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.