Explanation:

In electrical engineering, the normal component of the electric displacement vector (D) is always discontinuous at the boundary between two different media. This is due to the fact that the electric displacement vector depends on the charge density and polarization within the material, which can change abruptly at the boundary.

Electric Displacement Vector (D):
The electric displacement vector (D) represents the electric field created by free charges and polarization charges within a material. It is defined as the sum of the free charge density (ρf) and the polarization charge density (ρp) multiplied by the permittivity of the material (ε).

D = ε * E + P

Where:
- D is the electric displacement vector
- ε is the permittivity of the material
- E is the electric field
- P is the polarization vector

Discontinuity at the Boundary:
At the boundary between two different materials, the electric displacement vector can change abruptly. This is because the charge density and polarization within the materials may be different. When there is a discontinuity in these quantities, there will be a corresponding discontinuity in the electric displacement vector.

The normal component of the electric displacement vector (Dn) is the component that is perpendicular to the boundary. This component is discontinuous because the charge density and polarization may change abruptly at the boundary.

Importance:
The discontinuity in the normal component of the electric displacement vector is important in understanding the behavior of electromagnetic fields at boundaries. It is used in boundary conditions, such as the boundary conditions for electric fields, which relate the values of electric fields on either side of a boundary.

By understanding the discontinuity in the normal component of the electric displacement vector, engineers and scientists can accurately model and analyze electromagnetic fields at boundaries, which is crucial in designing and optimizing electrical systems and devices.