The total force on a charge due to several others is the vector sum of the forces each one exerts alone. This is the superposition principle, and it is confirmed by experiment.
Coulomb’s law gives the force between two point charges. In most situations, a charge is near several others. To find the total force on it, one more statement is needed.
The principle of superposition states: the total force on a charge q due to charges q₁, q₂, …, qₙ is the vector sum of the forces that each of those charges would exert on q if it were the only other charge present.
F = F₁ + F₂ + … + Fₙ
So the force between q and q₁ is not changed by the presence of q₂. Each pair interaction is calculated with Coulomb’s law on its own, and the results are added as vectors.
Superposition is not derived from Coulomb’s law. It is a separate fact, established by experiment. Measurements show it holds for electric forces to very high precision.
Why must the forces be added as vectors, not as plain numbers?
A force has a size and a direction. Two forces can add, cancel, or combine at an angle.
Two forces of 5 N each: in the same direction, the total is 10 N. In opposite directions, the total is 0 N. At right angles, the total is √(5² + 5²) ≈ 7.1 N.
The two sizes are the same in all three cases. Only vector addition, which takes direction into account, gives the correct total.
The resultant is the force the charge experiences. The method is the same for 3 charges or 3 million: Coulomb’s law for each pair, then vector addition.
A positive charge is at the centre of a square, with equal positive charges at all four corners. What is the net force on the central charge?
Zero.
Each corner charge repels the central charge, with a force directed from that corner towards the centre and beyond. The four corners are the same distance from the centre, so the four forces have the same size.
The forces form two pairs along the diagonals. In each pair, the two forces are equal in size and opposite in direction, so they cancel. The vector sum of all four is zero.
Each individual force is not zero. Adding their sizes would give four times one force. Only the vector sum shows that they cancel.
1The superposition principle states that the total force on a charge is…
2Using superposition, the force from each charge is calculated…
3Why must the forces be added as vectors?
4Two charges each exert 6 N on a third charge, in exactly opposite directions. The net force is…
5A positive charge is at the centre of a square with equal positive charges at the four corners. The net force on it is…
6How many charges can the superposition method handle?
Next: the electric field, the force per unit charge at each point in space.
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