Question

Find the dimensions of $RC$ ($R=$ Resistance, $C=$ Capacitance).

Answer 1

Hint:First derive the dimensional formula of resistance from ohm’s law and then derive the dimensional formula of capacitance. Therefore, find the dimensions of $RC$ by using the dimension of resistance and capacitance.

Complete step by step answer:
In this question, the resistance $R$ and the capacitance $C$ is given and we have to calculate the dimension of $RC$.
First, we obtain the dimension of resistance and the capacitance then we will calculate the dimension of $RC$.
As we know that from Ohm’s law, we can find the dimensions of $R$. According to Ohm’s law state that,
$V=IR$ [Where $I$ is current, $V$ is voltage and $R$ is resistance].
Hence, Resistance $\left(R\right)=$ Voltage/current
Since, Voltage $\left(V\right)$ $=$ Electric field $\times$ Distance $=$ Force/charge $\times$ Distance.
Now, charge $=$ current$\times$time $=I^1{T}^1$ and the dimension of force is $M^1{L}^1{T}^{-2}$
So, we can find the dimension of voltage $=$ force/charge $\times$ Distance$=\left[M^1{L}^1{T}^{-2}\right]\times {\left[I^1{T}^1\right]}^{-1}\times \left[L^1\right]=\left[M^1{L}^2{T}^{-3}{I}^{-1}\right]$
$\therefore$ Resistance $=$ Voltage/Current
Now we find the dimension of resistance
$R=[M^1{L}^1{T}^{-2}]\times [I{]}^{-1}=[M^1{L}^2{T}^{-3}{I}^{-2}]$
Now we will find the dimension of capacitance-
As we know that capacitance = charge/potential difference = charge/voltage.
Now, charge = current $\times$ time, hence the dimension of charge is $[I^1{T}^1]$ and voltage = electric field $\times$ distance = force/charge $\times$ Distance.
Dimensional formula of force is$[M^1{L}^1{T}^{-2}]$.Hence the dimension of voltage$=[M^1{L}^1{T}^{-2}]\times [I^1{T}^1{]}^{-1}\times [L{]}^1=[M^1{L}^2{T}^{-3}{I}^{-1}]$
No, the dimension formula of capacitance = charge/potential difference = charge/voltage
$C=[I^1{T}^1]\times [M^1{L}^2{T}^{-3}{I}^{-1}{]}^{-1}=[M^{-1}{L}^{-2}{T}^4{I}^2]$
Therefore, the dimension of capacitance is$[M^{-1}{L}^{-2}{T}^4{I}^2]$
Now we can determine the dimension of $RC$ by the dimensions of resistance$\left(R\right)$and capacitance$\left(C\right)$.
The dimension of RC is obtained as,
$RC=\left[M^1{L}^2{T}^{-3}{I}^{-2}\right]\times \left[M^{-1}{L}^{-2}{T}^4{I}^2\right]=\left[T\right]$
Therefore, the dimension formula of $RC$ is $\left[M^0{L}^{0}T\right]$.

Note:The electrical resistance of a circuit is mainly defined as the ratio of the applied voltage to the electric current that flows through it and its unit is Ohm. Similarly, capacitance of a capacitor is the amount of charge it can store per unit of voltage.