Dielectrics Knowledge-Based Study

Study notes

1. What is a Dielectric?

   A dielectric is an electrical insulator that can be polarized by an applied electric field. Unlike conductors, dielectrics contain no free charges that can move macroscopic distances through the material. Instead, all electrons remain bound to specific atoms or molecules.

   Molecular Polarization

   When placed in an external electric field, the microscopic charges within a dielectric undergo small displacements:

   • Non-polar molecules: In the absence of an external field, the centers of positive and negative charge coincide. When an external field is applied, the positive nuclei shift slightly in the direction of the field while the electron clouds shift in the opposite direction, creating induced dipoles.
   • Polar molecules: These molecules possess permanent electric dipole moments even without an external field (such as water, $\text{H}_2\text{O}$). In the absence of an applied field, thermal agitation causes random orientation, resulting in zero net macroscopic dipole moment. An applied electric field exerts a torque that aligns these permanent dipoles along the field direction.

   Induced Electric Field

   When a dielectric slab is placed in an external electric field $E_0$, polarization causes bound surface charges to accumulate on its opposing faces: negative charge appears near the positive plate, and positive charge near the negative plate.

   These bound surface charges generate an internal induced electric field $E_{\text{ind}}$ that opposes the original applied field $E_0$. Consequently, the net electric field $E$ inside the dielectric is significantly reduced:

   $$E = E_0 - E_{\text{ind}}$$

   Dielectric Constant and Permittivity

   The factor by which the electric field is reduced is defined as the dielectric constant (or relative permittivity), denoted by $\kappa$ or $\epsilon_r$:

   $$\kappa = \frac{E_0}{E}$$

   Since $E < E_0$ in any material medium, $\kappa > 1$ for all dielectrics (for vacuum, $\kappa = 1$; for dry air at STP, $\kappa \approx 1.0006$). The absolute permittivity $\epsilon$ of the dielectric medium is given by:

   $$\epsilon = \kappa \epsilon_0$$

   where $\epsilon_0$ is the permittivity of free space.

   Effect on Capacitance

   Consider a parallel-plate capacitor with plate area $A$ and separation $d$. When completely filled with a dielectric of constant $\kappa$, its capacitance increases by a factor of $\kappa$:

   $$C = \kappa C_0 = \frac{\kappa \epsilon_0 A}{d}$$

   where $C_0 = \frac{\epsilon_0 A}{d}$ is the vacuum capacitance.

   Behavior Under Different Operating Conditions

   The quantitative behavior of a capacitor when a dielectric is inserted depends on whether the capacitor remains connected to a voltage source:

   • Isolated Capacitor (Battery Disconnected): Charge $Q$ remains constant ($Q = Q_0$). Because capacitance increases to $C = \kappa C_0$, the potential difference drops to $V = \frac{V_0}{\kappa}$, the electric field decreases to $E = \frac{E_0}{\kappa}$, and the stored electrostatic energy decreases to $U = \frac{U_0}{\kappa}$. The lost energy is converted into mechanical work done by the capacitor drawing in the dielectric slab.
   • Connected Capacitor (Constant Potential Difference): Potential difference $V$ is held constant by the battery ($V = V_0$). The electric field $E = \frac{V}{d}$ remains unchanged. The stored charge increases to $Q = \kappa Q_0$, and the stored electrostatic energy increases to $U = \kappa U_0$. The battery supplies the necessary energy to increase both the stored energy and do work on the dielectric.

   Dielectric Breakdown

   Although dielectrics are insulating materials, their ability to prevent charge conduction is limited. If the applied electric field becomes excessively strong, the electric force exerts enough pull on bound electrons to strip them away from their parent atoms or molecules. This phenomenon is called dielectric breakdown.

   Once breakdown occurs, the material ionizes rapidly, conducting electricity via an electric spark or arc, which typically causes permanent physical damage to solid dielectrics.

   Dielectric Strength

   The maximum electric field that a material can withstand without undergoing electrical breakdown is called its dielectric strength. It is measured in volts per meter ($\text{V/m}$) or kilovolts per millimeter ($\text{kV/mm}$).

   • Dry Air: approximately $3 \times 10^6\text{ V/m}$ (or $3\text{ kV/mm}$).
   • Mylar: approximately $24 \times 10^6\text{ V/m}$.
   • Strontium Titanate: exhibits both a very high dielectric constant ($\kappa \approx 300$) and robust breakdown resistance.

   Therefore, introducing a suitable dielectric into a capacitor serves two critical engineering functions: it substantially increases capacitance for a given physical size, and it allows the device to operate at higher voltages without breakdown.