Charge Density

For a simple 1d charged rod of length L in meter has total charge 𝑄. Assuming there are 10 point charges in a row then

Electric Field of Charged Rod

The electric field of charged rod can be calcuted as following ∆𝑥=𝐿/10⇒∆𝑞=𝑄/10=(𝑄/𝐿)∆𝑥
𝐿=∑∆𝑥=∫𝑑𝑥
𝑄=∑∆𝑞=𝑄𝐿∑∆𝑥=𝑄𝐿∫𝑑𝑥

Electric Field in the Bisecting Plane

The electric field in the bisecting plance can be determined as following 𝑟=(𝑥2+𝑦2)1/2 and sin 𝜃=𝑥𝑟𝑖
𝐸𝑡𝑜𝑡=∑𝑖∆𝐸𝑖=∑𝑖∆𝐸𝑖,𝑥𝑥
∆𝐸𝑖,𝑥=|∆𝐸𝑖|cos(𝜃)  =14𝜋𝜀0∆𝑞𝑟2𝑖𝑥𝑟𝑖  =∆𝑞4𝜋𝜀0𝑥(𝑥2+𝑦2𝑖)3/2
𝐸𝑡𝑜𝑡=∑𝑖∆𝑞4𝜋𝜀0𝑥(𝑥2+𝑦2𝑖)3/2𝑥  =14𝜋𝜀0𝑄𝑥𝐿𝐿/2∫−𝐿/2𝑑𝑦(𝑥2+𝑦2𝑖)3/2𝑥  =14𝜋𝜀0𝑄𝑥𝑥2+(𝐿/2)2𝑥 For an infinite rod 𝐿→∞; 𝑄→∞ ⇒𝑄𝐿→𝜆 Where 𝜆 is defined as charge per unit length. 𝐸𝑡𝑜𝑡=14𝜋𝜀0𝑄𝑥𝐿∞∫−∞𝑑𝑦(𝑥2+𝑦2𝑖)3/2𝑥=14𝜋𝜀0𝑄𝑥𝐿2𝑥2𝑥=14𝜋𝜀02𝜆𝑥𝑥 Therefore, for a finite rod of length 𝐿 only on the bisecting plane 𝐸𝑡𝑜𝑡=14𝜋𝜀0𝑄𝑥𝑥2+(𝐿/2)2𝑥 And for an infinite rod of length 𝐿→∞, and 𝑄/𝐿 is with linear charge density and is not equal to infinite or zero 𝐸𝑡𝑜𝑡=14𝜋𝜀02𝜆𝑥𝑥

Procedure for Calculating Electric Field of Distributed Charges

• Cut the charge distribution into pieces for which the field is known.
• Write an expression for the electric field due to one piece
  • Choose origin
  • Write an expression for ∆𝐸 and its components
• Add up the contributions of all the pieces
  • Try to integrate symbolically
  • If impossible-integrate numerically
• Check the results
  • direction
  • Units
  • special cases

Source and Reference

https://www.youtube.com/watch?v=pJwg2Bk0BDE&list=PLZ6kagz8q0bvxaUKCe2RRvU_h7wtNNxxi&index=5