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Answer: The electric potential energy of any given charge or system of changes is termed 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 the 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.
Electric Potential
Denoted by V, ∆V, U, ∆U
Dimension: ML2T-3A-1
General Formula Voltage = Energy/Charge
SI Unit Volt
There are two key elements on which the electric potential energy of an object depends.
It’s own electric charge.
It’s relative position with other electrically charged objects.
⇒ Also Read:
Electrostatics
Equipotential Surface
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/(4πεo)] × [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 and Potential difference
Electric Potential Formula
Method 1:
electric potential at any point around a point charge q
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
electrostatic potential between any two arbitrary charges q1, q2
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 is at infinity is zero (as, r = ∞ in the above formula).
Electric Potential Derivation
Let us consider a charge q1. Let us say, 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, -∫ (ra→rb) F.dr = – (Ua – Ub)
Here, we see that the point rb is present at infinity and the point ra is r.
Substituting the values we can write, -∫ (r →∞) F.dr = – (Ur – U∞)
As we know that Uinfity is equal to zero.
Therefore, -∫ (r →∞) F.dr = -UR
Using Coulomb’s law, between the two charges we can write:
⇒ -∫ (r →∞) [-kqqo]/r2 dr = -UR
Or, -k × qqo × [1/r] = UR
Therefore, UR = -kqqo/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.
UE (r) = ke × [qQ/r]
where, ke = 1/4πεo = Columb’s constant
Let us consider a point charge ‘q’ in the presence of several point charges Qi with infinite separation between them.
UE (r) = ke q × ∑ni = 1 [Qi /ri]
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 = (1/4πεo) × [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 = (1/4πεo) × [(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 = (Vb – Va) × q = [1/4πεo × (Qq/b)] – [1/4πεo × (Qq/a)] = Qq/4πεo[1/b – 1/a] = (Qq/4πεo)[(a-b)/ab]
Explanation: hope it helps
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