Electric potential
Electrical "height"
- Potential is like height for charges: it measures energy per unit charge.
- Roll a positive charge "downhill" and it speeds up.
- It works just like gravitational potential — with one twist.
Electric potential
- $V = \dfrac{W}{q}$ — work per unit charge to bring a charge from infinity (zero there).
- For a point charge $V = \dfrac{Q}{4\pi\varepsilon_0 r}$ — note this is $\dfrac{1}{r}$, not $\dfrac{1}{r^{2}}$.
Electric potential at a point is the work per unit charge to bring a positive charge from:
Like gravitational potential, it is measured from infinity, where the potential is taken as zero.
The electric potential near a point charge varies as 1/r.
Yes — $V = \dfrac{Q}{4\pi\varepsilon_0 r}$ goes as $1/r$, while the field goes as $1/r^{2}$.
Field and potential
- The field is minus the potential gradient: $E = -\dfrac{dV}{dx}$.
- The field points toward lower potential. Between plates this gives $E = \dfrac{V}{d}$.
The electric field equals minus the potential ____ (E = −dV/dx).
The field points "downhill" in potential, toward lower V.
Electric potential energy
- A charge $q$ at potential $V$ has $E_{\text{P}} = qV$; for two charges $E_{\text{P}} = \dfrac{Qq}{4\pi\varepsilon_0 r}$.
- Like charges: positive PE (would fly apart). Opposite: negative PE (a bound system).
The electric potential energy of a charge $q$ at a point of potential $V$ is:
$E_{\text{P}} = qV$; for two point charges this becomes $\dfrac{Qq}{4\pi\varepsilon_0 r}$.
Two opposite charges have a negative electric potential energy (a bound system).
Negative PE means energy must be supplied to pull them apart — they are bound, like an electron and a nucleus.
Gravity vs electric
- Same shapes: $g = \dfrac{GM}{r^{2}}$ ↔ $E = \dfrac{Q}{4\pi\varepsilon_0 r^{2}}$; $\phi = -\dfrac{GM}{r}$ ↔ $V = \dfrac{Q}{4\pi\varepsilon_0 r}$.
- The twist: gravity is always attractive; electric forces can attract or repel.
Match each quantity to how it changes with distance from a point charge.
The field is inverse-square; the potential is inverse (one power of r gentler).
You've got it
- potential $V = \dfrac{Q}{4\pi\varepsilon_0 r}$ varies as $\dfrac{1}{r}$ (field varies as $\dfrac{1}{r^{2}}$)
- field = negative potential gradient: $E = -\dfrac{dV}{dx}$
- electric PE $= qV$; electric and gravitational fields share the same maths