Question

What is the relationship between electric flux density and volume charge density?

Answer 1

The relationship between electric flux density (\(\mathbf{D}\)) and volume charge density (\(\rho\)) is given by Gauss's law in the context of electromagnetism.

Gauss's law states that the divergence of the electric flux density \(\mathbf{D}\) is equal to the volume charge density \(\rho\):

\[ \nabla \cdot \mathbf{D} = \rho \]

In a vacuum or free space, where there are no free charges, the electric flux density \(\mathbf{D}\) is related to the electric field \(\mathbf{E}\) by:

\[ \mathbf{D} = \varepsilon_0 \mathbf{E} \]

where \(\varepsilon_0\) is the permittivity of free space.

In materials with dielectric properties, the relationship is:

\[ \mathbf{D} = \varepsilon \mathbf{E} \]

where \(\varepsilon\) is the permittivity of the material.

The volume charge density \(\rho\) is the source term in Gauss's law and represents the amount of electric charge per unit volume in a region of space.

Answer 2

The relationship between electric flux density (\(\mathbf{D}\)) and volume charge density (\(\rho\)) is given by Gauss's Law in its differential form. In the context of electromagnetism, this relationship is expressed by:

\[ \nabla \cdot \mathbf{D} = \rho \]

Here’s what each term represents:
- \(\mathbf{D}\) is the electric flux density or electric displacement field.
- \(\rho\) is the volume charge density, which represents the amount of charge per unit volume.

This equation states that the divergence of the electric flux density \(\mathbf{D}\) at a point in space is equal to the volume charge density \(\rho\) at that point. In other words, the presence of charge in a volume creates a divergence in the electric flux density at that location.