Electric Potential:

V	=	Vp = V(x,y,z)
	=	The Potential Energy per unit positive charge at a given location in space p = (x,y,z) due to the presence of an electric field at that point.
		SI: Volts = V = J/C

• If PE represents the Electrical Potential Energy then the potential energy (in Joules) at some point p on a charge q due to a voltage V_p at the location of the charge is,

The Electric Potential of a Point Charge Q is its Electrical Potential Energy divided by a unit charge,

Note that only make sense to talk about the voltage at some point if the voltage is measured relative to some reference point that we have set equal to be the zero energy of potential. For a point charge this is at infinity.


Also note that the sign of the charge is important and can not be neglected when calculating the electric potential of a set of point charges.

It is important to keep in mind that r is always a positive number. For example, if you place a positive charge at the origin then the electric potential at a distance of -5.00 cm on the x-axis is exactly the same at distance of +5.00 cm on the x-axis. The sign of the potential is determined by the sign of the charge and not the location distance.
A positive charge creates a positive potential around it in space while a negative charge creates a negative potential around it.

Connection between Work and Voltage:
W_ext = External Work needed to move a charge q from point A to point B.

It takes positive external work to move a positive charge into a region of higher voltage. In this case, you are doing external work against the electric force field. Likewise, it takes positive external work to move a negative charge into a region of lower voltage. Positive Charges that are free to move will move towards regions of lower voltage. Whereas negative free charges tend to move towards regions of lower voltage.

Work is path independent. The work to go from point A to point B is the same no matter which path is taken to go from A to B. The change in voltage around any closed path is always zero. (If you return to your starting point then the net external work you do is zero as is the net work done by the electric field.) For electrical circuits this means that the voltage gains and drops around any circuit loop is always zero. This is one of Kirchhoff's laws.

Connection between Electric Potential and the Electric Field:
W_field = Work done by the Electric Field.

Since the electric potential is work per unit charge,

We can write this more compactly as

Here C is a constant determined by were we set the zero of potential. Its inverse in one-dimension is

In three-dimensions this is the gradient; i.e. the electric field is the gradient of electric potential.

Here are the units vectors in the x,y,z direction.

• The Electric Field Lines point towards regions of lower voltage.
  
  
• The Electric Field Lines are always perpendicular to equal potential lines.
  
  
• The Electric Field is equal to the negative of the Voltage Gradient in space. This is why the E also has units of V/m.
  
  
• Because the potential V is a scalar, it is sometimes easier to determine V(x,y,z) first and then take its derivatives to find the electric field.