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Question: Find the dimensions of \(RC\) (\(R = \) Resistance, \(C = \) Capacitance)....

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

Explanation

Solution

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

Complete step by step answer:
In this question, the resistance RR and the capacitance CC is given and we have to calculate the dimension of RCRC.
First, we obtain the dimension of resistance and the capacitance then we will calculate the dimension of RCRC.
As we know that from Ohm’s law, we can find the dimensions of RR. According to Ohm’s law state that,
V=IRV = IR [Where II is current, VV is voltage and RR is resistance].
Hence, Resistance (R)=\left( R \right) = Voltage/current
Since, Voltage (V)\left( V \right) == Electric field ×\times Distance == Force/charge ×\times Distance.
Now, charge == current×\timestime =I1T1 = {I^1}{T^1} and the dimension of force is M1L1T−2{M^1}{L^1}{T^{ - 2}}
So, we can find the dimension of voltage == force/charge ×\times Distance=[M1L1T−2]×[I1T1]−1×[L1]=[M1L2T−3I−1] = \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=[M1L1T−2]×[I]−1=[M1L2T−3I−2]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 [I1T1][{I^1}{T^1}] and voltage = electric field ×\times distance = force/charge ×\times Distance.
Dimensional formula of force is[M1L1T−2][{M^1}{L^1}{T^{ - 2}}].Hence the dimension of voltage=[M1L1T−2]×[I1T1]−1×[L]1=[M1L2T−3I−1] = [{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=[I1T1]×[M1L2T−3I−1]−1=[M−1L−2T4I2]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−1L−2T4I2][{M^{ - 1}}{L^{ - 2}}{T^4}{I^2}]
Now we can determine the dimension of RCRC by the dimensions of resistance(R)\left( R \right)and capacitance(C)\left( C \right).
The dimension of RC is obtained as,
RC=[M1L2T−3I−2]×[M−1L−2T4I2]=[T]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 RCRC is [M0L0T]\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.