JEE PYQ: Units & Measurements - Question ID cf6e4a1936ad (JEE Main 2023)

Core Formula & Concept:

In dimensional analysis, every physical quantity is expressed in terms of the fundamental dimensions: mass ($\mathrm{M}$), length ($\mathrm{L}$), time ($\mathrm{T}$), electric current ($\mathrm{A}$), etc. The key formulas and concepts used here are:

• Pressure gradient: Pressure ($P$) has dimensions $\left[\mathrm{M}^{1} \mathrm{L}^{-1} \mathrm{T}^{-2}\right]$. A gradient is the rate of change with respect to distance, so dividing by length ($\mathrm{L}$) gives the dimensions of pressure gradient.
• Energy density: Energy ($E$) has dimensions $\left[\mathrm{M}^{1} \mathrm{L}^{2} \mathrm{T}^{-2}\right]$. Density implies division by volume ($\mathrm{L}^3$), so energy density has dimensions $\left[\mathrm{M}^{1} \mathrm{L}^{-1} \mathrm{T}^{-2}\right]$.
• Electric field ($\vec{E}$): Defined as force per unit charge, where force is $\left[\mathrm{M}^{1} \mathrm{L}^{1} \mathrm{T}^{-2}\right]$ and charge is $\left[\mathrm{A}^{1} \mathrm{T}^{1}\right]$. Thus, $\vec{E} = \frac{F}{q}$ has dimensions $\left[\mathrm{M}^{1} \mathrm{L}^{1} \mathrm{T}^{-3} \mathrm{A}^{-1}\right]$.
• Latent heat ($L$): Heat energy per unit mass, where energy is $\left[\mathrm{M}^{1} \mathrm{L}^{2} \mathrm{T}^{-2}\right]$ and mass is $\left[\mathrm{M}^{1}\right]$. Thus, latent heat has dimensions $\left[\mathrm{M}^{0} \mathrm{L}^{2} \mathrm{T}^{-2}\right]$.

Step-by-Step Derivation:

We derive the dimensional formula for each physical quantity in List I and match it with List II.

1. Pressure gradient (A):

   Pressure ($P$) has dimensions: $[P] = \left[\mathrm{M}^{1} \mathrm{L}^{-1} \mathrm{T}^{-2}\right]$. A gradient is the derivative of pressure with respect to distance ($x$), so: $\text{Pressure gradient} = \frac{dP}{dx}$. Dimensions: $\frac{\left[\mathrm{M}^{1} \mathrm{L}^{-1} \mathrm{T}^{-2}\right]}{\left[\mathrm{L}^{1}\right]} = \left[\mathrm{M}^{1} \mathrm{L}^{-2} \mathrm{T}^{-2}\right]$. This matches III in List II.

2. Energy density (B):

   Energy ($E$) has dimensions: $[E] = \left[\mathrm{M}^{1} \mathrm{L}^{2} \mathrm{T}^{-2}\right]$. Density implies division by volume ($V$), where $[V] = \left[\mathrm{L}^{3}\right]$. Thus, energy density has dimensions: $\frac{\left[\mathrm{M}^{1} \mathrm{L}^{2} \mathrm{T}^{-2}\right]}{\left[\mathrm{L}^{3}\right]} = \left[\mathrm{M}^{1} \mathrm{L}^{-1} \mathrm{T}^{-2}\right]$. This matches II in List II.

3. Electric Field (C):

   Electric field ($\vec{E}$) is force per unit charge. Force ($F$) has dimensions: $[F] = \left[\mathrm{M}^{1} \mathrm{L}^{1} \mathrm{T}^{-2}\right]$. Charge ($q$) has dimensions: $[q] = \left[\mathrm{A}^{1} \mathrm{T}^{1}\right]$. Thus, electric field has dimensions: $\frac{\left[\mathrm{M}^{1} \mathrm{L}^{1} \mathrm{T}^{-2}\right]}{\left[\mathrm{A}^{1} \mathrm{T}^{1}\right]} = \left[\mathrm{M}^{1} \mathrm{L}^{1} \mathrm{T}^{-3} \mathrm{A}^{-1}\right]$. This matches IV in List II.

4. Latent heat (D):

   Latent heat ($L$) is heat energy per unit mass. Heat energy has the same dimensions as energy: $[E] = \left[\mathrm{M}^{1} \mathrm{L}^{2} \mathrm{T}^{-2}\right]$. Mass ($m$) has dimensions: $[m] = \left[\mathrm{M}^{1}\right]$. Thus, latent heat has dimensions: $\frac{\left[\mathrm{M}^{1} \mathrm{L}^{2} \mathrm{T}^{-2}\right]}{\left[\mathrm{M}^{1}\right]} = \left[\mathrm{M}^{0} \mathrm{L}^{2} \mathrm{T}^{-2}\right]$. This matches I in List II.

The correct matching is:

• A → III
• B → II
• C → IV
• D → I

This corresponds to Option A.

Common Traps & Exam Tip:

Students often make the following mistakes:

• Confusing pressure gradient with pressure: Pressure gradient is pressure per unit length, not just pressure. Forgetting to divide by $\mathrm{L}$ leads to incorrect matching.
• Misidentifying energy density: Some confuse energy density with energy, forgetting to divide by volume. This leads to matching with $\left[\mathrm{M}^{1} \mathrm{L}^{2} \mathrm{T}^{-2}\right]$ instead of $\left[\mathrm{M}^{1} \mathrm{L}^{-1} \mathrm{T}^{-2}\right]$.
• Electric field dimensions: Students may forget that charge has dimensions of $\left[\mathrm{A}^{1} \mathrm{T}^{1}\right]$, leading to incorrect simplification. For example, they might omit $\mathrm{A}$ or miscalculate the exponents of $\mathrm{T}$.
• Latent heat vs. specific heat capacity: Latent heat is energy per unit mass, while specific heat capacity is energy per unit mass per unit temperature. Confusing the two leads to incorrect dimensions.

Exam Tip: Always write down the fundamental dimensions of each quantity involved (e.g., force, charge, volume) before deriving the final formula. This minimizes errors in dimensional analysis.