A dielectric slab of dielectric constant K is kept between the plates of a parallel plate capacitor. If net electric field inside the dielectric is ⃗ E then polarization vector ⃗ P is related to ⃗ E as

When a dielectric slab is introduced between the plates of a parallel plate capacitor, it affects the electric field and the polarization within the material. The relationship between the electric field, the polarization vector, and the dielectric constant is fundamental in understanding how dielectrics influence capacitors.

Understanding the Relationship

The polarization vector, denoted as ⃗ P, represents the dipole moment per unit volume in the dielectric material. When an electric field ⃗ E is applied, it induces polarization in the dielectric, which can be expressed mathematically. The relationship between the polarization vector and the electric field in a dielectric medium can be described by the equation:

⃗ P = ε₀(χ_e) ⃗ E

Here, ε₀ is the permittivity of free space, and χ_e is the electric susceptibility of the dielectric material. The electric susceptibility is related to the dielectric constant K by the equation:

χ_e = K - 1

Connecting the Concepts

Substituting the expression for χ_e into the polarization equation gives us:

⃗ P = ε₀(K - 1) ⃗ E

This equation shows how the polarization vector ⃗ P is directly proportional to the electric field ⃗ E and depends on the dielectric constant K of the material. The higher the dielectric constant, the greater the polarization for a given electric field.

Illustrative Example

Let’s consider a practical example. Imagine a parallel plate capacitor with a dielectric slab of K = 5 placed between its plates. If the electric field ⃗ E inside the capacitor is measured to be 1000 V/m, we can calculate the polarization vector ⃗ P:

• First, calculate the electric susceptibility: χ_e = K - 1 = 5 - 1 = 4
• Next, use the polarization formula: ⃗ P = ε₀(χ_e) ⃗ E
• Assuming ε₀ = 8.85 x 10⁻¹² F/m, we find: ⃗ P = (8.85 x 10⁻¹²)(4)(1000) = 3.54 x 10⁻⁸ C/m³

This result indicates that the dielectric material has a significant polarization, which enhances the capacitor's ability to store charge by reducing the effective electric field between the plates.

Final Thoughts

The relationship between the polarization vector and the electric field in a dielectric is crucial for understanding how capacitors operate with different materials. By manipulating the dielectric constant, engineers can design capacitors with desired properties for various applications, from small electronic devices to large power systems.