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Electric Potential
   
Electric Potential, V at a Point in an Electric Field  
The potential at a point in a field is the work done per unit charge moving a small positive charge from infinity to that point.  
Therefore, the units of V are JC-1  
As with gravitational potentials, the zero of electrical potential is at infinity* but electrical potentials can be positive or negative depending on whether work is done against the field or by the field.  
*Remember that this just means, far enough away for the field strength to have fallen to an undetectable value...  
   
If the potential at point A is VA and the potential at point B is VB then the potential difference, V, between the A and B is given, not surprisingly, by  
 
We tend to use V for potential differences and V plus subscript for potential at a point.  
If a charge, q, is moved through a potential difference, V, we can write  
   
As in the case of electric field strength, we usually miss out the "+" sign. It is included here as a reminder that the definitions of potential and potential difference are based on moving a positive charge.  
   
Potential at a Distance r from a Point Charge Q  
 
To find a figure for the potential at point p we have to imagine bringing a small positive test charge from infinity (a very long way away!) to p.  
This task is complicated by the fact that the force varies in strength as the charge is moved.  
This can be done by considering moving the test charge through very small steps (steps small enough to allow us to consider the force to be essentially constant) and then adding up the effects of all these small steps by the mathematical process of integration.  
Doing this we find  
   
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