Foundations — Potential of point charge, potential from field and vice versa

This page is a toolbox check. Before you can follow the parent note Potential of point charge, potential from field and vice versa, every single symbol it uses must feel obvious. We build them in order — each one uses only the ones before it.

1. A charge $q$ — the source of everything

The picture: a dot with a $+$ or $-$ label sitting at a point in space. Everything on the parent page radiates outward from such a dot.

Why the topic needs it: the whole story is "one charge $q$ creates a field and a potential everywhere around it." Without $q$ there is nothing to create anything. Its sign survives all the way to the answer $V=kq/r$ — a negative charge gives negative potential.

The size of the push is set by Coulomb's Law, which we lean on next.

2. Distance $r$ and the unit vector $\hat{r}$ — where and which-way

The picture (figure above): the charge sits at the centre. Pick any point $P$. Draw the straight line from charge to $P$ — its length is $r$, and the little arrow $\hat{r}$ sits on that line pointing outward.

Why we need both:

• $r$ answers "how far?" — and the field weakens as you go further, so distance is the input to that weakening.
• $\hat{r}$ answers "which way does the field point?" — for a positive charge the field arrows point straight out, exactly along $\hat{r}$.

Splitting an arrow into "how long" ($r$) and "which way" ($\hat{r}$) is the trick that lets us write the field cleanly in the next section.

3. The electric field $\vec{E}$ — force per charge

The picture: a whole field of arrows filling space around the charge — long near the charge, short far away, all pointing outward (for $+q$) or inward (for $-q$).

Read this as three factors multiplied:

• $\frac{q}{r^2}$ — a number saying how strong (grows with charge, shrinks with distance-squared),
• $\hat{r}$ — the direction (straight out),
• $\frac{1}{4\pi {\epsilon }_0}$ — a constant that just fixes the units (next section).

Why the topic needs it: $\vec{E}$ is half the story. The parent page's entire mission is to convert between this arrow-field and the height-landscape $V$. This exact formula is what gets integrated in the derivation of $V$. See Electric Field of Point Charge for its own derivation.

4. The constants ${\epsilon }_0$ and $k$ — the unit-fixers

The picture: think of $k$ as a fixed exchange rate. Charge and distance decide the shape of the answer; $k$ converts that shape into real volts and newtons.

Why the topic needs it: without $k$ the numbers come out in the wrong units. The parent page writes $V=kq/r$ precisely because carrying $\frac{1}{4\pi {\epsilon }_0}$ everywhere is clumsy — $k$ is the same thing, packaged.

5. The test charge $q_0$ and work $W$ — measuring potential

The picture (figure above): the red arrow is the field's push on $q_0$. The violet arrow is you dragging $q_0$ inward from far away. When you drag against the push, you do positive work — that stored effort is what becomes potential.

Why the topic needs it: potential is defined as work-per-charge: $V=W_{\text{ext}}/q_0$. You cannot understand the definition line of the parent page without knowing what "work by an external agent" means. This same idea powers Potential Energy of Charge System, where $U=qV$.

6. The dot product $\vec{E}\cdot d\vec{l}$ — how much field lies along the path

The picture (figure above): three cases side by side. Same direction → full product. At an angle → only the shadow (projection) counts. Perpendicular → nothing, zero.

Why the topic needs it: work is force along the direction of motion. As we drag $q_0$ through a tiny step $d\vec{l}$ (a tiny arrow of movement), only the part of $\vec{E}$ pointing along $d\vec{l}$ does any work. The dot product $\vec{E}\cdot d\vec{l}$ extracts exactly that part. This is why the parent's radial-path trick works: move straight along $\hat{r}$ and $\vec{E}\cdot d\vec{l}=Edr$ — perfectly aligned, full product, no wasted angle.

7. The line integral $\int \vec{E}\cdot d\vec{l}$ — adding up along a path

The picture: the inward drag of section 5, but now sliced into countless tiny steps, each contributing a sliver of work, all summed.

Why the topic needs it: the definition of potential is a line integral: $V(P)=-{\int }_{\infty }^P\vec{E}\cdot d\vec{l}$ "Sum up the field-slivers from infinity to $P$, flip the sign." Section 8 explains why we are even allowed to write a single such number.

8. Conservative field & the gradient — the two-way bridge

The picture: a hilly landscape. To go from base camp to a peak, it doesn't matter which trail you take — the height gained is the same. Loop back to start → zero net climb.

Why the topic needs it: this is the permission slip for the whole idea of potential. Only because every path gives the same answer can we assign one clean number $V(P)$ to each point. If loops didn't cancel, "height" would be ambiguous. The full story lives in Conservative Fields and Curl.

The picture (figure above): contour lines of a hill (curves of equal height). The gradient arrow crosses them at right angles, pointing to higher ground; it is longest where contours crowd together.

Why the topic needs it: the reverse bridge is $\vec{E}=-\nabla V$. Field = minus the uphill arrow = the downhill arrow. The minus sign is why the field points from high potential to low. See Gradient Operator and the perpendicular-to-contours fact in Equipotential Surfaces.

Prerequisite map

Equipment checklist

Cover the right side and test yourself. If any answer surprises you, reread that section above.

Connections

• Coulomb's Law — sets the strength of $q$'s push.
• Electric Field of Point Charge — the $\vec{E}$ we integrate.
• Potential Energy of Charge System — where the work $W=qV$ goes.
• Equipotential Surfaces — contour lines of the $V$ landscape.
• Conservative Fields and Curl — the permission slip for potential.
• Gradient Operator — the machinery behind $\vec{E}=-\nabla V$.