To determine the dimensions of the quantity $CV^2$, we can relate it to the energy stored in a capacitor.

1. Formula: The potential energy ($U$) stored in a capacitor is given by the formula $U = \frac{1}{2}CV^2$ .
2. Dimensional Analysis: Since the number $\frac{1}{2}$ is a dimensionless constant, the dimensions of $CV^2$ are exactly the same as the dimensions of Energy ($U$).
3. Dimensions of Energy: Energy (Work) has the dimensional formula $[ML^2T^{-2}]$ .
4. Verification using fundamental dimensions: Capacitance ($C$) = $[M^{-1}L^{-2}T^4A^2]$ Potential ($V$) = $[ML^2T^{-3}A^{-1}]$ $CV^2 = [M^{-1}L^{-2}T^4A^2] \times ([ML^2T^{-3}A^{-1}])^2$ $CV^2 = [M^{-1}L^{-2}T^4A^2] \times [M^2L^4T^{-6}A^{-2}]$

• $CV^2 = [M^{-1+2} L^{-2+4} T^{4-6} A^{2-2}] = [ML^2T^{-2}]$

Therefore, the dimensions of $CV^2$ correspond to those of energy: $[ML^2T^{-2}]$.