Physics - Unit & Dimension Question with Solution | TestHub

Dimensions of energy in terms of momentum (P). Area (A) and Time (T), is related of $\mathrm{E}={\mathrm{P}}^{\mathrm{a}}{\mathrm{A}}^{\mathrm{b}}{\mathrm{T}}^{\mathrm{C}}\dots (i)$ Writing dimensional formula of both side, $\begin{array}{rl}& \left[{\mathrm{ML}}^2{\mathrm{T}}^{-2}\right]={\left[{\mathrm{MLT}}^{-1}\right]}^a{\left[L^2\right]}^b[\mathrm{T}{]}^{\mathrm{c}}\\ & \left[{\mathrm{ML}}^2{\mathrm{T}}^{-2]}\right]=\left[{\mathrm{M}}^{\mathrm{a}}{\mathrm{L}}^{\mathrm{a}+}{\mathrm{T}}^{-\mathrm{a}+\mathrm{c}}\right]\\ & \end{array}$ Comparing the exponents, $\mathrm{a}=1,\mathrm{a}+2\mathrm{b}=2$ $\begin{array}{c}2\mathrm{b}=1\\ \mathrm{b}=\frac{1}{2}\\ -\mathrm{a}+\mathrm{c}=-2-1+\mathrm{c}=-2\\ \Rightarrow \mathrm{C}=-1\end{array}$ $\therefore$ Form Eq. (i), we have $\Rightarrow \mathrm{E}=\left[{\mathrm{PA}}^{1/2}{\mathrm{T}}^{-1}\right]$