Question

The dimensions of $\dfrac{L}{{RCV}}$​ (\[L = \] inductance, \[R = \]resistance, \[C = \]capacitance, \[V = \] potential difference) are
A) \[A\]
B) \[{A^{ - 1}}\]
C) \[AT\]
D) dimensionless

Answer 1

Hint: The dimensions of a physical quantity are the powers for which the fundamental units are raised to obtain a unit of that quantity. We can find the dimensional formula by substituting the dimensional formulas of inductance, capacitance and potential difference.

Complete step by step answer:
We know that:
\[V = IR\]
And \[L \times \dfrac{{di}}{{dt}}\]
And \[IR = \dfrac{{L \times I}}{t}\]
So, \[\dfrac{L}{R} = t\]
Then, \[L = \dfrac{{Vt}}{I}\]
Also, we know for capacitance \[C\]
\[
  I = C\dfrac{{dv}}{{dt}}
   \Rightarrow It = CV
   \Rightarrow It = AT
 \]
So,
\[
  \dfrac{L}{{RCV}} = \dfrac{T}{{AT}}
   \Rightarrow \dfrac{L}{{RCV}} = {A^{ - 1}}
 \]

Hence, the correct answer is option (B).

Additional Information:
Dimensional analysis is the practice of examining the relationship between physical quantities by identifying the dimensions of physical quantities. These dimensions are independent of numerical multiples and constants and all quantities in the world can be expressed as a function of fundamental dimensions.

An expression representing the power to lift the basic units to obtain a unit of a quantity is called the dimensional formula of that quantity.
If Q is the unit of derived quantity represented by \[Q = {M^a}{L^b}{T^c}\], then \[{M^a}{L^b}{T^c}\] is called the dimensional formula and the exponents a, b and, c are called dimensions.
The dimensions of a physical quantity are the powers for which the fundamental units are raised to obtain a unit of that quantity.

Note:
a) Capacitance: The amount of charge that is placed inside a capacitor at a given voltage is called capacitance. It is charged when the charges are forced into the positive (or upper) plate of the capacitor due to the EMF. Similarly, it is discharged when the chargers are forcefully pulled from the capacitor. Capacitance is measured in Farad.

b) Inductance: It is the property of a coil to resist any change in electric current. When a coil opposes the current change through itself, self-inductance occurs. Mutual induction occurs when a primary coil opposes a current change in a primary coil. Inductance is measured in Henry


c) Resistance: It is the property of a material that opposes the flow of electric current through it. The resistance of conductors is low while the resistance of insulators is very high. It is measured in ohms.