recall and use C = Q / V

Capacitors and Capacitance

Learning Objective

Recall and use the relationship

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

where C is capacitance, Q is the charge stored, and V is the potential difference across the plates.

What is a Capacitor?

A capacitor is a passive electrical component that stores energy in an electric field created between two conductors (plates) separated by an insulating material (dielectric).

Capacitance

Capacitance is a measure of a capacitor’s ability to store charge per unit voltage.

• Defined as C = Q / V.
• SI unit: farad (F), where 1 F = 1 C V⁻¹.
• Typical laboratory capacitors are in the microfarad (µF), nanofarad (nF) or picofarad (pF) range.

Factors Affecting Capacitance

For a parallel‑plate capacitor:

$$C = \varepsilon_0 \varepsilon_r \frac{A}{d}$$

• A: Area of each plate (larger area → larger C).
• d: Separation between plates (smaller distance → larger C).
• \(\varepsilon_0\): Permittivity of free space (≈ 8.85 × 10⁻¹² F m⁻¹).
• \(\varepsilon_r\): Relative permittivity (dielectric constant) of the material between the plates.

Energy Stored in a Capacitor

The electrical energy stored is

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

Series and Parallel Combinations

When more than one capacitor is used, the total capacitance depends on how they are connected.

Configuration	Formula for Total Capacitance	Explanation
Parallel	$$C_{\text{total}} = C_1 + C_2 + C_3 + \dots$$	All plates share the same voltage; charges add.
Series	$$\frac{1}{C_{\text{total}}} = \frac{1}{C_1} + \frac{1}{C_2} + \frac{1}{C_3} + \dots$$	Charge on each capacitor is the same; voltages add.

Example Calculation

Find the capacitance of a parallel‑plate capacitor with:

• Plate area, A = 0.02 m²
• Plate separation, d = 1.0 mm = 1.0 × 10⁻³ m
• Dielectric: air (≈ \(\varepsilon_r = 1\))

Solution:

$$C = \varepsilon_0 \varepsilon_r \frac{A}{d} = (8.85 \times 10^{-12}\,\text{F m}^{-1})(1)\frac{0.02}{1.0 \times 10^{-3}} = 1.77 \times 10^{-10}\,\text{F} = 177\,\text{pF}$$

Practical Considerations

• Dielectric breakdown: If the electric field exceeds a material‑specific limit, the dielectric becomes conductive and the capacitor fails.
• Leakage current: Real capacitors are not perfect insulators; a small current may flow over time.
• Temperature coefficient: Capacitance can change with temperature, especially for electrolytic types.

Summary Checklist

1. Capacitance definition: C = Q / V.
2. Units: farad (F) = coulomb per volt.
3. Parallel‑plate formula: C = ε₀εᵣA/d.
4. Energy stored: U = ½ CV².
5. Series combination reduces total capacitance; parallel combination adds capacitances.

Practice Questions

1. A 5 µF capacitor is charged to 200 V. Calculate the charge stored and the energy stored.
2. Two capacitors, 3 µF and 6 µF, are connected in series across a 12 V battery. Determine the voltage across each capacitor.
3. Three identical capacitors of 2 µF each are connected in parallel. What is the total capacitance? If the combination is then connected in series with a 4 µF capacitor, what is the final total capacitance?