Electric Potential - Important Concepts with Formulas for JEE

Electric potential is defined as the amount of work done to move a unit of positive electric charge from one reference point to a specific point acting against the electric field. In short, the electric potential is the potential energy per unit charge. The value of the electric potential can be calculated in a static or dynamic electric field at a specific time in units of joules per coulomb or volts. 

In simple words, the electric potential is work per unit of charge. For example, a battery of 1.5 V has an electric potential of 1.5 volts. This means the battery can do work or supply electric potential energy in the electric circuit of 1.5 joules per coulomb. Electric potential difference is used to control charge motion; for example in a TV screen or electron microscope. 

Electric Potential Formula

Ohm's law gives the electric potential formula: $V=R\times I$ 

Here, R is the resistance, measured in ohm $\left( \Omega  \right)$ , I - electric current measured in ampere (A), and V - voltage measured in volts (V). Here, Volt is equal to ohm multiplied by ampere, and the equation is $V=\Omega \times A$

What is Electric Potential Difference?

In an electrical circuit, the electric potential difference between two points (E) is the work done (W) by an outer agent considering the movement of a unit charge (Q) from one point to another point. The SI unit of potential difference is volt.

Mathematically, the potential difference formula is $E=\dfrac{W}{Q}$ 

(Here, E - electric potential difference, W- work done and Q - unit charge.)

Electric Potential at a Point Due to Point Charge

Point charges like electrons are the building blocks of matter. Furthermore, a spherical charge creates electric fields exactly like a point charge. The electric potential due to a point charge is found by considering important factors such as – work done, test charge, distance, and point charge. 

First, move a test charge 'q' from a distance away from a distance 'r' from a point charge 'Q.' Then notice the connection between the work and potential, which is derived as $W=-q\Delta V$ . It shows the electric potential of a point charge is;

The electric potential of a point charge is  

$V=\dfrac{kQ}{r}$ 

or

$V=\dfrac{1}{4\pi {{\varepsilon }_{\circ }}}\dfrac{Q}{r}$

Here, K is the coulomb constant, $k=\dfrac{1}{4\pi {{\varepsilon }_{\circ }}}=9\times {{10}^{9}}N{{m}^{2}}{{C}^{-2}}$, Q is the point charge and r is the distance of separation.

Electric potential is a scalar quantity, while the electric field is a vector. 

Electric Potential at a Point Due to Multiple Charges

The electric potential at a point in space which is produced by multiple point charges can be calculated by adding the point charges. 

The electric field from the multiple point charges is obtained by the vector sum of the electric fields of the charges. When there are a group of point charges, such as Q₁, Q₂, Q₃…Q_n from a distance of r₁, r₂, r₃,..r_n, you get the value of the electrostatic potential at any particular point. You can then add charges algebraically.

$V=\dfrac{k{{Q}_{1}}}{{{r}_{1}}}+\dfrac{k{{Q}_{2}}}{{{r}_{2}}}+\dfrac{k{{Q}_{3}}}{{{r}_{3}}}$ 

$V=\dfrac{1}{4\pi {{\varepsilon }_{\circ }}}\sum\limits_{i=1}^{n}{\dfrac{{{Q}_{i}}}{{{r}_{i}}}}$

Here, Q₁, Q₂, Q₃ are the charges and r₁, r₂ and r₃ are the distances between the charges.

What is Electric Potential Energy?

Electric potential energy is the form of energy needed to move the charges against an electric field. In the electric field, you need more energy to move the charge and also need the energy to move it via a stronger electric field. The unit of potential energy is Joules.

• Case 1 – Negatively Charged Plate

A negatively charged plate stuck through the electric force with a bit of positively charged particle. The presence of an electric field which is surrounding the plate pulls all positively charged objects towards it.

Take the positive particle and pull it off the plate against the electric field. It is hard work as the force is pulling them together. When the positive particle goes, it will snap back to the negative plate, which is pulled by the electric force. Here, the energy you utilise to move the particle from the plate is known as electrical potential energy.

While pulling the positive particle away from the plate, you need to use more energy, so that it can have more electrical potential energy. When the charge has doubled the charge on the plate, you will need even more energy to move the positive particle. And when you double the charge on the positive particle, you will need more energy to move it. 

• Case 2 – Positively Charged Plate

When you use a positively charged plate instead of a negative one, the positive particle will get pushed away from the plate because both carry positive. In this situation, you must put in the energy to move it closer to the plate instead of pulling action. The closer you try to move, the more energy you have to apply to have more electrical potential energy on the particle.

Electric Potential Energy Formula

When a charge is placed in an electric field, it possesses potential energy. It can be measured by the amount of work done in moving the electric charge from infinity to a point against the electric field. In the case of two charges, q₁ and q₂, which are separated at a distance of d, the total electrostatic potential energy formula is 

$U=\dfrac{1}{4\pi {{\varepsilon }_{\circ }}}\times \dfrac{{{Q}_{1}}{{Q}_{2}}}{d}$

$U=\dfrac{1}{4\pi {{\varepsilon }_{\circ }}}\times \dfrac{{{q}_{1}}{{q}_{2}}}{d}$

Here, U is Electric Potential Energy, q₁ and q₂ are charges and ‘d’ is the distance.

Conclusion

In this article, we have come across the concept of electric potential. We learnt the definition and formulae related to electric potential. Further, we saw that the electric fields are vectors that have magnitude and direction at each point. The electrical properties can be described through electric potential. Electrical potential is a simpler and more practical concept. It is a scalar quantity and has no direction. Electric potential is considered more practical than electric fields due to the differences in potential.