Capacitance

Electric Potential – Capacitance

1. Definition of Capacitance

Capacitance is the ability of a system to store electric charge per unit potential difference. For a capacitor the capacitance $C$ is defined as

$$C = \frac{Q}{V}$$

where $Q$ is the magnitude of charge on each plate and $V$ is the potential difference between the plates.

2. Units and Symbol

• Symbol: $C$
• SI unit: farad (F) = coulomb per volt (C V⁻¹)
• Common sub‑multiples: $\mu\text{F}$ (microfarad), $\text{nF}$ (nanofarad), $\text{pF}$ (picofarad)

3. Capacitance of a Parallel‑Plate Capacitor

For an ideal parallel‑plate capacitor filled with a dielectric of permittivity $\varepsilon$, the capacitance is

$$C = \frac{\varepsilon A}{d}$$

where $A$ is the plate area and $d$ is the separation between the plates.

4. Combination of Capacitors

4.1 Capacitors in Series

The total capacitance $C_{\text{eq}}$ for $n$ capacitors in series is given by

$$\frac{1}{C_{\text{eq}}} = \frac{1}{C_1} + \frac{1}{C_2} + \dots + \frac{1}{C_n}$$

4.2 Capacitors in Parallel

The total capacitance for $n$ capacitors in parallel is the sum of the individual capacitances:

$$C_{\text{eq}} = C_1 + C_2 + \dots + C_n$$

5. Energy Stored in a Capacitor

The electric potential energy $U$ stored in a charged capacitor can be expressed in three equivalent forms:

$$U = \frac{1}{2} QV = \frac{1}{2} C V^{2} = \frac{Q^{2}}{2C}$$

This energy is released when the capacitor discharges.

6. Example Calculation

1. Two capacitors, $C_1 = 4.0\;\mu\text{F}$ and $C_2 = 6.0\;\mu\text{F}$, are connected in series and then the combination is connected to a $12\;\text{V}$ battery.
2. Find the equivalent capacitance, the charge on each capacitor, and the energy stored.

Solution:

Series combination:

$$\frac{1}{C_{\text{eq}}} = \frac{1}{4.0\;\mu\text{F}} + \frac{1}{6.0\;\mu\text{F}} = \frac{3}{12\;\mu\text{F}} + \frac{2}{12\;\mu\text{F}} = \frac{5}{12\;\mu\text{F}}$$ $$C_{\text{eq}} = \frac{12\;\mu\text{F}}{5} = 2.4\;\mu\text{F}$$

Charge on each capacitor (same in series):

$$Q = C_{\text{eq}} V = 2.4\;\mu\text{F} \times 12\;\text{V} = 28.8\;\mu\text{C}$$

Energy stored:

$$U = \frac{1}{2} C_{\text{eq}} V^{2} = \frac{1}{2} \times 2.4\;\mu\text{F} \times (12\;\text{V})^{2} = 0.5 \times 2.4 \times 144\;\mu\text{J} = 172.8\;\mu\text{J}$$

7. Common Types of Capacitors

Type	Dielectric Material	Typical Applications	Capacitance Range
Ceramic	Metal oxide ceramic	High‑frequency circuits, decoupling	pF – $\mu$F
Electrolytic	Aluminium oxide (wet) or tantalum	Power supply filtering	$\mu$F – mF
Film	Polypropylene, polyester	Audio, precision timing	nF – $\mu$F
Mica	Natural mica	RF circuits, stable capacitance	pF – nF

8. Key Points to Remember

• Capacitance depends only on geometry and dielectric, not on charge or voltage.
• In series, the smallest capacitance dominates the equivalent value.
• In parallel, capacitances simply add.
• The energy stored grows with the square of the voltage.
• Always check the voltage rating of a capacitor before connecting it to a circuit.