Question

The dimensions of RC are:
A. ${\text{ML}}{{\text{T}}^{{\text{ - 1}}}}$
B.${{\text{M}}^0}{{\text{L}}^0}{{\text{T}}^{\text{1}}}$
C. ${{\text{M}}^0}{{\text{L}}^0}{{\text{T}}^{{\text{ - 1}}}}$
D. ${{\text{M}}^{ - 1}}{{\text{L}}^{ - 1}}{{\text{T}}^{{\text{ - 1}}}}$

Answer 1

Hint: We need to know the dimensional formulas for Resistance and Capacitance to solve this question. We can derive the dimensional formulas for Resistance and Capacitance by their basic formulas also. Multiplying the dimensions of R and C would give the dimension of RC.

Complete Step-by-Step solution:
A resistor-capacitor or RC circuit, is an electric circuit containing resistors and capacitors driven by a voltage or current source. This circuit is also known as RC filter or RC network. The simplest form of RC circuit includes one resistor and one capacitor.
Dimensional formula is an expression to denote the physical quantities in terms of certain fundamental quantities. There are 7 fundamental quantities of measurement. These fundamental quantities include Mass, Length, Time, Current, Temperature, Amount of substance or moles, and Luminous intensity.
The formula for Resistance is :
$R = \dfrac{V}{I}$
Dimensions of V: \[M{\text{ }}{L^2}\;{T^ - }^3\;{I^ - }^1\]
Therefore the dimension of resistance can be given as:
${{\text{M}}^{\text{1}}}{{\text{L}}^{\text{2}}}{{\text{T}}^{{\text{ - 3}}}}{{\text{I}}^{{\text{ - 2}}}}$ …Eq1
Similarly, the formula for Capacitance is
$C = \dfrac{Q}{V}$
Dimension of Q: ${M^0}{L^0}{T^{ - 1}}{I^1}$
Dimensions of V: \[M{\text{ }}{L^2}\;{T^ - }^3\;{I^ - }^1\]
Therefore the dimension of Capacitance can be given as:
${{\text{M}}^{{\text{ - 1}}}}{{\text{L}}^{{\text{ - 2}}}}{{\text{T}}^4}{{\text{I}}^{\text{2}}}$ …Eq2

Multiplying the dimensions of R and C from Eq1 and Eq2:
$
  {{\text{M}}^{\text{1}}}{{\text{L}}^{\text{2}}}{{\text{T}}^{{\text{ - 3}}}}{{\text{I}}^{{\text{ - 2}}}} \times {{\text{M}}^{{\text{ - 1}}}}{{\text{L}}^{{\text{ - 2}}}}{{\text{T}}^4}{{\text{I}}^{\text{2}}} \\
   = {{\text{M}}^0}{{\text{L}}^0}{{\text{T}}^1}{{\text{I}}^0} \\
   = {{\text{M}}^0}{{\text{L}}^0}{{\text{T}}^{\text{1}}} \\
$
Hence the dimension of RC is ${{\text{M}}^0}{{\text{L}}^0}{{\text{T}}^{\text{1}}}$.
The correct option is B.

Note- Dimensional analysis is an important concept that we should be aware of. Certain basic dimensions of quantities can be learnt. But they can also be derived at the moment. Example- Resistance is the ratio of voltage to current. If we know the dimensional formula of voltage, the dimensions of resistance can be derived.