Which of the following physical quantities have the same dimensions?

We need to identify which pair of physical quantities have the same dimensions.

For Option A, electric displacement $$\vec{D}$$ and surface charge density are considered.

From Gauss's law in dielectrics: $$\vec{D} = \varepsilon_0 \vec{E} + \vec{P}$$, and for a surface, $$\oint \vec{D} \cdot d\vec{A} = Q_{free}$$.

Therefore, $$[D] = \dfrac{[Q]}{[A]} = \dfrac{C}{m^2}$$

Surface charge density: $$[\sigma] = \dfrac{[Q]}{[A]} = \dfrac{C}{m^2}$$

Both have the same dimensions $$[AT L^{-2}]$$. This is correct.

For Option B, displacement current and the electric field are compared.

Displacement current has dimensions of current: $$[A]$$

Electric field has dimensions: $$[MLT^{-3}A^{-1}]$$

These are not the same.

For Option C, current density and surface charge density are compared.

Current density: $$[J] = \dfrac{[A]}{[L^2]} = [AL^{-2}]$$

Surface charge density: $$[\sigma] = \dfrac{[AT]}{[L^2]} = [ATL^{-2}]$$

These differ by a factor of $$[T]$$, so they are not the same.

For Option D, electric potential and energy are compared.

Electric potential: $$[ML^2T^{-3}A^{-1}]$$

Energy: $$[ML^2T^{-2}]$$

These are not the same.

Hence, the correct answer is Option A.