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}{C1} + \frac{1}{C2} + \dots + \frac{1}{Cn}\$

4.2 Capacitors in Parallel

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

\$C{\text{eq}} = C1 + C2 + \dots + Cn\$

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, \$C1 = 4.0\;\mu\text{F}\$ and \$C2 = 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

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.