Dielectrics and polarization under a uniform electric field A linear, homogeneous dielectric with relative permittivity εr is subjected to a uniform electric field E. What is the polarization vector P (dipole moment per unit volume) expressed in terms of ε0, εr, and E?

Introduction / Context:Polarization connects microscopic dipole formation to macroscopic electric-field response. In linear dielectrics, this relationship lets engineers compute capacitance, stored energy, and field distributions in cables, capacitors, and high-voltage insulation.

Given Data / Assumptions:

• Material is linear, isotropic, and homogeneous.
• Relative permittivity is εr (dimensionless); ε0 is the permittivity of free space.
• Applied electric field is uniform E.

Concept / Approach:

The constitutive relation for a linear dielectric is D = εE = ε0εrE. Polarization is defined by D = ε0E + P. Eliminating D gives P directly in terms of εr and E, revealing how far the material’s response deviates from vacuum behavior.

Step-by-Step Solution:

Verification / Alternative check:

Define electric susceptibility χe by P = ε0 χe E. Since εr = 1 + χe, we obtain P = ε0 (εr − 1) E, confirming consistency with susceptibility definitions.

Why Other Options Are Wrong:

Option A ignores field dependence; B omits the material contribution (only vacuum). D uses εr without subtracting unity; E has incorrect dimensions. Only C matches the constitutive framework.

Common Pitfalls:

Confusing D with P; forgetting εr is unitless; missing the (εr − 1) factor that isolates material polarization from vacuum.

Final Answer:

P = ε0 (εr − 1) E