Electric Field - Electric Field Due to a System of Charges

The electric field at a point due to a system (group) of charges is determined using the Principle of Superposition.

"The electric field at any point due to a group of charges is the vector sum of the electric fields at that point due to each individual charge, calculated as if the other charges were not present."

• Each charge in the system contributes its own independent electric field at the point of interest.
• These individual fields are then added vectorially to give the total (resultant) field.

For a system of n point charges q₁, q₂, q₃,…, qn, the total electric field at point P is:

E(r) = \[{\frac{1}{4\pi\varepsilon_0}\sum_{i=1}^n\frac{q_i}{r_{iP}^2}\hat{\mathbf{r}}_{iP}}\]

Symbol Reference

The total field is built up as follows:

• Step 1: Find the field E₁ at P due to q₁ alone, as if all other charges were absent.
• Step 2: Find the field E₂ at P due to q₂ alone, and so on for each charge.
• Step 3: Add all individual fields vectorially:
  E = E₁ + E₂ + E₃ + ⋯ + E_n 

Note: This is a vector sum, not a simple arithmetic (scalar) sum. Both the magnitude and direction of each individual field must be considered before adding.

• The resultant field E is the vector sum of all individual fields.
• Each individual field E_i is calculated independently, as if no other charges exist.
• The unit vector \[\hat r_i\]P points from each charge q_i toward point P.
• The principle holds for any number of charges in any configuration.
• This is a direct application of the Superposition Principle to electric fields.