Determining the Dimensional Formula for Pressure Multiplied by Volume
This solution explains how to find the dimensional formula for the product of two physical quantities: pressure and volume. We will break down the dimensions of each quantity and then combine them.
Understanding Dimensional Formulas
Dimensional formulas are used in physics to express a physical quantity in terms of its fundamental dimensions, typically Mass (M), Length (L), and Time (T). They help us check the consistency of physical equations and understand the relationships between different quantities.
Dimensions of Pressure
Pressure ($P$) is defined as force ($F$) acting per unit area ($A$). The formula for pressure is: $P = \frac{F}{A}$ We know the dimensions of force are $[F] = [M L T^{-2}]$ (Mass × Acceleration). The dimensions of area are $[A] = [L^2]$. Therefore, the dimensions of pressure are: $[P] = \frac{[M L T^{-2}]}{[L^2]} = [M L^{1-2} T^{-2}] = [M L^{-1} T^{-2}]$
Dimensions of Volume
Volume ($V$) is the amount of space occupied by an object. For a cube or rectangular prism, it's length × width × height. The dimensions of volume are: $[V] = [L \times L \times L] = [L^3]$
Calculating the Product of Pressure and Volume Dimensions
The question asks for the dimensional formula of the product of pressure and volume ($P \times V$). To find this, we multiply the individual dimensional formulas:
Dimensional Formula of ($P \times V$) = [Dimensions of P] $\times$ [Dimensions of V]
Substituting the dimensions we found:
$[P \times V] = [M L^{-1} T^{-2}] \times [L^3]$Now, we combine the powers of L:
$[P \times V] = [M L^{-1+3} T^{-2}]$ $[P \times V] = [M L^2 T^{-2}]$Interpreting the Result
The resulting dimensional formula, $[M L^2 T^{-2}]$, represents quantities that have dimensions of energy or work. For example, Work = Force × Displacement, and its dimensions are $[M L T^{-2}] \times [L] = [M L^2 T^{-2}]$. Energy also has the same dimensions.
Identifying the Correct Option
Comparing our calculated dimensional formula $[M L^2 T^{-2}]$ with the given options:
- Option 1: $ML^2T^{-2}$
- Option 2: $ML^{-1}T^{-3}$
- Option 3: $MLT^3$
- Option 4: $ML^{-1}T^{-2}$
Our calculated dimensions match Option 1.
Important Questions from Dimensional formulae and dimensional equations
- Considering the Lorentz force $\vec{F} = q(\vec{v} \times \vec{B})$, where $F$ is force, $q$ is electric charge, and $v$ is velocity, what is the dimensional formula for magnetic flux density $B$?
Given that the energy stored in an inductor is expressed as $U_L = \frac{1}{2}LI^2$ and the power dissipated in a resistor is $P = I^2R$, where $L$ is inductance, $R$ is resistance, and $I$ is current, determine the dimension of the ratio $\frac{L}{R}$.
The dimensions of energy are:
If force $[F]$, acceleration $[A]$ and time $[T]$ are chosen as the fundamental physical quantities. Find the dimensions of pressure.
- The characteristic impedance of free space, $Z_0$, is given by the expression $Z_0 = \sqrt{\frac{\mu_0}{\epsilon_0}}$. If $\mu_0$ represents the magnetic permeability and $\epsilon_0$ represents the electric permittivity, what are the dimensions of $Z_0$?
