Capacitors and Capacitance: Principles, Formulas, and Circuit Applications

Capacitors & Capacitance

Introduction to Capacitors

Capacitors are devices formed by two conductors separated in space, carrying equal and opposite charges. The electric fields generated fill the space between the conductors, and a potential difference is established. The energy is stored in the electric fields.

• Capacitor: A system of two conductors with equal and opposite charges.

• Potential Difference: The voltage between the conductors is proportional to the charge.

• Capacitance (C): The proportionality constant relating charge and potential difference.

Formula:

$V_{ab} \propto Q$

$C = \frac{Q}{V_{ab}}$   (units: Farads, F)

Electric Field and Capacitance in Parallel Plate Capacitors

Parallel plate capacitors consist of two parallel metal plates of area A separated by a distance d. The electric field between the plates is uniform.

• Electric Field: $E = \frac{Q}{\varepsilon_0 A}$

• Potential Difference: $V = Ed = \frac{Qd}{\varepsilon_0 A}$

• Capacitance: $C = \frac{\varepsilon_0 A}{d}$

Example: Increasing the area A or decreasing the separation d increases the capacitance.

The Permittivity of Free Space ($\varepsilon_0$)

$\varepsilon_0$ is a physical constant describing the strength of the electric field in vacuum.

• Value: $\varepsilon_0 = 8.854 \times 10^{-12} \; \frac{C^2}{N m^2}$

• Relation to Coulomb's Law: $k = \frac{1}{4 \pi \varepsilon_0}$

Parallel Plate Capacitors: Geometry and Effects

Capacitance and Plate Separation

For a parallel plate capacitor with circular plates of radius R:

• If the plates are moved closer together (with fixed potential difference), capacitance increases.

• If the plates are moved apart, capacitance decreases.

Charge on Plates

• With fixed potential difference, moving plates closer together increases the charge on each plate.

• If the capacitor is disconnected from the battery, the charge remains the same as the plates are moved.

Potential Difference Changes

• If disconnected from the battery and plates are moved closer, the potential difference decreases.

• Formula: $V = Ed = \frac{Qd}{\varepsilon_0 A}$

Capacitor Circuits: Diagrams and Rules

Basic Circuit Rules

Wires are treated as resistance-less and act as equipotentials. Potential changes occur only when components are attached.

• Batteries: Set fixed potential differences between wires.

• Capacitors: Cause potential drops across their plates.

• Charge Conservation: Total charge in a circuit remains zero; charges are redistributed but not created or destroyed.

Charge Movement in Circuits

• Positive charges build up on one plate, pushing positive charge off the other plate.

• In a closed circuit, pushed-off charges travel around the circuit, forming the first set of positive charges.

Capacitors in Series

Series Combination

When capacitors are connected in series, the total capacitance is less than any individual capacitance.

• Formula: $\frac{1}{C} = \frac{1}{C_1} + \frac{1}{C_2}$

• Key Point: The formula for series is NOT $C = C_1 + C_2$.

• Example: If $C_1 = 1$ F and $C_2 = 1$ F, then $C = 0.5$ F.

Potential Drops in Series

• The total potential difference is the sum of the drops across each capacitor.

• Charge on each capacitor in series is the same.

Summary Table: Capacitance Formulas

Configuration	Capacitance Formula
Single Parallel Plate	$C = \frac{\varepsilon_0 A}{d}$
Series	$\frac{1}{C} = \frac{1}{C_1} + \frac{1}{C_2}$
Parallel	$C = C_1 + C_2$ Additional info: Parallel formula inferred from standard physics context.

Key Terms

• Capacitance (C): Ability to store charge per unit potential difference.

• Farad (F): SI unit of capacitance.

• Permittivity of Free Space ($\varepsilon_0$): Physical constant for electric field strength in vacuum.

• Potential Difference (V): Voltage between two points.

• Electric Field (E): Force per unit charge in a region.

Examples and Applications

• Increasing Plate Area: Increases capacitance.

• Decreasing Plate Separation: Increases capacitance.

• Series Connection: Reduces total capacitance.

• Parallel Connection: Increases total capacitance.

Additional info: Parallel formula and circuit rules expanded for completeness and academic clarity.