Visual walkthrough — Superposition principle for forces

We are chasing this target (don't panic — by the last step it will feel obvious): ${\vec{F}}_{\text{net}}=\frac{q_0}{4\pi {\epsilon }_0}{\sum }_{i=1}^n\frac{q_i({\vec{r}}_0-{\vec{r}}_i)}{|{\vec{r}}_0-{\vec{r}}_i{|}^3}$

Step 1 — Draw the players: charges live at positions

WHAT. Before any force, we need a way to say where each charge sits. We drop a grid (an $x$–$y$ map) and mark every charge with a little dot and an arrow from the origin to that dot. That arrow is called a position vector.

WHY. A force has a direction, and direction only makes sense once we know where things are. Numbers alone ("charge is 2 metres away") lose the direction; an arrow keeps both the distance and the way it points.

PICTURE. Look at the figure. The target charge $q_0$ (blue dot) has a position arrow ${\vec{r}}_0$. A source charge $q_i$ (orange dot) has its own arrow ${\vec{r}}_i$. Both arrows start at the origin $O$.

Nothing physical yet — just bookkeeping of where. Next we make one charge feel another.

Step 2 — The separation arrow ${\vec{r}}_0-{\vec{r}}_i$

WHAT. We build a brand-new arrow that points from the source $q_i$ straight to the target $q_0$. We get it by subtracting position vectors: ${\vec{r}}_0-{\vec{r}}_i$.

WHY subtraction, and why this order? Subtracting arrows is the "how do I get from tail to tip" operation: ${\vec{r}}_0-{\vec{r}}_i$ is the arrow you'd walk starting at $q_i$, ending at $q_0$. We want this order because the force we are about to compute acts on $q_0$, so it should point along the line into $q_0$.

PICTURE. In the figure, tip-to-tail: the green arrow ${\vec{r}}_0-{\vec{r}}_i$ closes the triangle, reaching from the orange source dot to the blue target dot. Its length is the physical distance between the charges.

Step 3 — Strip out pure direction: the unit vector ${\hat{r}}_{0i}$

WHAT. We shrink the separation arrow down until it has length exactly $1$, keeping the same direction. We do that by dividing the arrow by its own length. The result wears a little hat: ${\hat{r}}_{0i}$.

WHY. A force needs a strength and a direction, and it is cleanest to handle these separately. The unit vector is the "pure direction" — a compass needle with no length of its own. Later we multiply it by the strength (a plain number) to rebuild the full force arrow.

PICTURE. The figure shows the long green separation arrow and, laid on top of it, the short red unit arrow of length $1$ pointing the same way. Same heading, standardised length.

Step 4 — Coulomb's law for ONE pair (strength × direction)

WHAT. Now attach the strength. Coulomb's Law gives the size of the push between two charges; we multiply that size by the pure-direction arrow ${\hat{r}}_{0i}$ to get the full force arrow ${\vec{F}}_{0i}$ on $q_0$ from just $q_i$.

WHY this exact strength? Experiment: the pull/push grows with each charge (double a charge, double the force) and weakens with the square of distance (twice as far → one-quarter the force). The constant $\frac{1}{4\pi {\epsilon }_0}$ just fixes the units (recall ${\epsilon }_0$ is the vacuum permittivity defined at the top).

PICTURE. The figure draws the full force arrow ${\vec{F}}_{0i}$ sitting on $q_0$, pointing along the red unit direction, with a length proportional to the strength. Two mini-panels contrast the repel (arrow out) and attract (arrow in) cases.

Step 5 — The superposition principle: many arrows, one crowd

WHAT. Add a second source $q_j$. It produces its own force arrow ${\vec{F}}_{0j}$ on $q_0$, computed exactly as in Step 4, ignoring $q_i$ entirely. This "each source acts as if alone" rule is the superposition principle.

WHY it is a separate fact. Coulomb's law only ever mentions two charges — there is no place in it for a third. So how the two-charge forces combine is an extra experimental law: nature turns out to be linear — forces just add, no charge screens or weakens another. It could have been otherwise; measurement says it is not.

PICTURE. Three source charges each fire their own coloured arrow onto $q_0$. Crucially, the length and heading of each arrow is fixed by its own source alone — the arrows do not "notice" each other.

Step 6 — Adding arrows the right way: tip-to-tail

WHAT. To actually do the sum, lay each force arrow tip-to-tail. The single arrow from the very first tail to the very last tip is ${\vec{F}}_{\text{net}}$. This is Vector Addition and Resolution in action.

WHY not just add the lengths? Because arrows in different directions partly cancel. A push right and a push up do not make a push of size (right + up); they make a diagonal push, and its length is less than the two lengths added. Only arrows on the same line may have their numbers simply added.

PICTURE. The figure builds the tip-to-tail chain of the three arrows from Step 5 and draws the fat resultant ${\vec{F}}_{\text{net}}$ closing the polygon.

Step 7 — Resolve into components, then recombine

WHAT. Give each force arrow a shadow on the $x$-axis and a shadow on the $y$-axis — its components. Add all the $x$-shadows to get $F_x$; add all the $y$-shadows to get $F_y$. Then rebuild the single arrow from those two totals.

WHY. Shadows on one axis are all on the same line, so they can be added as plain numbers. This is the trick that turns messy arrow-addition into ordinary arithmetic done twice.

PICTURE. The figure drops the dashed shadows of one force arrow onto both axes, labels the angle ${\theta }_i$, and shows the final ${\vec{F}}_{\text{net}}=(F_x,F_y)$ rebuilt with its tilt angle $\varphi$.

Step 8 — Degenerate cases (never get surprised)

WHAT & WHY. A good derivation must survive its own edge cases. Check them:

• Collinear charges (all on one line). Every arrow has zero shadow on the other axis, so $F_y=0$ and you do just add signed numbers along $x$. The general recipe collapses to plain addition — exactly the 1-D case.
• A source at the same spot as $q_0$ (${\vec{r}}_i={\vec{r}}_0$). Then $|{\vec{r}}_0-{\vec{r}}_i|=0$ and we divide by zero — the formula blows up. Physically two point charges can't sit at one point; this is a warning, not a value.
• A zero charge ($q_i=0$). Its term is $0$; it contributes nothing. Superposition quietly ignores empty seats.
• Exact cancellation. Two equal arrows pointing opposite ways sum to the zero arrow: ${\vec{F}}_{\text{net}}=\vec{0}$. The charge feels no net force even though each source pushes hard. (Its angle $\varphi$ is then undefined — there is no arrow to point.)

PICTURE. Four mini-panels: collinear (add numbers), coincident (explosion symbol), zero charge (ghost dot), and perfect cancellation (arrows annihilate).

The one-picture summary

Every step, compressed: positions → separation arrow → unit direction → one-pair force → repeat for all sources → tip-to-tail sum → components → net arrow.

$\boxed{{\vec{F}}_{\text{net}}=\frac{q_0}{4\pi {\epsilon }_0}\sum _{i=1}^n\frac{q_i({\vec{r}}_0-{\vec{r}}_i)}{|{\vec{r}}_0-{\vec{r}}_i{|}^3}}$

Active Recall

Connections

• Superposition principle for forces — the parent this walkthrough visualises.
• Coulomb's Law — the one-pair force built in Step 4.
• Vector Addition and Resolution — Steps 6–7, the arrow arithmetic.
• Electric Field — the same superposition, for fields instead of forces.
• Continuous Charge Distributions — the sum becomes an integral.
• Electric Potential — scalar superposition (just add numbers).
• Principle of Linear Superposition (Waves) — same idea, different physics.

Concept Map