Foundations — Capacitance — parallel plate derivation, cylindrical, spherical

Before you can watch that machine run, you need every gear it uses. This page builds each symbol and idea from absolute zero, in the order they stack. Nothing here is assumed — if the parent note used it, we forge it below.

0. Charge — the thing we pump

In a capacitor we always put $+Q$ on one conductor and $-Q$ on the other — the same size, opposite sign. Why equal and opposite? Because to charge a capacitor you move electrons from one plate to the other: whatever one plate loses, the other gains. Nothing is created.

1. Electric field — the invisible push

Why the topic needs it. Charge alone is invisible. The field is how that charge reaches across the gap to talk to the other plate. Every derivation goes charge → field → voltage, so the field is the middle man we cannot skip.

2. The vector arrow and the dot product $\vec{E}\cdot d\vec{l}$

The parent note writes $\vec{E}\cdot d\vec{l}$ and $\vec{E}\cdot d\vec{A}$. Those dots are not multiplication — they are the dot product, and you must know what it asks.

Two cases you must recognise instantly:

• Same direction ($\varphi =0°$, $\cos 0=1$): $\vec{E}\cdot d\vec{l}=Edl$ — full effect. This happens when you walk straight along the field, as we do between capacitor plates.
• Perpendicular ($\varphi =90°$, $\cos 90=0$): $\vec{E}\cdot d\vec{l}=0$ — no effect. Walking across the field costs nothing.

Why the topic needs it. Voltage is built by adding up $\vec{E}\cdot d\vec{l}$ along a path. The dot product is the tool that answers "how much of the field actually points along my walk?" — no other operation asks exactly that question.

3. Tiny pieces: $d\vec{l}$, $d\vec{A}$, and the integral sign $\int$

The little arrow on $d\vec{l}$ and $d\vec{A}$ just reminds us each tiny piece also has a direction — the step direction, or the direction a patch faces — so it can enter a dot product.

4. Potential difference — the voltage $V$

Why the topic needs it. Voltage is the thing a battery sets and a meter reads. Capacitance links the cause ($Q$) to this measurable effect ($V$). See Electric Potential for the full machinery — here we only need that $V$ is the field summed along a path.

5. Charge densities: $\sigma$ and $\lambda$

When charge spreads over a surface or a wire, we describe how crowded it is with a density.

Why the topic needs it. Gauss's law counts enclosed charge. Densities let us convert "charge on the whole plate" into "charge inside my little Gaussian box" cleanly — the $A$ or $L$ then cancels, which is exactly why $C$ turns out to be pure geometry.

6. The constant ${\epsilon }_0$

Sometimes you'll see $k=\frac{1}{4\pi {\epsilon }_0}=8.99\times 10^9$ — the Coulomb constant. It's the same ${\epsilon }_0$ wearing a different hat, convenient for spheres.

7. Gauss's law — the engine

Now the pieces combine into the tool that finds the field.

Why this tool and not another? We could add up Coulomb's law over every speck of charge — brutal integrals. But when the charge is symmetric (flat plane, long cylinder, sphere), Gauss's law lets us pull the constant $E$ out of the integral and solve in one line. It's the shortcut that only works when symmetry hands us a surface where $E$ is constant. Full treatment: Gauss's Law.

8. Why three geometries, three field powers

Each shape gives Gauss a surface of a different area formula, and that decides how the field falls off:

And the field's shape then decides the integral that makes $V$:

• constant field → $V=Ed$ (just multiply),
• $1/r$ field → $\int dr/r=\ln$ (a logarithm), see Coaxial Cable Transmission Lines,
• $1/r^2$ field → $\int dr/r^2=-1/r$ (the $1/a-1/b$ combo).

That single chain — area → field power → integral type — is the skeleton of all three derivations.

The prerequisite map

Everything upstream feeds the two arrows $Q$ and $V$ that define $C$. Master the upstream boxes and the parent note's three derivations become the same walk, three times.

Related building blocks

• Dielectrics and Polarization — what the factor $\kappa$ does to ${\epsilon }_0$.
• Energy Stored in a Capacitor — turns $C$ and $V$ into stored joules.
• Capacitors in Series and Parallel — combining several $C$'s.
• Back to the parent: the Hinglish version.

Equipment checklist

Test yourself — cover the right side. If any answer is fuzzy, re-read its section above before touching the derivations.