# Electrostatic Potential

**Electrostatic Potential**

The electrostatic potential, also known as the electric field potential, electric potential or potential drop is defined as “The amount of work done to move a unit charge from a reference point to a specific point inside the field without producing acceleration.”

### SI Unit of Electrostatic Potential

*The SI unit of electrostatic potential is volt.*

SI unit of electrostatic potential |
volt |

Other units |
statvolt |

Symbol of electrostatic potential |
V or φ |

Dimensional formula |
ML |

**Electric Potential Formula**

A charge placed in an electric field possesses potential energy and is measured by the work done in moving the charge from infinity to that point against the electric field. If two charges, q_{1} and q_{2} are separated by a distance d, the electric potential energy of the system is:

U = [($\frac{1}{4\pi \epsilon o}$)] × [$\frac{{q}_{1}.{q}_{2}}{d}$]

If two like charges (two protons or two electrons) are brought towards each other, the potential energy of the system increases. If two unlike charges, i.e., a proton and an electron, are brought towards each other, the electric potential energy of the system decreases.

## Electric Potential Derivation

Let us consider a charge q_{1}. Let us say that they are placed at a distance ‘r’ from each other. The total electric potential of the charge is defined as the total work done by an external force in bringing the charge from infinity to the given point.

We can write it as, -∫ (r_{a}→r_{b}) F.dr = – (U_{a} – U_{b})

Here, we see that the point r_{b }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 U_{infity} is equal to zero.

Therefore, -∫ (r →∞) F.dr = -U_{R}

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

⇒ -∫ (r →∞) [$\frac{-kq{q}_{o}}{{r}^{2}}$]dr = -U_{R}

Or, -k × qq_{o} × [$\frac{1}{r}$] = U_{R}

Therefore, **U _{R} = $\frac{-kq{q}_{o}}{r}$**

## Electric Potential of a Point Charge

Let us consider a point charge ‘q’ in the presence of another charge ‘Q’ with infinite separation between them.

U_{E }(r) = k_{e }× [$\frac{qQ}{r}$]

where, k_{e} =$\frac{1}{4\pi {\epsilon}_{o}}$ = Columb’s constant

Let us consider a point charge ‘q’ in the presence of several point charges Q_{i} with infinite separation between them.

**U _{E }(r) = k_{e }q × ∑^{n}_{i = 1 } [$\frac{{Q}_{i}}{{r}_{i}}$]**

## Electric Potential for Multiple Charges

### In the case of 3 charges:

If three charges, q_{1}, q_{2} and q_{3,} are situated at the vertices of a triangle, the potential energy of the system is,

**U =U _{12} + U_{23} + U_{31 }**= ($\frac{1}{4\pi {\epsilon}_{o}}$) × [$\frac{{q}_{1}{q}_{2}}{{d}_{1}}$+ $\frac{{q}_{2}{q}_{3}}{{d}_{2}}$+ $\frac{{q}_{3}{q}_{1}}{{d}_{3}}$]

### In the case of 4 charges:

If four charges, q_{1}, q_{2}, q_{3} and q_{4,} are situated at the corners of a square, the electric potential energy of the system is,

U = ($\frac{1}{4\pi {\epsilon}_{o}}$) × [($\frac{{q}_{1}{q}_{2}}{d}$) + ($\frac{{q}_{2}{q}_{3}}{{d}_{}}$) + ($\frac{{q}_{3}{q}_{4}}{d}$) + ($\frac{{q}_{4}{q}_{1}}{{d}_{}}$) + ($\frac{{q}_{4}{q}_{2}}{\sqrt{2}{d}_{}}$) + ($\frac{{q}_{3}{q}_{1}}{\sqrt{2}{d}_{}}$)]

### Special Case:

In the field of a charge Q, if a charge q is moved against the electric field from a distance ‘a’ to a distance ‘b’ from Q, the work done is given by,

W = (V_{b }– V_{a}) × q = [$\frac{1}{4\pi {\epsilon}_{o}}$_{ }× ($\frac{Qq}{b}$)] – [$\frac{1}{4\pi {\epsilon}_{o}}$ × ($\frac{Qq}{a}$)]

= $\left(\frac{Qq}{4{\mathrm{\pi \epsilon}}_{\mathrm{o}}}\right)[\frac{1}{b}-\frac{1}{a}]$

= $\left(\frac{Qq}{4{\mathrm{\pi \epsilon}}_{\mathrm{o}}}\right)\left[\frac{a-b}{ab}\right]$

**Potential Energy**

An object can store energy because of its location. In the case of the bow and arrow, when the bow is drawn, it stores a certain amount of energy that accounts for the kinetic energy when it is released.

Similarly, in the case of a spring, when it moves from its equilibrium position, it gains energy, which we perceive as the tension we feel in our hands as we stretch it. We can define potential energy as a form of energy that results from a change in its location or state.

After understanding the definition of potential energy and potential energy, we will learn the formula, unit and examples of potential energy.

**Potential Energy Formula**

The formula for potential energy depends on the force acting on the two objects. For the gravitational force, the formula is:

W = m×g×h = mgh

Where,

m is the mass in kilograms

g is the acceleration due to gravity

h is the height in meters

**Types of Potential Energy**

Potential energy is one of the two main forms of energy. There are two main types of potential energy, and they are:

- Gravitational Potential Energy
- Elastic Potential Energy

**Gravitational Potential Energy**

The gravitational potential energy of an object is defined as the energy possessed by an object that rose to a certain height against gravity. We shall formulate gravitational energy with the following example.

Consider an object of mass = m.

Placed at a height h from the ground, as shown in the figure.

As we know, the force required to raise the object equals m×g, that is, the object’s weight.

As the object is raised against the force of gravity, some amount of work (W) is done on it.

Work done on the object = force × displacement.

So,

W = m×g×h = mgh

**Elastic Potential Energy**

Elastic potential energy is stored in objects that can be compressed or stretched, such as rubber bands, trampolines and bungee cords. The more an object can stretch, the more flexible its potential energy. Many objects are specially designed to store elastic potential energy, such as the following:

- A twisted rubber band that powers the toy plane
- · Archer's drawn bow
- · Curved diving board immediately in front of diver
- · Clock winding spring

An object that stores elastic potential energy usually has a high elastic limit. However, all elastic objects have their tolerable load threshold. If an object deforms beyond its elastic limit, it will not return to its original shape.

Elastic potential energy can be calculated using the following formula:

Where,

*U*is the elastic potential energy*k*is the spring force constant*x*is the string stretch length in m