Capacitance and Dielectrics Notes

Chapter 26 Capacitance and Dielectrics Lecture 6

History of Capacitors

  • Condensers patented by Nikola Tesla in U.S. Patent 567,818, Electrical Condenser, in 1896 on September 15.
  • In October 1745, Ewald Georg von Kleist of Pomerania in Germany invented the first recorded capacitor: a glass jar with water inside as one plate was held on the hand as the other plate. A wire in the mouth of the bottle received charge from an electric machine, and released it as a spark.
  • Benjamin Franklin investigated the Leyden jar and proved that the charge was stored on the glass, not in the water as others had assumed.

Direct Current (DC) and Electromotive Force (EMF)

  • Direct Current (DC): When the current in a circuit has a constant direction.
  • Circuits are often analyzed in a steady state, with constant magnitude and direction.
  • The potential difference between the terminals of a battery is constant.
  • The battery supplies free electrons with potential energy, which is converted into kinetic energy.
  • The battery is a source of electromotive force (EMF).
  • Electromotive Force (EMF):
    • The electromotive force (E) of a battery is the maximum possible voltage that the battery can provide between its terminals.
    • EMF supplies energy but does not apply a force.
    • The battery is the source of energy in the circuit.
    • The positive terminal of the battery is at a higher potential than the negative terminal.
    • Wires in the circuit analysis are considered to have no resistance.

Capacitors

  • Capacitors are devices that store electric charge.
  • The amount of stored charge is finite, unlike a battery where charges are assumed to be infinite.
  • Examples of capacitor applications:
    • Radio receivers
    • Filters in power supplies
    • Eliminating sparking in automobile ignition systems
    • Energy-storing devices in electronic flashes

Definition of Capacitance

  • The capacitance (C) of a capacitor is defined as the absolute value of the ratio of the magnitude of the charge on either conductor to the potential difference between the conductors.
  • C = \frac{Q}{\Delta V}, where Q is the charge and \Delta V is the potential difference.
  • Units: Farad (F)
  • 1 \ F = 1 \frac{C}{V}
  • A Farad is a very large unit; common prefixes include µF (10^{-6}), nF (10^{-9}), and pF (10^{-12}).
  • The amount of charge that can be stored is given by: Q = C \Delta V.
  • \Delta V is generated by a battery.
  • Capacitance indicates the amount of charge that can be stored for a given battery and capacitance.

Makeup of a Capacitor

  • A capacitor consists of one or two conductors called plates.
  • When two conductors are used, they are charged with equal magnitude and opposite directions.
  • A potential difference exists between the plates due to the charge.

More About Capacitance

  • Capacitance (C) is always a positive quantity.
  • The capacitance of a given capacitor is constant.
  • Capacitance measures a capacitor’s ability to store charge.
  • Capacitors are split into two groups:
    • Polarized
    • Unpolarized

Charging of a Parallel Plate Capacitor

  • Each plate is connected to a terminal of the battery.
  • The battery is a source of potential difference.
  • If the capacitor is initially uncharged, the battery establishes an electric field in the connecting wires.
  • Some capacitors are ‘polarized’ and some are not.

Parallel Plate Capacitor: Charging

  • An electric field applies a force on electrons in the wire just outside of the plates.
  • The force causes the electrons to move onto the negative plate.
  • This continues until equilibrium is achieved.
  • The plate, the wire, and the terminal are all at the same potential.
  • At this point, there is no field present in the wire, and the movement of the electrons ceases.

Parallel Plate Capacitor, final configuration

  • The plate is now negatively charged.
  • A similar process occurs at the other plate, with electrons moving away from the plate, leaving it positively charged.
  • In its final configuration, the potential difference across the capacitor plates is the same as that between the terminals of the battery.
  • \Delta V{\text{capacitor}} = \Delta V{\text{battery}}

Example 2: Capacitance of Two Parallel Plates

  • The charge density on the plates is \sigma = \frac{Q}{A}.
  • A is the area of each plate, which are equal.
  • Q is the charge on each plate, equal with opposite signs.
  • The electric field is uniform between the plates and zero elsewhere.
  • Falstad 3D Vector applet: http://www.falstad.com/vector3de/

Example 1: Capacitance of Two Parallel Plates

  • The capacitance is proportional to the area of its plates and inversely proportional to the distance between the plates.

Example 2: Capacitance of an Isolated Sphere

  • Assume a spherical charged conductor with radius a.
  • The sphere will have the same capacitance as it would if there were a conducting sphere of infinite radius, concentric with the original sphere.
  • Assume V = 0 for the infinitely large shell.

Example 3: Capacitance of a Cylindrical Capacitor

  • \Delta V = -2 k_e \lambda \ln \left(\frac{b}{a}\right)
  • \lambda = \frac{Q}{l}
  • The capacitance is

Example 4: Capacitance of a Spherical Capacitor

  • The potential difference is
  • The capacitance is

Contributions to Capacitance

  • The capacitance contains two main contributions:
    • The space around the charged object represented by the permittivity \epsilon
    • Geometrical properties

Energy in a Capacitor – Overview

  • Consider the circuit to be a system.
  • Before the switch is closed, the energy is stored as chemical energy in the battery.
  • When the switch is closed, the energy is transformed from chemical energy in the battery to electric potential energy in the capacitor.

Energy Stored in a Capacitor

  • Assume the capacitor is being charged and, at some point, has a charge Q on it.
  • The work needed to transfer a charge from one plate to the other is
  • The total work required is

Energy Considerations

  • The work done in charging the capacitor appears as electric potential energy U:
  • This applies to a capacitor of any geometry.
  • The energy stored increases as the charge increases and as the potential difference increases.
  • In practice, there is a maximum voltage before discharge occurs between the plates.

Energy per Unit Volume

  • The energy can be considered to be stored in the electric field.
  • For a parallel-plate capacitor, the energy can be expressed in terms of the field as
    • U = \frac{1}{2} (\epsilon_0 A d) E^2
  • It can also be expressed in terms of the energy density (energy per unit volume = Ad)
    • U = \frac{1}{2} Q \Delta V = \frac{1}{2} Q(E d)
    • E = \frac{\sigma}{\epsilon0} = \frac{Q}{A \epsilon0} \implies Q = E(A \epsilon_0)
    • U = \frac{1}{2} E (A \epsilon0) (E d) = \frac{1}{2} \epsilon0 E^2 (A d)
    • \frac{U}{Vol} = uE = \frac{1}{2} \epsilon0 E^2

Dielectrics, Motivation

  • For a parallel-plate capacitor, C = \epsilon_0 (A/d).
  • In theory, d could be made very small to create a very large capacitance.
  • In practice, there is a limit to d.
  • d is limited by the electric discharge that could occur through the dielectric medium separating the plates, also called current leakage.
  • For a given d, the maximum voltage that can be applied to a capacitor without causing a discharge depends on the dielectric strength of the material.
  • Solution: Include an insulator in between the plates, also called a dielectric.

Capacitors with Dielectrics

  • A dielectric is a non-conducting material that, when placed between the plates of a capacitor, increases the capacitance.
  • Dielectrics include rubber, glass, and waxed paper.
  • With a dielectric, the capacitance becomes
    • C = \epsilon (A/d) = \epsilon0 \kappa (A/d) / \epsilon0
    • C = \kappa C_0
  • The capacitance increases by the factor \kappa (kappa) when the dielectric completely fills the region between the plates.
  • \kappa is the dielectric constant of the material.

Charging a Capacitor with and without battery

*Charging a Capacitor, remove the battery and insert a dielectric

  • The dielectric being an insulator, the voltage across the two plates \Delta V can change is dropped while the total number of charges Q cannot:

  • The capacitance is multiplied by the dielectric constant \kappa when the dielectric completely fills the region between the plates. The dielectric effect on charge and voltage
    *Charging a Capacitor, keeping the battery connected and insert a dielectric

  • The dielectric being an insulator, the voltage across the two plates \Delta V cannot change equal to that of the battery But the total number of charges Q can.

  • The capacitance is also multiplied by the dielectric constant \kappa when the dielectric completely fills the region between the plates.

Dielectrics benefits

  • Dielectrics provide the following advantages:
    • Increase in capacitance
    • Increase the maximum operating voltage
    • Possible mechanical support between the plates
    • This allows the plates to be close together without touching
    • This decreases d and increases C

Dielectrics – An Atomic View

  • The molecules that make up the dielectric are modeled as dipoles.
  • The molecules are randomly oriented in the absence of an electric field.
  • When an external electric field is applied:
    • This produces a torque on the molecules.
    • The molecules partially align with the electric field, and the insulator becomes polarized.

Polar vs. Non-polar Molecules

  • Molecules are said to be polarized when a separation exists between the average position of the negative charges and the average position of the positive charges.
  • Polar molecules are those in which this condition is always present.
  • Molecules without a permanent polarization are called non-polar molecules.

Example: Water Molecules

  • A water molecule is an example of a polar molecule.
  • The center of the negative charge is near the center of the oxygen atom.
  • The x is the center of the positive charge distribution.
  • The average positions of the positive and negative charges act as point charges.
  • Therefore, polar molecules can be modeled as electric dipoles.

Induced Polarization

  • A linear symmetric molecule has no permanent polarization (a).
  • Polarization can be induced by placing the molecule in an electric field (b).
  • Induced polarization is the effect that predominates in most materials used as dielectrics in capacitors.

How a dielectric increases the capacitance of a capacitor

  • The dielectric becomes polarized by the electric field.
  • The presence of positive charge on the dielectric effectively reduces some of the effective negative charge on the metal.
  • This allows more negative charge on the plates for a given applied voltage, so Q increases due to the additional flow of charges from the dielectric.
  • So, the capacitance increases when the dielectric is inserted between the plates.

Dielectrics, the induced electric field

  • An external field can polarize the dielectric whether the molecules are polar or non-polar.
  • The charged edges of the dielectric act as a second pair of plates producing an induced electric field in the direction opposite the original electric field.