Potential Energy of a System of Point Charges
The electric potential energy of a point charge system is the work required to assemble the system by bringing the charges together from infinitely far away.
Definition of Point Charge
A point charge is an idealized charge concentrated at a single point in space. It is used to analyze complex systems with multiple charges more easily.
Coulomb’s Law and Force Between Point Charges
Coulomb’s Law describes the force between two point charges. It states that the force is directly proportional to the product of the charges and inversely proportional to the square of the distance between them.
`F\ =\ \frac{k\ q_1q_2}{r^2}`
Where,
K = Coulomb’s Constant
` q_1` and ` q_2` = magnitudes of the two charges and
r = distance between them.
Concept of Electric Potential Energy
The electric potential energy represents the energy stored in a system due to its position of configuration. Potential energy refers to the energy associated with the relative positions of charged particles.
Mathematical Formula for Electrostatic Potential Energy
The electrostatic potential energy U between two points charges is given by –
`U = \frac{k q_1 q_2}{r}`
The electrostatic potential energy U is scalar and its SI unit is Joule (J). It is also expressed in terms of electron volts (eV) in atomic and particles.
System of Point Charges
Electrostatic Potential Energy of a System of Two-Point Charges
In assembling two-point charges, work is done against the electrostatic forces this work done is stored in the form of electrostatic potential energy.
Energy is required to bring two-point charges from infinite separation to a specific distance apart.
We imagine that `q_1` and `q_2` are two-point charges and similar in charge and are at a distance r.
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Electrostatic Potential Energy of a System of Two-Point Charges |
We again imagine that `q_1` and `q_2` were initially at rest at infinite distance. If charge `q_1` is taken out from infinity to its present position, then there was no electric field present. Thus, neither there was any field nor any work done by the external agent.
When charge `q_2` was taken out from infinity to its present position, then the external agent has to work against the electrostatic force.
Potential due to `q_1` at the position of point charge `q_2` .
`V = \frac{k\ q_1}{r}`
From the definition of potential
`U= q_2 V`
So that
`U = \frac{k q_1q_2}{r}`
Important Points
* If both charges are of same sign, then the work done by the external agent will be positive.
* If both charges are of opposite sign, then the work done by the external agent will be negative.
* The work done is stored in the form of the potential energy of the system.
* Electric potential energy is different from gravitational potential energy.
* Electric potential energy can be positive or negative according to the nature of the charge whereas gravitational potential energy is always negative.
Electrostatic Potential Energy of a System of Three Point Charges
To determine the electrostatic potential energy of a system of three point charges (or more than two point charges), we do the algebraic sum of electric potential energies of all possible pairs of two charges. The net electric potential energy of the system is equal to sum of these energies.
Calculation of Electric Potential Energy
Consider that the three point charges `q_1` , `q_2` and `q_3` are forms a system of three point charges
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Electrostatic Potential Energy of a System of Three Point Charges |
The total electrostatic potential energy is calculated by sunning the potential energies between each pair.
Electrostatic potential energy between point charges `q_1` and `q_2`
Work done in bringing the charge `q_2` from infinity to point B is stored in the form of electrostatic potential energy.
`U_{12} = \frac{k\ q_1q_2}{r_{12}}`
Electrostatic potential energy between point charges `q_1` `q_2` and `q_3`
Work done in bringing the charge `q_3` from infinity to point C is stored in the form of electrostatic potential energy.
`U_{123} = \frac{k\ q_1q_3}{r_{13}} + \frac{k\ q_2q_3}{r_{23}}`
Total electrostatic potential energy is
`U\ =\ U_{12}\ +\ U_{123}`
`U_123 = `` \frac{k\ q_1q_2}{r_{12}}` + ` \frac{k\ q_1q_3}{r_{13}} + \frac{k\ q_2q_3}{r_{23}}`
Questions and Answers
Question -
Define point charge, and why is it used in analyzing complex systems?
A point charge is an idealized charge concentrated at a single point in space. It is used to analyze complex systems with multiple charges more easily.
Question -
According to Coulomb’s Law, what factors determine the force between two point charges?
According to that the force is directly proportional to the product of the charges and inversely proportional to the square of the distance between them.
`F = \frac{k q_1q_2}{r^2}`
Where,
K = Coulomb’s Constant
`q_1` and `q_2` = magnitudes of the two charges and
r = distance between them.
There are three factors to determine the force between two charges
* The magnitude of the charges
* The distance between the charges
* Medium between the charges
Question -
Explain the concept of electric potential energy and how it is stored in a system of point charges.
Electric potential energy is the energy stored in a system due to the arrangement of charges. This type of energy is related to the relative positions of charged particles within the system.
Question -
Write the formula of the electrostatic potential energy of a system of three point charges calculated?
The formula of the electrostatic potential energy of a system of three point charges is
`U_123 = `` \frac{k\ q_1q_2}{r_{12}}` + ` \frac{k q_1q_3}{r_{13}} + \frac{k q_2q_3}{r_{23}}`
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