Relation Cross-Scans

Coulomb Product Distance Cross-Scan

charge product and distance

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

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∝qleftqrightr2F\propto \frac{q_{\text{left}}q_{\text{right}}}{r^2}
Coulomb cross-scanThe pair force arrows are equal and opposite.+2 Cleft+2 Cright16 N16 N

Scan one charge at fixed distance

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

qleftqrightrF1 C2 C1 m8 N2 C2 C1 m16 N3 C2 C1 m24 N\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}
Charge-product scanThe middle row is the checked diagram.+2 Cleft+2 Cright16 N16 N

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.

qleftqrightrF2 C2 C1 m16 N2 C2 C2 m4 N2 C2 C4 m1 N\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.

charge product: directdistance: inverse square\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.

Fleft=Fright=16 NF_{\text{left}}=F_{\text{right}}=16\ \text{N}
Coulomb cross-scanEqual and opposite force arrows match the table.+2 Cleft+2 Cright16 N16 N
electrostatics Compare charge-product rows against distance rows without mixing the two relationships.