Question

If R and C represent the resistance and capacitance respectively, then give the dimension of $RC$.

Answer 1

Although known that the time constant of an $R - C$ circuit is given by $\tau = RC$ , but before we use that approach, it is important to find the dimensions of the individual elements and combining them to find the dimension of $RC$

Formulae used:
$R = \dfrac{V}{I}$
Where $V$ is the voltage of the circuit and $I$ is the current in the circuit and is dimensionally represented by $A$ .
$C = \dfrac{Q}{V}$
Where $Q$ is the charge in the circuit and $V$ is the voltage of the circuit.
$\tau = RC$
Where $\tau$ is the time constant of an $R - C$ circuit, $R$ is the resistance of the circuit and $C$ is the capacitance of the circuit.

Complete step by step solution:
$R = \dfrac{V}{I}$
Where $V$ is the voltage of the circuit whose dimensional formula is found to be $M{\text{ }}{L^2}\;{T^ - }^3\;{A^ - }^1$ by relating it with energy and charge. $I$ is the current in the circuit and is dimensionally represented by $A$ . $R$ is the resistance of the circuit
Therefore the dimensional formula of $R$ is
$\dfrac{{M{\text{ }}{L^2}\;{T^ - }^3\;{A^ - }^1}}{A} = M{\text{ }}{L^2}\;{T^ - }^3\;{A^ - }^2$ $...\left( 1 \right)$
Similarly in the case of the capacitance of the circuit
$C = \dfrac{Q}{V}$
Where $Q$ is the charge in the circuit and $V$ is the voltage of the circuit. The dimensions of $Q$ are $A{T^{ - 1}}$ and the dimensions of $V$ are $M{\text{ }}{L^2}\;{T^ - }^3\;{A^ - }^1$ .
Therefore the dimensional formula of $C$ is
$\dfrac{{A{T^{ - 1}}}}{{M{\text{ }}{L^2}\;{T^ - }^3\;{A^ - }^1}} = {M^{ - 1}}{L^{ - 2}}{T^2}{A^2}$
To find the dimensional formula of $RC$ we simply multiply the individual dimensions, that is,
$\Rightarrow \left( {M{\text{ }}{L^2}\;{T^ - }^3\;{A^ - }^2} \right)\left( {{M^{ - 1}}{L^{ - 2}}{T^2}{A^2}} \right)$
$\Rightarrow {M^0}{L^0}{T^1}{A^0}$
$\Rightarrow T$

Therefore, the dimension of $RC$ is dependent solely on time. Therefore dimensions of $RC$ is $T$.

Alternatively:
Since the time constant, $\tau = RC$ , therefore you can tell that the dimensions of $RC$ will be similar to that of $\tau$ as dimensional equality is only possible if the dimensions of both quantities are equal.

Therefore dimensions of $RC$ is $T$.

Note: Dimensional analysis questions usually have multiple approaches possible, depending entirely on the ease of your application and knowledge. Dimensions of a particular quantity can be solved in multiple ways by using the right formula to relate that quantity to those quantities whose dimensions you’re sure of. In this question, you could’ve further expanded the formula or Resistance and Capacitance to get to the four basic units: mass $\left( M \right)$ , time $\left( T \right)$ , length $(L)$ and current $\left( A \right)$.