Gauge Transformations

For a number of purposes it is more convenient to work in terms of field-related potentials rather than the fields B and E themselves. To find these potentials, we start with the two homogeneous Maxwell equations. The simplest of these is the equation in B  

Using the relation in (3.5), we see that the magnetic field may be expressed as the curl of another vector A  

With this choice for B, (3.8) is automatically satisfied, and the corresponding properties of the field are built into any further theoretical developments. 

This time we can use another orthogonality relation involving differential operators. For any scalar we have 

Using this relation, we can introduce a scalar - p defined by 

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