Coulomb Proportionality

Charge Product Scales Force

double charge, double force

With distance unchanged, electric force scales directly with either charge. A stated reference case lets the comparison stay exact.

Example

With distance unchanged, electric force scales directly with either charge. A stated reference case lets the comparison stay exact. Exact arithmetic here means exact results for the stated model inputs; measured inputs still carry uncertainty and significant-figure limits.

highlighted = computed this step

Start from a reference case

Use a local reference case. The left charge is 2 coulombs, the right charge is 3 coulombs, and the force size is 6 newtons.

qleft=2 Cqright=3 CF=6 Nq_{\text{left}} = 2\ \text{C}\qquad q_{\text{right}} = 3\ \text{C}\qquad F = 6\ \text{N}
Reference charge pairTwo positive charges repel with the reference force.+2 Cleft+3 Cright6 N6 N

Double one charge

Keep the distance and the right charge the same. Change the left charge from 2 coulombs to 4 coulombs. The charge product doubles.

qleft,new=4 Cinput factor=2q_{\text{left,new}} = 4\ \text{C}\qquad \text{input factor} = 2
Doubled chargeOne charge is doubled, so the repelling force doubles.+4 Cleft+3 Cright12 N12 N

The force doubles

When the charge product doubles and the distance is unchanged, the force doubles from 6 newtons to 12 newtons.

Fnew=6 N⋅2=12 NF_{\text{new}} = 6\ \text{N}\cdot 2 = 12\ \text{N}
Doubled chargeOne charge is doubled, so the repelling force doubles.+4 Cleft+3 Cright12 N12 N

One charge gives a direct row pattern

Hold the right charge at 3 coulombs and keep the distance fixed. The diagram shows the last row; the table shows the direct scaling.

qleftqrightF1 C3 C3 N2 C3 C6 N4 C3 C12 N\begin{array}{c|c|c}q_{\text{left}}&q_{\text{right}}&F\\1\ \text{C}&3\ \text{C}&3\ \text{N}\\2\ \text{C}&3\ \text{C}&6\ \text{N}\\4\ \text{C}&3\ \text{C}&12\ \text{N}\\\end{array}
Doubled chargeThe last table row is the checked diagram.+4 Cleft+3 Cright12 N12 N

The other charge has the same direct pattern

Now hold the left charge at 2 coulombs and vary the right charge. Either charge scales the product directly; the force follows the product.

qleftqrightF2 C1 C2 N2 C3 C6 N2 C5 C10 N\begin{array}{c|c|c}q_{\text{left}}&q_{\text{right}}&F\\2\ \text{C}&1\ \text{C}&2\ \text{N}\\2\ \text{C}&3\ \text{C}&6\ \text{N}\\2\ \text{C}&5\ \text{C}&10\ \text{N}\\\end{array}
Reference charge pairThe middle table row is the reference diagram.+2 Cleft+3 Cright6 N6 N
electrostatics Coulomb proportionality compares exact changes from a stated reference case.