Question

Verify that the equation \(I=\langle u\rangle c\) is dimensionally consistent (i.e., check the units).

Short answer

Answer: Yes, the equation is dimensionally consistent since the dimensions on both sides of the equation are equal (\([M L^2 T^{-3}] = [M L^0 T^{-3}]\)).

Step-by-step solution

Identify the dimensions of each variable in the equation.

Intensity (I) has dimensions of power per unit area, which can be expressed as energy per unit area per unit time (\([M L^2 T^{-3}]\)). The average energy density (\(\langle{u}\rangle\)) has dimensions of energy per unit volume (\([M L^{-1} T^{-2}]\)). Lastly, the speed of light (c) has dimensions of length per unit time (\([L T^{-1}]\)).

Evaluate the dimensions of \(\langle u \rangle c\).

Multiply the dimensions of average energy density (\([M L^{-1} T^{-2}]\)) with the dimensions of the speed of light (\([L T^{-1}]\)): $$ \langle{u}\rangle c = [M L^{-1} T^{-2}] \cdot [L T^{-1}] = [M L^0 T^{-3}] $$

Compare dimensions on both sides.

Finally, compare the dimensions obtained in step 2 with the dimensions of intensity (I). $$ I = [M L^2 T^{-3}] $$ $$ \langle{u}\rangle c = [M L^0 T^{-3}] $$ Notice that the dimensions of both sides of the equation are equal (\([M L^2 T^{-3}] = [M L^0 T^{-3}]\)), so the equation is dimensionally consistent.