Question 5: Show that the famous Einstein’s equation E = mc^{2 is dimensionally consistent.}

Answer

Dimensional consistency means the dimensions on both sides of the equation are same.

Consider the given equation E = mc² and we calculate dimensions on both sides and then see whether they come out to be the same or not. Take the LHS,

Unit of E = Joule = N m = (kg m s^-2) m

In case of dimensions, the unit is = [M][L][T^-2][L] = [MLLT^-2] = [ML²T^-2] —– (A)

Similarly, on the right hand side,

Unit of mc² = kg. ms^-1 .ms^-1

In case of dimensions, the unit is = [M][L][T^-1][L][T^-1] = [MLLT^-1T^-1]

= [ML²T^-2] —– (B)

Equations (A) and (B) give dimensions of LHS and RHS respectively. It can be seen that the dimensions on both sides of the equation are the same. Therefore, the famous Einstein equation E = mc² is dimensionally consistent.