# Electric Potential

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Electric potential energy is possessed by an object through two elements, the charge possessed by an object itself and the relative position of an object with respect to other electrically charged objects. The amount of work required to move an item from one place to another against an electric field determines the size of the electric potential(EP). Topics like EP are important for various entrance examinations such as the IIT JEE, therefore, having sound knowledge of these topics is crucial. Read the full blog on Electric Potential and learn about its derivation!

## Electric Potential Energy

As per the definition, Electric potential energy is defined as the total potential energy a unit charge will possess if located at any point in outer space.

In other words, the total work done by an external agent in bringing a charge or system of charges from infinity to the original setup without undergoing any acceleration is referred to as the electric potential energy of that charge or system of charges.

It is a scalar quantity with no direction and just magnitude. It is represented by the letter V and is measured in Joules. It has the ML2T-3A-1 dimensional formula.

The energy of an item is determined by two main factors.

• It energy produced
• Its location in relation to other electrically charged objects.

## Electric Potential Formula

A charge in an electric field has potential energy, which is measured by the amount of work required to move the charge from infinity to that point in the electric field. The electric potential energy of the system is; (if two charges q1 and q2 are separated by a distance d):
U = [1/(4πεo)] × [q1q2/d]
When two similar charges (two protons or two electrons) are brought together, the system’s potential energy increases. When two opposite charges, such as a proton and an electron, are brought together, the system’s electric potential energy decreases.

Formula Method 1:

The electric potential at any place in the area of a point charge q is calculated as follows:
V = k × [q/r]
Where,

• V = EP energy
• q = point charge
• r = distance between any point around the charge to the point charge
• k = Coulomb constant; k = 9.0 × 109 N

Formula Method 2:

Coulomb’s law states that the EP between any two arbitrary charges q1 and q2 separated by a distance r and is mathematically expressed as:
U = k × [q1q2/r2]
Here,

• The electrostatic potential energy is denoted by U.
• The two charges are q1 and q2.

Note that the EP at infinity is 0 (as shown by r = in the formula above).

## Electric Potential Derivation

Consider the following charge: q1. Let’s assume they’re separated by a distance of “r” from one another. The entire work done by an external force in moving the charge from infinity to the given point.

It can be written as, -∫ (ra→rb) F.dr = – (Ua – Ub)
The point rb is at infinity, while the point ra is r, as we can see.
Substituting the values we can write, -∫ (r →∞) F.dr = – (Ur – U∞)
Uinfinity is equivalent to zero, as we all know.
As a result, – (r) F.dr = -UR
Using Coulomb’s law, we can write the following between the two charges:
⇒ -∫ (r →∞) [-kqqo]/r2 dr = -UR
Alternatively, -k × qqo × [1/r] = UR
Therefore, UR = -kqqo/r

## Electric Potential for a Point of Change

Consider a point charge ‘q’ in the presence of another charge ‘Q’ separated by an infinite distance.
UE (r) = ke × [qQ/r]
Here, ke = 1/4πεo = Coulomb’s constant

Consider a single point charge ‘q’ in the presence of many point charges Qi separated by an indefinite distance.
UE (r) = ke q × ∑ni = 1  [Qi /ri]
Electric Potential for Multiple Charges

### In the case of 3 Charges:

The potential energy of a system with three charges q1, q2, and q3 at the vertices of a triangle is,
U =U12 + U23 + U31 = (1/4πεo) × [q1q2/d1 + q2q3/d2 + q3q1/d3]

### In the case of 4 Charges:

The electric potential energy of the system is, if four charges q1, q2, q3, and q4 are placed at the four corners of a square.
U = (1/4πεo) × [(q1q2/d) + (q2q3/d) + (q3q4/d) + (q4q1/d) + (q4q2/√2d) + (q3q1/√2d)]

Special Cases:

The work done in the field of a charge Q is given by, if a charge q is moved against the electric field from a distance ‘a’ to a distance ‘b’ from Q.
W = (Vb – Va) × q = [1/4πεo × (Qq/b)] – [1/4πεo × (Qq/a)] = Qq/4πεo[1/b – 1/a] = (Qq/4πεo)[(a-b)/ab]

## Important Points to Remember

• The electric potential is zero at a point midway between two equal and opposing charges, but the electric field is not.
• If one joule of effort is spent in pushing one Coulomb of the charge against the electric field, the electric potential at that point is said to be one volt.
• When a negative charge is transferred from point A to point B, the system’s electric potential increases.
• Infinity is the reference level used to describe EP at a point. It denotes that at the reference level, the force on a test charge is zero.
• Because the earth is so massive that adding or subtracting charge from it does not affect its electrical state, the surface of the earth is assumed to be at zero potential.

## Electrical Potential Difference

The potential between two points (E) in an electrical circuit is defined as the amount of work (W) done by an external agent in transferring a unit charge (Q) from one point to another.

On a mathematical basis, we may state,
E = W/Q
Here,
E = the difference in electrical potential between two locations.
Q = Quantity of charge in coulombs
W = Work done in transferring a charge from one place to another

## FAQs

Q. What is the difference in electric potential?

A. The potential between two points (E) in an electrical circuit is defined as the amount of work (W) done by an external agent in moving a unit charge (Q) from one point to another. On a mathematical basis, we may state, W/Q = E

What is Electric Potential Energy?

The total potential energy a unit charge will have if it is located anywhere in space is described as electric potential energy.

In this blog, we discussed Electric Potential along with its formula and the derivation. We hope the information provided was helpful. Stay connected with Leverage Edu for more educational content and amazing quizzes!

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