Electric Field of a Ring

Electric Field along axis of a Ring

The electric field 𝐸 at ⟨0,0,𝑧⟩ with radius 𝑟 due to charged particle at ⟨-𝑎,0,0⟩ is 𝑟=⟨𝑎,0,𝑧⟩⇒|𝑟|=(𝑎2+𝑧2)1/2⇒𝑟=𝑟|𝑟|=⟨𝑎,0,𝑧⟩(𝑎2+𝑧2)1/2
𝐸 at ⟨0,0,𝑧⟩: 𝐸=14𝜋𝜀0𝑞|𝑟|2𝑟  =14𝜋𝜀0𝑞(𝑎2+𝑧2)⟨𝑎,0,𝑧⟩(𝑎2+𝑧2)1/2=14𝜋𝜀0𝑞(𝑎2+𝑧2)3/2⟨𝑎,0,𝑧⟩ The total electric field 𝐸_𝑡𝑜𝑡 at ⟨0,0,𝑧⟩ due to a ring of charged particles. By symmetry, the electric field 𝐸_𝑡𝑜𝑡 𝐸𝑡𝑜𝑡=  ∑aroundthe ring∆𝐸𝑧𝑧
∆𝐸𝑧=14𝜋𝜀0𝑧∆𝑞(𝑎2+𝑧2)3/2 Let total net charge 𝑄 is uniformlly distributed on the ring with total length 𝐿 of radius 𝑎. The total electric field 𝐸_𝑡𝑜𝑡 at ⟨0,0,𝑧⟩ is 𝐿=2𝜋𝑎; ∆𝑞=𝑄2𝜋𝑎∆𝐿=𝑄𝐿∆𝐿
∆𝐿=𝑎∆𝜃⇒∑∆𝐿→𝑎2𝜋∫0𝑑𝜃=2𝜋𝑎=𝐿
⇒∆𝑞=𝑄2𝜋𝑎𝑎∆𝜃=𝑄2𝜋∆𝜃
⇒∑∆𝑞→𝑄2𝜋2𝜋∫0𝑑𝜃
⇒𝐸𝑡𝑜𝑡 𝑧=14𝜋𝜀0∑𝑧∆𝑞(𝑎2+𝑧2)3/2→14𝜋𝜀0𝑄2𝜋2𝜋∫0𝑧𝑑𝜃(𝑎2+𝑧2)3/2  =14𝜋𝜀0𝑄2𝜋𝑧(𝑎2+𝑧2)3/22𝜋∫0𝑑𝜃  =14𝜋𝜀0𝑄𝑧(𝑎2+𝑧2)3/2

Electric Field along Axis of a disk

The electric field 𝐸 at ⟨0,0,𝑧⟩ with radius 𝑟 due to charged particle ∆𝑞 of ∆A=∆𝑎∆𝐿 at ⟨-𝑎,0,0⟩ is similar to that of a ring. Let total net charge 𝑄 is uniformly distributed on the disk with total area 𝐴 of radius 𝑎. Total 𝑄 in area A=𝜋𝑎2
Elemental ∆𝑞𝑖 in area element ∆A𝑖=(𝑎𝑖∆𝜃)(∆𝑎)
⇒∆𝑞𝑖=𝑄𝜋𝑎2∆A𝑖=𝑄𝜋𝑎2(𝑎𝑖∆𝜃)(∆𝑎)
⇒ ∑𝑖∆𝑞𝑖= ∑𝑖𝑄𝜋𝑎2∆A𝑖=𝑄𝜋𝑎2 ∑𝑖(𝑎𝑖∆𝜃∆𝑎)  =𝑄𝜋𝑎2∬(𝑎𝑖𝑑𝜃𝑑𝑎)=𝑄𝜋𝑎2𝑎∫0𝑎𝑑𝑎2𝜋∫0𝑑𝜃=𝑄𝜋𝑎212𝑎22𝜋=𝑄
⇒𝐸𝑡𝑜𝑡 𝑧=14𝜋𝜀0∑𝑧∆𝑞(𝑎2+𝑧2)3/2→14𝜋𝜀0𝑄𝜋𝑎2∬(𝑎𝑖𝑑𝜃𝑑𝑎)𝑧(𝑎2+𝑧2)3/2  =14𝜋𝜀0𝑄𝜋𝑎2𝑎∫0𝑎𝑧(𝑎2+𝑧2)3/2𝑑𝑎2𝜋∫0𝑑𝜃=14𝜋𝜀0𝑄𝜋𝑎21−𝑧𝑎2+𝑧22𝜋  =12𝜀0𝑄𝜋𝑎21−𝑧𝑎2+𝑧2 ∴𝐸𝑡𝑜𝑡 𝑧=12𝜀0𝑄𝜋𝑎21−𝑧𝑎2+𝑧2𝑧

Infinity Charged Plane

Assume 𝑎 go to infinity, e.g. 𝑎≫𝑧 𝑎→∞; 𝑄→∞ ⇒𝑄𝜋𝑎2=𝜎 charge per unit area
𝐸𝑡𝑜𝑡 𝑧=12𝜀0𝑄𝜋𝑎21−𝑧𝑎2+𝑧2𝑧=12𝜀0𝜎1−0𝑧=𝜎2𝜀0𝑧 For two parallel opposite charged infinite plane

Source and Reference

https://www.youtube.com/watch?v=WqSa620ln9M&list=PLZ6kagz8q0bvxaUKCe2RRvU_h7wtNNxxi&index=6