Animated Solution for Physics - Physics and Measurement: Match the physical quantities given in Column I with dimensions expressed in terms of mass ($M$), length ($L$), time ($T$), and charge ($Q$) given in Column II and write the correct answer against the matched quantity in a tabular form in your answer book.

Dimensional analysis is a powerful tool in physics. It allows us to verify equations, convert units, and even derive relationships between physical quantities. In this problem, we are tasked with finding the dimensional formulas for six different physical quantities. The twist here is that we must express them in terms of mass ($M$), length ($L$), time ($T$), and charge ($Q$), rather than the standard SI base units which use Ampere ($A$) for current.

Let's break down each quantity step by step.

Angular Momentum

Angular momentum ($L$) is the rotational analog of linear momentum. It is defined as the product of moment of inertia ($I$) and angular velocity ($\omega$).

The moment of inertia has the dimension of mass times distance squared, $[M{L}^2]$. Angular velocity is angle over time, so its dimension is $[T^{-1}]$. Multiplying these gives:

Latent Heat

Latent heat ($L$) is the heat energy required to change the state of a unit mass of a substance without changing its temperature.

Heat ($H$) is a form of energy, so its dimension is $[M{L}^2{T}^{-2}]$. Dividing this by the dimension of mass ($[M]$) gives:

Torque

Torque ($\tau$) is the turning effect of a force. It is calculated as the cross product of the position vector and the force vector.

The dimension of force is $[ML{T}^{-2}]$, and the dimension of distance is $[L]$. Therefore:

Notice that torque has the same dimensional formula as energy, although they are fundamentally different physical quantities.

Capacitance

Capacitance ($C$) is the ability of a system to store an electric charge. To find its dimension, we can use the formula for the energy stored in a capacitor:

Here, $q$ is the charge with dimension $[Q]$, and $U$ is the energy with dimension $[M{L}^2{T}^{-2}]$. Substituting these in:

Inductance

Inductance ($L$) is the property of an electrical conductor by which a change in current induces an electromotive force. We can find its dimension using the energy stored in an inductor:

Current ($i$) is the rate of flow of charge, so its dimension is $[Q{T}^{-1}]$. Substituting the dimensions of energy and current:

Resistivity

Resistivity ($\rho$) is a fundamental property of a material that quantifies how strongly it resists electric current. It is related to resistance ($R$) by the formula:

To find the dimension of resistance, we can use Joule's law of heating:

Substituting this into the resistivity formula:

Now, we plug in the dimensions: heat $[M{L}^2{T}^{-2}]$, area $[L^2]$, current $[Q{T}^{-1}]$, time $[T]$, and length $[L]$.

By systematically applying fundamental formulas, we have successfully derived the dimensional formulas for all six quantities in terms of $M$, $L$, $T$, and $Q$. This exercise highlights the interconnectedness of physical laws and the elegance of dimensional analysis.