Electric Displacement

Definition

Electric displacement, also called electric flux density, is a vector quantity defined as the electric field intensity multiplied by the permittivity of the medium.

In vacuum:

$$\mathbf{D} = \varepsilon_0 \mathbf{E}$$

where:

• $\mathbf{D}$ = electric displacement vector
• $\varepsilon_0$ = permittivity of free space
• $\mathbf{E}$ = electric field intensity

Its SI unit is coulomb per square meter $(\text{C/m}^2)$. It represents the electric flux per unit area normal to the field.

Main Content

1. Electric Displacement in Vacuum

• In vacuum, electric displacement is directly proportional to the electric field, and the proportionality constant is the permittivity of free space, $\varepsilon_0$.
• Since vacuum contains no material polarization, $\mathbf{D}$ depends only on free charges and is often used to express Gauss’s law in a more convenient form: $$\nabla \cdot \mathbf{D} = \rho_f$$ where $\rho_f$ is the free charge density.

In electrostatics, the field $\mathbf{D}$ is useful because it separates the effect of free charge from the influence of the medium. For example, around a point charge $Q$ in vacuum:

$$\mathbf{D} = \frac{Q}{4\pi r^2}\hat{r}$$

This shows that electric displacement behaves like flux density radiating outward from the charge.

2. Relation with Electric Field and Electric Flux

• Electric displacement is related to the electric field by: $$\mathbf{D} = \varepsilon_0 \mathbf{E}$$ in vacuum, and in a linear dielectric: $$\mathbf{D} = \varepsilon \mathbf{E}$$

• The total electric flux through a closed surface is: $$\oint \mathbf{D}\cdot d\mathbf{A} = Q_{\text{free}}$$

This means the flux of $\mathbf{D}$ through any closed surface gives only the enclosed free charge, not bound charge. That is why $\mathbf{D}$ is extremely useful in electrostatics problems involving conductors and dielectric boundaries. For example, if a spherical surface encloses a charge $Q$, the total flux of $\mathbf{D}$ through that surface is exactly $Q$, regardless of the shape or size of the surface.

3. Physical Significance and Boundary Behavior

• Electric displacement helps describe how electric fields interact with matter and how charges are distributed at surfaces and interfaces.
• At the boundary between two media, the normal component of $\mathbf{D}$ changes according to the free surface charge density: $$D_{2n} - D_{1n} = \sigma_f$$

This boundary condition is extremely important in solving electrostatic problems involving parallel plate capacitors, layered dielectrics, and charged interfaces. In vacuum, it simplifies because $\mathbf{D}$ depends only on the electric field and permittivity of free space. For instance, at the surface of a charged conductor in vacuum, the electric displacement is normal to the surface and equals the free surface charge density.

Working / Process

1. Identify the free charge distribution in the region under study and note whether the medium is vacuum or another material.

2. Use Gauss’s law for electric displacement: $$\oint \mathbf{D}\cdot d\mathbf{A} = Q_{\text{free}}$$ to find $\mathbf{D}$ based on symmetry.

3. Relate $\mathbf{D}$ to the electric field using $\mathbf{D}=\varepsilon_0\mathbf{E}$ in vacuum, then apply boundary conditions or field equations if needed.

For example, for a point charge in vacuum, symmetry tells us $\mathbf{D}$ is radial. Applying Gauss’s law on a spherical surface gives:

$$D(4\pi r^2)=Q \Rightarrow D=\frac{Q}{4\pi r^2}$$

Then the electric field is found from:

$$E=\frac{D}{\varepsilon_0} = \frac{Q}{4\pi \varepsilon_0 r^2}$$

This is the standard Coulomb field in vacuum.

Advantages / Applications

• It simplifies electrostatic analysis by focusing on free charges rather than both free and bound charges.
• It is widely used in capacitor calculations, especially when dealing with dielectrics and layered media.
• It provides a convenient way to apply Gauss’s law and boundary conditions in problems with symmetry.

Summary

• Electric displacement is the electric flux density associated with the electric field.
• In vacuum, it is given by $\mathbf{D}=\varepsilon_0\mathbf{E}$.
• It is especially useful because its flux depends only on free charge.
• Electric displacement is an essential tool in solving electrostatic problems in vacuum and in dielectric media.