Electric Potential

For two fixed opposite charged plates, the electric field due to plates is as following If a charged particle is placed in the capacitor, an induced force will drive the charged particle to move. The eletric pontential energy acting on the charge particle 𝑞_𝑖 is 𝑈ele= ∑𝑗𝑈𝑖𝑗= ∑𝑗14𝜋𝜀0𝑞𝑖𝑞𝑗𝑟𝑖𝑗 The electric potential is defined as the potential or the ability to have potential energy if a test charge enters the system. In other words, the electric potential can be expressed as 𝑉≡𝑈ele𝑞=JoulesCoulomb=[Volts]

Potential Difference

When the charge particle is moved from 𝑎 to 𝑏 as following the change in potential energy of the system is ∆𝑈=−𝑊internal=−∫𝑞𝐸∙𝑑𝑟=−𝑞∫𝐸𝑥𝑑𝑥
⇒∆𝑈=−𝑞𝐸∆𝑥 In which the systems tends to lower their potential energy by moving the charge particle from a higher potential energy position to a lower potential energy position. In other words, the potential difference drive the charge particle to move from one position to another. ∆𝑈=−𝑞𝐸∆𝑥≡𝑞∆𝑉⇒∆𝑉=∆𝑈𝑞=−𝐸∆𝑥
𝑉=[Volt]=JouleCoulomb
∆𝑈/𝑞=−𝐸∆𝑥≡∆𝑉⇒𝐸=−∆𝑈/𝑞∆𝑥=−∆𝑉/∆𝑥
|𝐸|=JouleCoulomb-meter=NewtonCoulomb=VoltMeter An electron-Volt, 𝑒𝑉 is the energy required to move a charge particle, 𝑞=1𝑒=1.6×10⁻¹⁹𝐶 through 1𝑉. That is 1.6×10⁻¹⁹𝐽=1𝑒𝑉

Non-uniform Electric Field

For a uniform electric field, i.e. 𝐸∥𝑥, ∆𝑉=−𝐸∆𝑥. And for a uniform electric field pointing in any direction. ∆𝑉=−(𝐸𝑥∆𝑥+𝐸𝑦∆𝑦+𝐸𝑧∆𝑧)≡−𝐸∙𝑑𝑟 If the electric field 𝐸 is not uniform, but varies in space. The potential difference can be determined by a line integral like stepping along a path. At each step, use component of electric field parallel to the step direction. ∆𝑉=−𝑓∑𝑖𝐸∙𝑑𝑙=−𝑓∫𝑖𝐸∙𝑑𝑙 The electric field can therefore be expressed as 𝐸=−𝑑𝑉𝑑𝑥=−∂∂𝑥𝑉𝑥+∂∂𝑦𝑉𝑦+∂∂𝑧𝑉𝑧=−∇𝑉

Path of Potential Difference

Since electricity is a conservative force the potential difference ∆𝑉 is independent of the path taken. For example, in a simple uniform electric field ∆𝑉=−𝑓∑𝑖𝐸∙𝑑𝑙=−𝑓∑𝑖𝐸𝑥𝑑𝑙𝑥+𝑓∑𝑖𝐸𝑦𝑑𝑙𝑦+𝑓∑𝑖𝐸𝑧𝑑𝑙𝑧=𝑉𝑓−𝑉𝑖

Source and Reference

https://www.youtube.com/watch?v=8HSwBeHZoao&list=PLZ6kagz8q0bvxaUKCe2RRvU_h7wtNNxxi&index=9