Electric Potential

         I spend little time on thinking about Electric Potential while reading about Electrodynamics. It is really a great amazement the concept of Field and Potential is just created for the sake of Mathematical Simplicity. But it was realized later only that the field concept is a real Phenomenon.  

Electric Potential is defined as the work required per unit charge to move it in some linear path.

The definition and the sign of the potential functions are based only on our conventional sign systems. Eg. The potential of a positive charge is chosen to be a positive quantity with respect to the reference frame of point at infinity.

It is just our convention to use that the potential of positive charge is hill and the potential of a negative charge is a valley. But it can be chosen alternatively if we desire. For the sake of conventional mathematics we chose some sign conventions. These sign conventions also enter in Thermodynamics.

In mathematical form Potential difference between two points is defined as

Va – Vb = - ∫ E . dl

where E is Electric field vector

dl is linear integral path vector

V is called the Potential. The integral is from point “a” to point “b”.

The interesting relation between the potential function and the Electric field is that they are related in the following way

E = - ∂V/∂r

or                                            E  = - ΔV

From this equation we can simply deduce the following result that

if Electric field in any space is zero then it implies the potential at those points is const. And that is why equipotential surfaces are always perpendicular to the direction of the Electric field. If it is not then it will be contradictory to the above Equations.

But the great thing about the potential is, it is a scalar quantity. A scalar quantity in Nature simply turns out to be vector quantity in Nature by a simple mathematical formulation. Isn’t it just great?

It says that though the potential is scalar one, it always follows some rule from the Nature itself. And the rule is just expressed as the “gradient” operator “Δ” in mathematics.

This is the reason why Mathematics serves as the language for Physics.