Question
If $$C$$ and $$R$$ denote capacitance and resistance respectively, then the dimensional formula of $$CR$$ is
A.
$$\left[ {{M^0}{L^0}T} \right]$$
B.
$$\left[ {{M^0}{L^0}{T^0}} \right]$$
C.
$$\left[ {{M^0}{L^0}{T^{ - 1}}} \right]$$
D.
Not expressible in terms of $$\left[ {MLT} \right]$$
Answer :
$$\left[ {{M^0}{L^0}T} \right]$$
Solution :
$$\eqalign{
& \because C = \frac{q}{V} = \frac{q}{{\frac{W}{q}}} = \frac{{{q^2}}}{W} = \frac{{{{\left( {it} \right)}^2}}}{{F \cdot x}} = \frac{{{{\left[ {AT} \right]}^2}}}{{\left[ {M{L^2}{T^{ - 2}}} \right]}} \cr
& = \left[ {{M^{ - 1}}{L^{ - 2}}{T^4}{A^2}} \right]\,\,{\text{and }}R = \frac{V}{i} = \frac{W}{{qi}} = \frac{{F \cdot x}}{{{i^2}t}} \cr
& = \frac{{\left[ {M{L^2}{T^{ - 2}}} \right]}}{{\left[ {AT} \right]\left[ A \right]}} = \left[ {M{L^2}{T^{ - 3}}{A^{ - 2}}} \right] \cr
& \therefore {\text{Dimensional formula of }}CR = \left[ {{M^{ - 1}}\;{L^{ - 2}}\;{T^4}\;{A^2}} \right]\left[ {M{L^2}\;{T^{ - 3}}\;{A^{ - 2}}} \right] \cr
& = \left[ {{M^0}\;{L^0}\;T} \right] \cr} $$