Electric Potential Energy: Formula, Definition, Solved Examples - JEE PDF Download

What is Electric Potential Energy?

The electric potential energy of any given charge or system of changes is defined as the total work done by an external agent in bringing the charge or the system of charges from infinity to the present configuration without undergoing any acceleration.
Definition: Electric potential energy is defined as the total potential energy a unit charge will possess if located at any point in outer space.

Overview

Electric potential energy is a scalar quantity and possesses only magnitude and no direction. It is measured in terms of Joules and is denoted by V. It has the dimensional formula of ML²T^-3A^-1.

There are two key elements on which the electric potential energy of an object depends:

• Its own electric charge.
• Its relative position with other electrically charged objects.

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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, q₁ and q₂, are separated by a distance d, the electric potential energy of the system is:
U = [1/(4πε_o)] × [q₁q₂/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 Formula
Method 1:

The electric potential at any point around a point charge q is given by:
V = k × [q/r]

Where,

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

Method 2: Using Coulomb’s Law

The electrostatic potential between any two arbitrary charges q₁, q₂ separated by distance r is given by Coulomb’s law and mathematically written as:
U = k × [q₁q₂/r²]

Where,

• U is the electrostatic potential energy
• q₁ and q₂ are the two charges

Note: The electric potential at infinity is zero (as r = ∞ in the above formula).

Electric Potential Derivation

Let us consider a charge q₁. 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 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 U_infity 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

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 × [qQ/r]
where, k_e = 1/4πε_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 × ∑ⁿi = 1  [Q_i /r_i]

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Electric Potential for Multiple Charges

In the case of 3 charges:
If three charges, q₁, q₂ and q₃, are situated at the vertices of a triangle, the potential energy of the system is,
U =U₁₂ + U₂₃ + U₃₁ = (1/4πε_o) × [q₁q₂/d₁ + q₂q₃/d₂ + q₃q₁/d₃]
In the case of 4 charges:
If four charges, q₁, q₂, q₃ and q₄, are situated at the corners of a square, the electric potential energy of the system is,
U = (1/4πε_o) × [(q₁q₂/d) + (q₂q₃/d) + (q₃q₄/d) + (q₄q₁/d) + (q₄q₂/√2d) + (q₃q₁/√2d)]
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 = [1/4πε_o × (Qq/b)] – [1/4πε_o × (Qq/a)] = Qq/4πε_o[1/b – 1/a] = (Qq/4πε_o)[(a-b)/ab]

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What is Electric Potential Difference?

In an electrical circuit, the potential between two points (E) is defined as the amount of work done (W) by an external agent in moving a unit charge (Q) from one point to another.
Mathematically we can say that, 
E = W/Q
Where,

• E = Electrical potential difference between two points
• W = Work done in moving a charge from one point to another
• Q = Quantity of charge in coulombs

Example: Let us say we have two charges of magnitude 1C and 2C placed at a distance of 2 metres from each other. Calculate the electric potential between these two charges. (Take: k = 1)
Solution:
Given that, the magnitude of charges is q₁ = 1C and q₂ = 2C.
The distance between these two charges is r = 2m.
The electric potential between these two charges is given by, Ur = -[kqq_o]/r
Substituting the given values in the above equation, we get,
U_r = -1 J.