Electric Potential & Capacitance
Every point in space around a charge has an invisible "electrical height" — potential is what tells you how far a charge would fall.
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The big idea: Electric potential is potential energy per unit charge — a scalar "elevation map" of space that exists whether or not a charge is actually there to feel it. The field points downhill on that map, and capacitors are simply devices that store energy by separating charge to create a controlled potential difference.
Height, but for charge
Think about a ball sitting on a hillside. It has gravitational potential energy that depends on how high up the hill it is — and that energy exists whether or not the ball is actually there. The hill has a "height" at every point, independent of any particular ball.
Electric potential works the same way. Around any charge, space has an "electrical height" at every point — call it \(V\). Drop a test charge anywhere nearby and it will have potential energy \(U = qV\) simply by being at that spot. Move it to a point of lower \(V\) and, if it's positive, it will accelerate there on its own, the same way a ball rolls downhill. The whole idea of voltage — the volts on a battery, an outlet, a neuron — is just this potential-energy-per-charge idea in disguise.
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Potential is a scalar map, not a push
Electric potential \(V\) is potential energy per unit charge, measured in volts (1 V = 1 J/C). Unlike the electric field \(\vec{E}\), potential has no direction — it's just a number at every point in space. That's what makes it so useful: you can add up potentials from many charges with plain arithmetic, no vector components required.
\[ U = \dfrac{kQq}{r} \qquad\qquad V = \dfrac{U}{q} = \dfrac{kQ}{r} \]
Potential energy of two point charges, and the potential \(V\) created by a single charge \(Q\) at distance \(r\) (with \(k = 8.99\times10^{9}\ \text{N·m}^2/\text{C}^2\)). Notice \(V\) belongs to the source charge \(Q\) alone — it's defined even before a second charge \(q\) shows up to feel it.
\[ E = -\dfrac{dV}{dr} \qquad\text{(uniform field: } E = \dfrac{\Delta V}{d}\text{)} \]
The field is the rate at which potential drops with distance — it always points from high \(V\) toward low \(V\), along the steepest descent. Where the potential "hill" is steep, the field is strong; where potential is flat, the field is zero.
🎮 Interactive: Map the Potential Around a Charge LIVE
Drag charges and watch the equipotential contours and field arrows respond together — notice the field always crosses the contour lines at a right angle, and points toward lower potential.
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Equipotential surfaces: the contour lines of charge
An equipotential surface is just a set of points that all share the same value of \(V\) — like a contour line on a topographic map connecting points of equal elevation. Around a single point charge, these surfaces are concentric spheres. Two rules make them powerful: (1) moving a charge along an equipotential surface takes zero work, since \(V\) never changes; and (2) the electric field is always perpendicular to these surfaces, because the fastest way downhill is never sideways along a level line.
📝 Worked example: A tiny charged sphere carries +2.0 nC. What electric potential does it create at a point 3.0 cm away?
- Use the point-charge potential formula: V = kQ/r, with k = 8.99×10⁹ N·m²/C².
- Convert units: Q = 2.0×10⁻⁹ C, r = 3.0 cm = 0.030 m.
- Plug in: V = (8.99×10⁹ × 2.0×10⁻⁹) / 0.030 = 17.98 / 0.030.
- V ≈ 599 V.
✓ ≈ 599 V (about 600 volts) — and every other point 3.0 cm from the charge shares this same value, forming a spherical equipotential surface around it.
Capacitance: paying charge to buy voltage
A capacitor is two conductors separated by a gap. Pump charge \(+Q\) onto one plate and \(-Q\) onto the other, and a potential difference \(V\) builds up between them — the more charge you pack on, the higher \(V\) climbs. For a given geometry, that relationship is always proportional, so we define a constant that captures it: capacitance.
\[ C = \dfrac{Q}{V} \qquad\qquad U = \tfrac{1}{2}QV = \tfrac{1}{2}CV^2 = \dfrac{Q^2}{2C} \]
Capacitance \(C\) (in farads) tells you how much charge a device stores per volt of potential difference. The three energy expressions are algebraically identical — pick whichever variables you already know.
📝 Worked example: A 100 pF capacitor is charged until the voltage across it reaches 12 V. How much energy does it store?
- Use U = ½CV² for energy stored in a capacitor.
- Convert units: C = 100 pF = 1.00×10⁻¹⁰ F, V = 12 V.
- U = 0.5 × (1.00×10⁻¹⁰) × (12)² = 0.5 × 1.00×10⁻¹⁰ × 144.
- U = 7.2×10⁻⁹ J.
✓ 7.2 nJ. Small, but notice the V² dependence — double the voltage and the stored energy quadruples, which is why capacitor energy storage grows so fast as you charge them up.
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Don't confuse potential with field, or potential energy with potential
Three mix-ups trip up almost everyone at first: (1) \(V\) is a scalar, \(\vec{E}\) is a vector — they have different units (volts vs. volts/meter) and answer different questions. (2) Potential energy \(U = qV\) depends on the charge you place there; potential \(V\) itself does not — it's a property of the source and the location. (3) Only differences in potential are physically meaningful; the zero point of \(V\) is a choice (often "infinitely far away" or "ground"), so a single potential value alone tells you nothing until you know what it's measured relative to.
Check your understanding
1. Electric potential at a point is best defined as:
Potential (voltage) is potential energy per unit charge, V = U/q — a scalar property of location, just like the electric field is force per unit charge.
2. A point charge of +4.0 nC sits alone in space. What potential does it create 6.0 cm away? (k = 8.99×10⁹ N·m²/C²)
V = kQ/r = (8.99×10⁹ × 4.0×10⁻⁹) / 0.060 = 35.96/0.060 ≈ 600 V.
3. Which statement about equipotential surfaces is correct?
The field points along the steepest drop in potential, and along an equipotential surface potential doesn't change at all — so the two must always be perpendicular.
4. A 200 pF capacitor is charged to 9.0 V. How much energy does it store?
U = ½CV² = 0.5 × (200×10⁻¹² F) × (9.0 V)² = 0.5 × 2.00×10⁻¹⁰ × 81 = 8.1×10⁻⁹ J = 8.1 nJ.
✅ Key takeaways
- Electric potential V is potential energy per charge (U = qV) — a scalar "elevation map" of space that exists at a point whether or not a charge sits there to feel it.
- For a point charge, V = kQ/r: potential falls off with distance and, unlike the field, has no direction to worry about.
- The electric field always points from high potential toward low potential and crosses equipotential surfaces at a right angle; its strength is how fast potential changes with distance.
- Capacitors store charge and energy through C = Q/V, with stored energy U = ½QV = ½CV² = Q²/2C — energy that grows with the square of the voltage.