Question

Let Q denote the charge on the plate of a capacitor of capacitance C. Find the dimensional formula for \[\dfrac{{{Q^2}}}{C}\]
A. \[\left[ {{L^2}{M^2}T} \right]\]
B. \[\left[ {LM{T^2}} \right]\]
C. \[\left[ {{L^2}M{T^{ - 2}}} \right]\]
D. \[\left[ {{L^2}{M^2}{T^2}} \right]\]

Answer 1

Hint: An equation, which gives the relation between the fundamental units and derived units in terms of dimensions is known as dimensional formula or equation. In mechanics, the three base dimensions are length, mass, and time and are represented by the letters L, M, and T respectively.

Complete step by step solution:
Here, they have denoted Q as the charge on the plate of a capacitor of capacitance C. We need to find the dimensional formula for \[\dfrac{{{Q^2}}}{C}\]. For that, first, let us see what the term\[\dfrac{{{Q^2}}}{C}\]represents. As we know, the term \[\dfrac{{{Q^2}}}{C}\] is nothing but the energy stored in the capacitor, which can also be called work done.

The formula for the work done is,
\[W = F \times d\]
\[\Rightarrow W = m \times a \times d\]
Now, if we write the dimensional formula for mass, acceleration, and displacement then, it will be,
\[\left[ {ML{T^{ - 2}} \times L} \right]\]
\[\therefore \left[ {M{L^2}{T^{ - 2}}} \right]\]
Therefore, the dimensional formula for \[\dfrac{{{Q^2}}}{C}\] is \[\left[ {M{L^2}{T^{ - 2}}} \right]\].

Hence, Option C is the correct answer

Additional informationDimensionless quantities are those which do not have any dimensions but have a fixed value. These dimensional quantities are of two types;
Dimensionless quantities without units: Pure numbers and the trigonometric functions are the dimensionless quantities without units.

Dimensionless quantities with units: Angular displacement, Joule’s constant, etc are the quantities having units.

Dimensional variables: These are the physical quantities that have dimensions and do not have a fixed value. For example, acceleration, work, force, etc.

Dimensionless variables: These are the physical quantities that do not have dimensions and do not have a fixed value. For example, refractive index, the Poisson’s ratio, coefficient of friction, etc.

Note: Here before we go to solve this problem it is important to remember the dimensional formula of force and displacement and then we will be able to solve this problem as shown in the solution.