Capacitors and Capacitance: Principles, Formulas, and Circuit Applications

Capacitors & Capacitance

Introduction to Capacitors

Capacitors are electrical components formed by two conductors separated in space, carrying equal and opposite charges. They store energy in the electric field created between the conductors.

• Capacitor: Device that stores electric charge and energy.

• Electric Field: The space between the conductors is filled with electric field lines, representing the force experienced by a charge.

• Potential Difference: A voltage is established between the conductors due to the separation of charge.

• Energy Storage: The energy is stored in the electric field between the plates.

Key Formula:

• Potential difference is proportional to charge: $V_{ab} \propto Q$

• Capacitance is defined as: $C = \frac{Q}{V_{ab}}$

• Units of capacitance: Farads (F)

Parallel Plate Capacitors

Structure and Electric Field

A parallel plate capacitor consists of two parallel metal plates of area A, separated by a distance d. The electric field between the plates is uniform.

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

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

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

• Capacitance depends only on the geometry of the capacitor (area and separation).

Physical Constants

• Permittivity of Free Space: $\varepsilon_0 = 8.854 \times 10^{-12} \ \frac{C^2}{N \cdot m^2}$

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

Behavior of Plate Capacitors

Changing 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 and Potential Difference

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

• If the capacitor is disconnected from the battery after charging, moving plates closer keeps the charge the same, but the potential difference decreases.

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

Capacitor Circuit Diagrams and Rules

Basic Circuit Principles

In ideal circuits, wires are treated as resistance-less and act as equipotentials. Potential changes occur only when electrical components are attached.

• Batteries: Set fixed potential differences between wires.

• Capacitors: Cause potential drops across their plates.

• Charge does not accumulate; total charge in a circuit remains zero.

Charge Movement in Circuits

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

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

Capacitors in Series

Series Connection Principles

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

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

• Charge on each capacitor is the same; total voltage is the sum of voltages across each capacitor.

• Common mistake: Series capacitance is NOT simply the sum of individual capacitances.

Example Calculation

• If $C_1 = 1 \text{ F}$ and $C_2 = 1 \text{ F}$, then $C = \frac{1}{2} \text{ F}$

Summary Table: Capacitance Formulas

Configuration	Capacitance Formula
Single Parallel Plate	$C = \frac{\varepsilon_0 A}{d}$
Series (2 capacitors)	$\frac{1}{C} = \frac{1}{C_1} + \frac{1}{C_2}$
General (n capacitors in series)	$\frac{1}{C_{eq}} = \sum_{i=1}^n \frac{1}{C_i}$

Key Terms and Definitions

• Capacitance (C): Measure of a capacitor's ability to store charge per unit voltage.

• Farad (F): SI unit of capacitance.

• Permittivity ($\varepsilon_0$): Physical constant describing how electric fields interact with the vacuum.

• Series Connection: Capacitors connected end-to-end, sharing the same charge.

• Parallel Connection: Capacitors connected with both plates joined, sharing the same voltage (not covered in detail in these slides).

Example Application

• Energy Storage: Capacitors are used in electronic circuits for energy storage, filtering, and timing applications.

• Physics Experiment: Measuring the change in capacitance as plate separation is varied demonstrates the dependence on geometry.

Additional info: Parallel connection formulas and energy in capacitors are important extensions for further study.