The normal component of which quantity is always discontinuous at the boundary?

The normal component of electric displacement field (D) is always discontinuous at the boundary when there is a free surface charge. This is because the normal component of the electric displacement field, which is directly related to the presence of free charges, changes sharply at the boundary where the density of free charges changes. This principle is encapsulated in one of Maxwell’s equations, specifically the boundary condition for electric fields at the interface between two media, which states that the component of the electric displacement field perpendicular to the boundary differs by an amount equal to the surface charge density ((sigma)) present at the boundary. Mathematically, this is expressed as:

[

D_1^perp – D_2^perp = sigma

]

Here, (D_1^perp) and (D_2^perp) are the perpendicular (normal) components of the electric displacement fields in the two media on either side of the boundary, and (sigma) is the free surface charge density at the boundary. This equation highlights the discontinuity of the normal component of (D) across a boundary with surface charge.