# Electric Field and Electric Potential

Two charged objects exert a force on one another in proportion to the magnitude of their charges and the inverse square of their distance apart. If one of the charges was reduced to a single positive unit charge, then the force on single unit of charge would be only dependent on the magnitude of the other charge and inversely proportional to the square of their distance apart.

If we move the unit charge to a different position we would obtain another value for the force on the unit charge. As we move the unit charge to many places around the charge we would build up a picture of the force acting on the charge. This picture of the force around a charged object we call an electric field and the force that we measure on the unit charge is called the electric potential.

The electric field **E** is a vector quantity defined as the force per unit charge.

**E** = **F**/*q*

We know a charged object creates an electric field which has an effect on another charged object which is brought close to the other. The electric potential is the force per unit charge. Electric field is measured in Volts.

The force between point charges is given by Coloumb's law, where **r^** is a unit vector along the line of force between *AB*

**F** = *Qq***r^**/(4πε_{0}*r*^{2})

For a unit charge *q* = 1, therefore **F** = *Q***r^**/(4πε*r*^{2})

The work done is the force multiplied by the distance. However, since the force is not a constant with distance we need to integrate the force over the path.

## Potential Difference

To move a charge from *A* to *B* in an electric field requires that work is done. The potential difference is

*V*_{AB} = ∫_{A}^{B}**E**.*d***l** = ∫_{A}^{B}*q*/(4πε_{0}*r*^{2}) =[-*q*/(4πε_{0}*r*)] _{A}^{B}

= *q*/(4πε_{0}*A*) - *q*/(4πε_{0}*B*)

= *q*/(4π&epsilon_{0}) [(1/*A*) - (1/*B*)]

Alternatively, **E** = -( *dV**dx* + *dV**dy* + *dV**dz* ) = -∇*V*

## Zero of Potential

We are usually interested in the differences in the potential between to points, but if we agree on a single point of reference to take as a zero of potential we can talk about potential with respect to this point. In theoretical problems where we are considering point charges it is conventional to take the zero of potential as being at infinity. In more practical situations we often use the Earth as another convenient point reference of zero potential.