Deriving Electric Field From Potential

Electric potential energy is the process that happens due to two elements-one which is possessed by the object itself, the other is the relative position of the object. The result of the electric potential completely depends on the total work done in moving the object from one point to another.

Imagine you have a negative charged plate, with a little positive charged particle stuck to it through the electric force. You will witness an electric field around the plate pulling positively charged objects towards it. Now take the positive particle, and pull it off the plate against the electric field. This could be hard work because the electric force is pulling all together. The energy used for moving the particle from the plate is stored in the particle as the electrical potential energy. 

What is Electrical Potential Energy?

To understand about the deriving electric field from potential, it is important to know the meaning of the electrical potential energy. Electric potential energy of the given charge or system of changes is termed as the total work done by the external agent to bring the charge or the system of charges. In other words, electric potential energy is defined as the total potential energy a unit charge will possess if located at any point in the exterior space. 

Overview

Electric potential energy is scalar quantity and possesses only magnitude and no direction. 

Electric Potential Formula:

Charge placed in an electric field possesses potential energy and is measured by the work done in moving the charge from infinity to the point against the electric field. If two charges q₁ and q₂ are separated by the distance D, electric potential energy of the system is- U = 1/ (4πε_o) × [q₁q₂/d].

What is Electric Potential?

Electric potential or voltage is the difference in potential energy per unit charge between two locations in an electric field. It is important to know that the amount of charge you are pushing or pulling makes a huge difference to the electrical potential energy. So is why physicists use single positive charge as our imaginary charge to test out the electrical potential. 

Electric Potential Derivation:

To understand this, you need to consider a charge q₁. Let's say, they are placed at the distance "r" from each other. Total electric potential of the charge is defined as the total work done by an external force.

We can write it as, -∫ (r_a → r_b) F.dr = – (U_a – U_b)

Here, we see that the point rb is present at infinity and the point r_a is r.

Substituting the values we can write, -∫ (r →∞) F.dr = – (U_r – U_∞)

As we know that Infinity is equal to zero.

Therefore, -∫ (r →∞) F.dr = -U_R

Using Coulomb’s law, between the two charges we can write:

⇒ -∫ (r →∞) [-kqq_o]/r² dr = -U_R

Or, -k × qq_o × [1/r] = U_R

Therefore, U_R = -kqq_o/r

What is Meant By Electric Potential Difference?

In an electrical circuit, potential between two points (E) is defined as the amount of work done (W) by the external agent in moving the unit charge. 

In mathematical way we can say that: E = W/Q.

E = Electrical potential difference between two points.

W = Work done in moving a change from one point to another.

Q = Quantity of charge in coulombs.