FAD1022 L7-L9 — Capacitors — Formula Sheet
Comprehensive formula sheet extracted from FAD1022 L7-L9 — Capacitors.
1. Capacitance Definition
Fundamental Definition of Capacitance
$$C = \frac{Q}{\Delta V}$$
| Symbol | Meaning | Units |
|---|---|---|
| $C$ | Capacitance | farad (F) |
| $Q$ | Charge stored on one plate | coulomb (C) |
| $\Delta V$ | Potential difference between plates | volt (V) |
Key insight: A capacitor does not create charge — it separates and stores existing charges on opposite plates, creating a potential difference between them.
2. Parallel-Plate Capacitor (Vacuum / Air)
Electric Field Strength Between Plates
$$E = \frac{\sigma}{\varepsilon_0} = \frac{Q}{A\varepsilon_0}$$
| Symbol | Meaning | Units |
|---|---|---|
| $E$ | Electric field strength | V m$^{-1}$ or N C$^{-1}$ |
| $\sigma$ | Surface charge density ($Q/A$) | C m$^{-2}$ |
| $\varepsilon_0$ | Permittivity of free space | $8.85 \times 10^{-12}$ F m$^{-1}$ |
| $Q$ | Charge on the plate | C |
| $A$ | Area of one plate | m$^2$ |
Electric Field from Potential Difference
$$E = \frac{\Delta V}{d}$$
| Symbol | Meaning | Units |
|---|---|---|
| $E$ | Electric field strength (uniform between plates) | V m$^{-1}$ |
| $\Delta V$ | Potential difference between plates | V |
| $d$ | Distance between the two plates | m |
For $d \ll A$, the electric field $E$ is uniform between the plates and zero elsewhere.
Capacitance of a Parallel-Plate Capacitor (Vacuum/Air)
$$C = \frac{A\varepsilon_0}{d}$$
| Symbol | Meaning | Units |
|---|---|---|
| $C$ | Capacitance | F |
| $A$ | Area of one plate | m$^2$ |
| $\varepsilon_0$ | Permittivity of free space | $8.85 \times 10^{-12}$ F m$^{-1}$ |
| $d$ | Separation distance between plates | m |
Factors Affecting Capacitance
- Directly proportional to plate area $A$
- Inversely proportional to plate separation $d$
To increase accumulated charge $Q$:
- Increase the voltage $\Delta V$
- Increase the capacitance $C$ (by increasing $A$ or decreasing $d$)
3. Capacitors with Dielectrics
Dielectric Constant
$$\kappa = \frac{\varepsilon}{\varepsilon_0}$$
| Symbol | Meaning | Units |
|---|---|---|
| $\kappa$ | Dielectric constant (dimensionless) | — |
| $\varepsilon$ | Permittivity of the dielectric material | F m$^{-1}$ |
| $\varepsilon_0$ | Permittivity of free space | $8.85 \times 10^{-12}$ F m$^{-1}$ |
Capacitance with Dielectric
$$C = \kappa C_0 = \kappa \frac{A\varepsilon_0}{d}$$
| Symbol | Meaning | Units |
|---|---|---|
| $C$ | Capacitance with dielectric | F |
| $C_0$ | Capacitance without dielectric (vacuum/air) | F |
| $\kappa$ | Dielectric constant | — |
| $A$ | Area of one plate | m$^2$ |
| $\varepsilon_0$ | Permittivity of free space | F m$^{-1}$ |
| $d$ | Separation distance between plates | m |
Effect on Isolated Capacitor (Battery Disconnected)
When a dielectric is inserted into a charged, isolated capacitor:
| Quantity | Relation | Explanation |
|---|---|---|
| Charge | $Q = Q_0$ | Remains constant |
| Potential difference | $\Delta V = \dfrac{\Delta V_0}{\kappa}$ | Decreases by factor $\kappa$ |
| Capacitance | $C = \kappa C_0$ | Increases by factor $\kappa$ |
| Electric field | $\vec{E} = \dfrac{\vec{E}_0}{\kappa}$ | Decreases by factor $\kappa$ |
Subscript $0$ denotes the original vacuum-filled capacitor values.
Atomic-Level Mechanism
- Polar molecules align with the applied electric field $\vec{E}_0$
- The polarized dielectric creates an induced electric field $\vec{E}_{\text{ind}}$ opposite to $\vec{E}_0$
- Net field inside: $\vec{E} = \dfrac{\vec{E}_0}{\kappa}$
- Weaker field means lower voltage for the same charge $\rightarrow$ higher capacitance
Dielectric Strength
- Dielectric strength is the maximum electric field that can exist in a dielectric without electrical breakdown.
- Exceeding this value causes the dielectric to fail and conduct.
4. Energy Stored in a Charged Capacitor
Fundamental Relationship
$$\text{Energy stored} = \text{Work done to charge it}$$
Work is required because electrons are forced onto a plate that already contains electrons (repulsion). The fuller the plate, the more work is required. This work is stored as electrical potential energy in the electric field between the plates.
Energy Stored — Three Equivalent Forms
$$U = W = \frac{1}{2}\frac{Q^2}{C} = \frac{1}{2}C(\Delta V)^2 = \frac{1}{2}Q,\Delta V$$
| Symbol | Meaning | Units |
|---|---|---|
| $U$ (or $W$) | Energy stored / work done | joule (J) |
| $Q$ | Charge on the capacitor | C |
| $C$ | Capacitance | F |
| $\Delta V$ | Potential difference across the capacitor | V |
Note: $U = \frac{1}{2}Q,\Delta V$ corresponds to the area under the $\Delta V$ vs $Q$ graph (a triangle).
5. RC Circuits — Time Constant
Time Constant
$$\tau = RC$$
| Symbol | Meaning | Units |
|---|---|---|
| $\tau$ | Time constant | second (s) |
| $R$ | Resistance in the circuit | ohm ($\Omega$) |
| $C$ | Capacitance | F |
The time constant $\tau = RC$ governs the transient behavior of charging and discharging RC circuits.
6. Quick Reference: Symbols & Constants
| Symbol | Value / Meaning |
|---|---|
| $\varepsilon_0$ | $8.85 \times 10^{-12}\ \text{F m}^{-1}$ (permittivity of free space) |
| F | farad — unit of capacitance |
| C | coulomb — unit of charge |
| V | volt — unit of potential difference |
| J | joule — unit of energy |
| $\kappa$ | dielectric constant (dimensionless) |
| $\tau$ | time constant ($RC$) |
7. Common Dielectric Materials (Reference)
| Material | Dielectric Constant $\kappa$ | Dielectric Strength ($10^6$ V m$^{-1}$) |
|---|---|---|
| Paper | 3.7 | 16 |
| Mylar | 3.2 | 7 |
| Rubber | 6.7 | 12 |
| Silicone oil | 2.5 | 15 |
| Nylon | 3.4 | 14 |
| Teflon | 2.1 | 60 |
Formula sheet compiled from FAD1022 L7-L9 — Capacitors lecture notes.