Polarization

When a dielectric material is placed inside an electric field, the bound charges within the material shift slightly from their equilibrium positions. This shift creates small electric dipoles throughout the material, and the collective effect of all these dipoles is described by a quantity called dielectric polarization. Understanding polarization is essential for analyzing capacitors, dielectric materials, and electromagnetic boundary problems in GATE and university exams.

Core Concept: What Is Polarization

In a dielectric material, atoms and molecules are electrically neutral overall, but the positive nucleus and the surrounding negative electron cloud can be displaced relative to each other when an external electric field is applied. This displacement creates a small electric dipole moment at each atomic site, with the negative charge side pointing against the field and the positive side pointing along the field direction.

The polarization vector P is defined as the dipole moment per unit volume of the dielectric material. If there are N dipoles per unit volume, each with dipole moment p (a vector quantity), then P = N times p. The unit of P is C/m², which is the same as surface charge density. This is not a coincidence: the bound surface charges that appear on the faces of a polarized dielectric are directly related to P.

For most linear isotropic dielectric materials, polarization is proportional to the applied electric field. The constant of proportionality is epsilon_0 times the electric susceptibility chi_e. So P = epsilon_0 * chi_e * E. The susceptibility chi_e is dimensionless and is always positive for dielectrics. It represents how easily a material polarizes in response to an applied field.

Mathematical Expression

The relationship between the electric displacement vector D, the electric field E, and the polarization P is the most important equation in dielectric theory. The electric displacement D is defined as D = epsilon_0 * E + P. For a linear dielectric, substituting P = epsilon_0 * chi_e * E gives D = epsilon_0 * (1 + chi_e) * E = epsilon_r * epsilon_0 * E = epsilon * E, where epsilon_r = 1 + chi_e is the relative permittivity.

The bound polarization charge density inside the material is given by rho_b = negative divergence of P. On the surface, the bound surface charge density is rho_sb = P dot n_hat, where n_hat is the outward unit normal. These bound charges are responsible for reducing the net internal electric field compared to what it would be in free space.

Practical Understanding

Polarization explains why inserting a dielectric slab between capacitor plates increases the capacitance. The bound surface charges created by polarization partially cancel the free charges on the plates. This reduces the effective electric field inside the dielectric, which means more free charge can be stored on the plates for the same potential difference, increasing capacitance by a factor of epsilon_r.

In practice, different dielectrics polarize by different mechanisms. Ionic crystals polarize by displacement of positive and negative ion sublattices. Polar molecules like water polarize by orientation of permanent dipoles along the field. Non-polar materials polarize by distortion of electron clouds. The net effect is always the same: a macroscopic P that opposes the internal field.

Given:
A dielectric with chi_e = 4, E = 2 x 10^5 V/m, epsilon_0 = 8.85 x 10^-12 F/m

Why this formula applies:
Linear dielectric: P = epsilon_0 * chi_e * E

Formula:
P = epsilon_0 * chi_e * E

Substitution:
P = 8.85e-12 * 4 * 2e5

Calculation:
P = 8.85e-12 * 8e5
P = 7.08e-6 C/m2

Final Answer:
P = 7.08 uC/m2
Also: epsilon_r = 1 + chi_e = 5, D = epsilon_r * epsilon_0 * E = 8.85e-6 C/m2

Exam Tip: In GATE, D = epsilon_0*E + P always holds in any medium. For free space, P = 0 so D = epsilon_0*E. Never confuse chi_e (susceptibility) with epsilon_r (relative permittivity): epsilon_r = 1 + chi_e, not chi_e alone.

• Polarization P is dipole moment per unit volume, in units of C/m2.
• For linear dielectrics: P = epsilon_0 * chi_e * E, where chi_e is always positive.
• The displacement vector D accounts for both free and bound charge effects.
• Bound surface charges appear on dielectric faces: rho_sb = P dot n_hat.
• Polarization reduces the net internal field, which is why capacitance increases with dielectrics.

Quick Revision

• P = N*p = dipole moment per unit volume (C/m2).
• Linear dielectric: P = epsilon_0 * chi_e * E.
• D = epsilon_0*E + P = epsilon_r * epsilon_0 * E.
• epsilon_r = 1 + chi_e (trap: chi_e is not epsilon_r).
• Bound charges: rho_b = -div(P), rho_sb = P . n_hat.
• D depends only on free charges; E depends on both free and bound charges.
• GATE trap: In free space chi_e = 0, P = 0, D = epsilon_0*E only.