Definition
Given two points in space, $x_{A}$ and $x_{B}$, voltage is the difference in electric potential between those two points. From the definition of electric potential it follows that:
 $\Delta V_{BA}=V(x_{B})V(x_{A})=\int _{r_{0}}^{x_{B}}{\vec {E}}\cdot d{\vec {l}}\left(\int _{r_{0}}^{x_{A}}{\vec {E}}\cdot d{\vec {l}}\right)$


 $=\int _{x_{B}}^{r_{0}}{\vec {E}}\cdot d{\vec {l}}+\int _{r_{0}}^{x_{A}}{\vec {E}}\cdot d{\vec {l}}=\int _{x_{B}}^{x_{A}}{\vec {E}}\cdot d{\vec {l}}$
The electric field around the rod exerts a force on the charged pith ball, in an
electroscope
In a static field, the work is independent of the path
Electric potential is electric potential energy per unit charge, measured in joules per coulomb (volts). "Electric potential" must be distinguished from "electric potential energy" by noting that the "potential" is a "perunitcharge" quantity. Like mechanical potential energy, the zero of electric potential can be chosen at any point, so the difference in potential, i.e. the voltage, is the quantity which is physically meaningful. The voltage between point A to point B is equal to the work which would have to be done, per unit charge, against or by the electric field to move the charge from A to B. The voltage between the two ends of a path is the total energy required to move a small electric charge along that path, divided by the magnitude of the charge. Mathematically this is expressed as the line integral of the electric field and the time rate of change of magnetic field along that path. In the general case, both a static (unchanging) electric field and a dynamic (timevarying) electromagnetic field must be included in determining the voltage between two points.
Historically this quantity has also been called "tension" and "pressure". Pressure is now obsolete but tension is still used, for example within the phrase "high tension" (HT) which is commonly used in thermionic valve (vacuum tube) based electronics.
Voltage is defined so that negatively charged objects are pulled towards higher voltages, while positively charged objects are pulled towards lower voltages. Therefore, the conventional current in a wire or resistor always flows from higher voltage to lower voltage. Current can flow from lower voltage to higher voltage, but only when a source of energy is present to "push" it against the opposing electric field. This is the case within any electric power source. For example, inside a battery, chemical reactions provide the energy needed for ion current to flow from the negative to the positive terminal.
The electric field is not the only factor determining charge flow in a material, and different materials naturally develop electric potential differences at equilibrium (Galvani potentials). The electric potential of a material is not even a well defined quantity, since it varies on the subatomic scale. A more convenient definition of 'voltage' can be found instead in the concept of Fermi level. In this case the voltage between two bodies is the thermodynamic work required to move a unit of charge between them. This definition is practical since a real voltmeter actually measures this work, not a difference in electric potential.