8.4 Capacitor with a Dielectric

Capacitors with dielectrics are game-changers in electrical engineering. By inserting an insulating material between the plates, we can boost a capacitor's charge-storing ability and tweak its voltage characteristics. This simple modification opens up a world of possibilities in circuit design.

The magic lies in how dielectrics polarize in an electric field. This polarization reduces the field between the plates, allowing more charge storage. It's like adding an extra lane to a highway – suddenly, you can fit more cars without increasing congestion.

Capacitors with Dielectrics

Dielectrics and capacitor properties

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• Inserting a dielectric between capacitor plates increases the capacitance
  • Dielectric material becomes polarized in the presence of an electric field
    • Positive charges slightly displaced toward the negative plate (electrons in atoms)
    • Negative charges slightly displaced toward the positive plate (nuclei of atoms)
  • Polarized dielectric reduces the effective electric field between the plates
    • Internal electric field created by polarized dielectric opposes the applied field
  • Reduced electric field allows more charge to be stored on the plates for the same voltage (capacitance is charge per volt)
• Inserting a dielectric between capacitor plates decreases the voltage across the capacitor
  • Voltage across the capacitor given by $V = \frac{Q}{C}$
  • With a dielectric, capacitance increases while charge remains constant
    • Increased capacitance results in a decreased voltage across the capacitor (voltage is energy per charge)

Capacitance with dielectric materials

• Capacitance of a capacitor with a dielectric given by $C = \kappa C_0$
  • $\kappa$ is the dielectric constant, also known as relative permittivity, a material property of the dielectric (vacuum has $\kappa = 1$)
  • $C_0$ is the capacitance of the capacitor without the dielectric
    • $C_0 = \frac{\varepsilon_0 A}{d}$ for a parallel-plate capacitor
    • $A$ is the area of the plates and $d$ is the distance between them
    • $\varepsilon_0$ is the permittivity of free space, equal to $8.85 \times 10^{-12} \frac{F}{m}$
• Stored energy in a capacitor with a dielectric given by $U = \frac{1}{2}CV^2$
  • $C$ is the capacitance with the dielectric (higher than without dielectric)
  • $V$ is the voltage across the capacitor (lower than without dielectric for same charge)
• Stored energy can also be calculated using $U = \frac{1}{2}\frac{Q^2}{C}$
  • $Q$ is the charge stored on the capacitor plates (same as without dielectric)

Dielectric polarization effects

• Dielectric materials are insulators that can be polarized by an external electric field
  • Polarization occurs due to the displacement of positive and negative charges within the material
    • Positive charges slightly displaced in the direction of the electric field (nuclei)
    • Negative charges slightly displaced opposite to the direction of the electric field (electrons)
  • The sum of these displacements creates an electric dipole moment in the material
• Polarization of the dielectric creates an internal electric field that opposes the external field
  • The opposing internal field reduces the net electric field within the dielectric
    • Reduced net electric field allows more charge to be stored on the capacitor plates (higher capacitance)
• The effect of dielectric polarization on capacitor properties includes:
  • Increased capacitance compared to a capacitor without a dielectric (more charge stored per volt)
  • Decreased voltage across the capacitor for the same amount of stored charge (less energy per charge)
  • Increased maximum stored energy for the same voltage (more charge and lower voltage)
• Dielectrics have a maximum electric field they can withstand, known as their dielectric strength