Energy of a Single Particle

The energy of a single particle is 𝐸particle= 
𝑚𝑐21−𝑣2𝑐2=𝑚𝑐2+𝐾≈𝑚𝑐2+12𝑚𝑣2=Rest energy+Kinetic Energy (𝑣≪𝑐)

Energy Principle for a Particle

work done on particle=∆𝐸particle=𝑊=∫𝐹∙𝑑𝑟 Assume rest energy does not change ∆𝐸particle=𝑊=∆𝐾

Potential Energy

Potential energy is an interaction energy. ∆𝐸1=𝑊on,1=∫𝐹1,surr∙𝑑𝑟+∫𝐹12∙𝑑𝑟  =𝑊1,surr+𝑊12 Work done on particle 1 ∆𝐸2=𝑊on,2=∫𝐹2,surr∙𝑑𝑟+∫𝐹21∙𝑑𝑟  =𝑊2,surr+𝑊21 work done on particle 2 Consider the two particles as a system. internal work happens when the system changes shape. Potential energy is the interaction energy in a system of more than one particle. Therefore potential energy does not exist for a single particle system. ∆(𝐸1+𝐸2)=𝑊surr+𝑊internal Total change in single particle energies ∆(𝐸1+𝐸2)=∆𝐾=𝑊surr+𝑊internal 𝑊internal+∆𝐾=𝑊surr ∆𝑈+∆𝐾=𝑊surr Multiparticle energy principle ⇒∆𝑈=−𝑊internal

Work

The work done on charge particle 𝑞₂ to against the influence due to 𝑞₁ is 𝑊𝑎,𝑏=𝑏∫𝑎𝐹1,2𝑑𝑥=−𝑏∫𝑎𝑞1𝑞24𝜋𝜀0𝑟2𝑟∙𝑑𝑟=𝑞1𝑞24𝜋𝜀01𝑟𝑏｜𝑎=𝑞1𝑞24𝜋𝜀01𝑏−1𝑎
⇒∆𝑈=𝑊surr=𝑞1𝑞24𝜋𝜀01𝑏−1𝑎≡𝑈𝑏−𝑈𝑎

Electric Potential Energy

According to ∆𝑈=−𝑊_internal, the energy difference of a charged partcle in different position is equal to the difference of electric potential energy at different position with respect to the relative postion of the system. In other words, the electric potential energy between two charged particles can be expressed as Electric Potential Energy 𝑈=𝑞1𝑞24𝜋𝜀0𝑟 Assume 𝑣_𝑓−𝑣_𝑖. Then ∆𝐾=0. Doing work on the system increases its potential energy. Work always changes 𝐸_sys, so the potential energy must have changed. The electric potential energy is similar to gravitational potential energy 𝐹=−𝐺𝑚1𝑚2𝑟2𝑟
𝑈grav=−𝐺𝑚1𝑚2𝑟
𝑈grav=−𝐺𝑚Earth𝑚𝑅Earth+ℎ
For 𝑅≫ℎ; 𝑈grav≈−𝐺𝑚Earth𝑚𝑅Earth+𝑚𝑔ℎ+𝑂ℎ𝑅Earth2 For multiple electric charges The electric potential energy of the system is 𝑈ele= ∑𝑖<𝑗𝑈𝑖𝑗= ∑𝑖<𝑗14𝜋𝜀0𝑞𝑖𝑞𝑗𝑟𝑖𝑗

Source and Reference

https://www.youtube.com/watch?v=-uUoR3DDwmo&list=PLZ6kagz8q0bvxaUKCe2RRvU_h7wtNNxxi&index=8