Capacitance Study Notes

INTRODUCTION TO CAPACITANCE

8.1 Capacitors and Capacitance

• Capacitors are key components in electrical circuits, found in devices such as pacemakers, cell phones, and computers.

• Function of Capacitors: Store electrical energy and release it when needed. Can also act as filters in circuits, allowing certain electrical signals to pass while blocking others.

• Capacitors are crucial for many electronic applications including filtering and energy storage.

Learning Objectives

• Explain the concepts of a capacitor and its capacitance.

• Describe how to evaluate the capacitance of a system of conductors.

Definition of a Capacitor

• Capacitor: A device that stores electrical charge and electrical energy, consisting of two electrical conductors (also referred to as capacitor plates) separated by a distance, which can be a vacuum or filled with an insulating material known as a dielectric.

• Vacuum Capacitor: A capacitor with a vacuum between its plates.

• Dielectric Material: An insulating material between capacitor plates affecting capacitance.

Key Points on Capacitance

• Capacitance (C): Defined as the ratio of the maximum charge (Q) that can be stored on the capacitor to the applied voltage (V) across its plates:
    $C = rac{Q}{V}$

• The SI unit of capacitance is the farad (F), where 1 farad = 1 coulomb per 1 volt:
    $1 ext{ F} = 1 rac{C}{V}$

• Typical ranges of capacitance values: picofarads (pF), microfarads (μF), millifarads (mF).

• The capacitance is influenced by:
    - Size and shape of the plates.
    - Distance between the plates.
    - Dielectric material between the plates.

Calculating Capacitance

• Calculation Steps: To find the capacitance of conductors:
    1. Assume capacitor charge Q.
    2. Determine the electric field (E) between conductors. Use Gauss’s law if the arrangement is symmetrical.
    3. Calculate potential difference (V) between conductors based on the electric field and geometry.
    4. Substitute values into the capacitance formula.

Examples of Capacitors

• Parallel-Plate Capacitor: Consists of two conducting plates of equal area (A) separated by distance (d). The electric field strength (E) between plates is given by:
    $E = rac{<br>ho}{ ext{perm}}$,
    where $
  ho$ is surface charge density.

• Potential difference (V) across the plates can be calculated as:
    $V = E imes d$.

• The capacitance of a parallel-plate capacitor:
    $C = rac{ ext{perm} imes A}{d}$,
    where perm = permittivity of free space.

Other Capacitor Types

• Spherical Capacitor: Composed of two concentric spherical shells, defined by their radius. The capacitance for a spherical capacitor:
    $C = 4 ext{π} ext{perm} rac{r_{1} r_{2}}{r_{2} - r_{1}}$,
    where $r_1$ is the inner and $r_2$ is the outer radius.

• Cylindrical Capacitor: Designed from concentric cylinders where the capacitance expression depends on the length and the radii of the cylinders.

• Coaxial Cable: Used in transmission, also designed as a cylindrical capacitor with proper capacitive properties to shield signals from external interference.

8.2 Capacitors in Series and in Parallel

Learning Objectives

• Explain how to determine the equivalent capacitance of capacitors in series and parallel.

• Compute voltage and charge on plates for capacitor networks.

Series Combination of Capacitors

• In a series configuration:
    - Each capacitor has the same charge Q.
    - Potential differences across capacitors vary depending on their individual capacitances.
    - The equivalent capacitance (C_eq) for n capacitors in series:
      $rac{1}{C_ ext{eq}} = rac{1}{C_1} + rac{1}{C_2} + … + rac{1}{C_n}$.

Parallel Combination of Capacitors

• In a parallel configuration:
    - Each capacitor experiences the same voltage V.
    - Total charge stored can be found by summing individual charges:
      $Q_ ext{total} = Q_1 + Q_2 + … + Q_n$.
    - The equivalent capacitance for n capacitors in parallel:
      $C_ ext{eq} = C_1 + C_2 + … + C_n$.
      

Example Calculations

• There are examples of calculating total capacitance for both series and parallel configurations through combining capacitors.

8.3 Energy Stored in a Capacitor

Learning Objectives

• Explain how energy is stored in a capacitor.

• Use energy relations to determine energy in a capacitor network.

Energy Stored Equation

• Energy stored in a capacitor (U):
    $U = rac{1}{2} C V^2$
    where C is capacitance and V is voltage across plates.

• Applications of Stored Energy: Used in electronic flash units, defibrillators, etc.

8.4 Capacitor with a Dielectric

Learning Objectives

• Describe effects of dielectrics in capacitors on capacitance and other properties.

• Calculate capacitance with dielectric aids.

Effect of Dielectric:

• Inserting a dielectric material increases capacitance by a multiplicative factor known as the dielectric constant (K):
    $C' = K imes C$.

• The stored energy changes:
    $U' = U imes rac{1}{K}$.

8.5 Molecular Model of a Dielectric

Learning Objectives

• Explain polarization of a dielectric in a uniform electric field.

• Describe the effect of polarized dielectrics on fields between plates.

Polarization Effect

• Dielectric materials can be polar (with permanent dipole moments) or nonpolar (induced polarization).

• Dielectric strength describes the electric field strength that causes breakdown (ionization).

• Typical dielectric constants and breakdown values compared.

Chapter Review

Key Terms

• Capacitance: Amount of charge stored per unit volt.

• Capacitor: Device that stores electrical charge and energy.

• Dielectric: Insulating material between capacitor plates.

Key Equations

1. Capacitance: $C = rac{Q}{V}$.

2. Energy stored: $U = rac{1}{2} CV^2$.

3. Capacitance with dielectric: $C' = K imes C$.

4. Series/Equivalent Capacitance: $rac{1}{C_ ext{eq}} = rac{1}{C_1} + rac{1}{C_2} + …$.

5. Parallel Capacitance: $C_ ext{eq} = C_1 + C_2 + …$.

Conceptual Questions

1. Discuss how capacitance changes under different configurations.

2. Explain the effects of dielectric materials and their properties.