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# Content

`Charge Density Electric Field of Charged Rod  Electric Field in the Bisecting Plane Procedure for Calculating Electric Field of Distributed Charges Source and Reference`

# 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

ID: 191101802 Last Updated: 11/18/2019 Revision: 0 Home 5

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