Coulomb Product Distance Cross-Scan - Electric Fields, Electrostatics & Capacitors Step by Step

Coulomb proportionality has a direct charge-product side and an inverse-square distance side. The two scans stay separate.

electrostatics Compare charge-product rows against distance rows without mixing the two relationships.

Example

Coulomb proportionality has a direct charge-product side and an inverse-square distance side. The two scans must stay separate. Exact arithmetic here means exact results for the stated model inputs; measured inputs still carry uncertainty and significant-figure limits.

highlighted = computed this step

Keep the force model proportional

Use the same one-dimensional pair model as the earlier Coulomb chapter. The rendered middle case has force magnitude 16 newtons on each charge.

F\propto \frac{q_{\text{left}}q_{\text{right}}}{r^2}

Scan one charge at fixed distance

Hold the right charge and distance fixed. Three rows show the force growing directly with the left charge.

\begin{array}{c|c|c|c}q_{\text{left}}&q_{\text{right}}&r&F\\1\ \text{C}&2\ \text{C}&1\ \text{m}&8\ \text{N}\\2\ \text{C}&2\ \text{C}&1\ \text{m}&16\ \text{N}\\3\ \text{C}&2\ \text{C}&1\ \text{m}&24\ \text{N}\\\end{array}

Scan distance at fixed charges

Now hold both charges fixed and move the pair farther apart. Doubling distance quarters the force; quadrupling distance leaves one sixteenth of the reference force.

\begin{array}{c|c|c|c}q_{\text{left}}&q_{\text{right}}&r&F\\2\ \text{C}&2\ \text{C}&1\ \text{m}&16\ \text{N}\\2\ \text{C}&2\ \text{C}&2\ \text{m}&4\ \text{N}\\2\ \text{C}&2\ \text{C}&4\ \text{m}&1\ \text{N}\\\end{array}

Do not mix the two scans mentally

The charge rows are direct-product rows. The distance rows are inverse-square rows. Keeping the rows separate makes the two relationships scanable.

\text{charge product: direct}\qquad \text{distance: inverse square}

The middle row is structurally drawn

The final check is not a caption. The rendered arrows are the same 16 newton quantity shown in the middle rows.

F_{\text{left}}=F_{\text{right}}=16\ \text{N}