JEE PYQ: Units & Measurements - Question ID 207577dfc8e8 (JEE Main 2022)

In dimensional analysis, every physical quantity is expressed in terms of the fundamental dimensions: mass ($M$), length ($L$), time ($T$), electric current ($I$), thermodynamic temperature ($\Theta$), amount of substance ($N$), and luminous intensity ($J$). The dimensions of derived quantities are obtained by combining these fundamental dimensions using the defining equations.

Key formulas and their dimensions:

• Wave number ($k$): Defined as $k = \frac{2\pi}{\lambda}$, where $\lambda$ is wavelength. Since wavelength has dimension $L$, wave number has dimension $L^{-1}$.
• Rydberg’s constant ($R$): Appears in the formula for atomic spectra: $\frac{1}{\lambda} = R \left(\frac{1}{n_1^2} - \frac{1}{n_2^2}\right)$. Since $\frac{1}{\lambda}$ has dimension $L^{-1}$, Rydberg’s constant also has dimension $L^{-1}$.
• Stress ($\sigma$): Defined as force per unit area: $\sigma = \frac{F}{A}$. Force has dimension $MLT^{-2}$, area has dimension $L^2$, so stress has dimension $ML^{-1}T^{-2}$.
• Coefficient of elasticity ($E$ or $Y$): Defined as stress per unit strain: $E = \frac{\sigma}{\epsilon}$. Strain ($\epsilon$) is dimensionless, so the coefficient of elasticity has the same dimension as stress: $ML^{-1}T^{-2}$.
• Coercivity ($H_c$): Defined as the magnetic field strength required to reduce magnetization to zero. Magnetic field strength ($H$) has dimension $L^{-1}I$.
• Magnetisation ($M$): Defined as magnetic moment per unit volume. Magnetic moment has dimension $L^2I$, volume has dimension $L^3$, so magnetization has dimension $L^{-1}I$.
• Specific heat capacity ($c$): Defined as heat energy per unit mass per unit temperature: $c = \frac{Q}{m \Delta T}$. Heat energy ($Q$) has dimension $ML^2T^{-2}$, mass ($m$) has dimension $M$, temperature ($\Delta T$) has dimension $\Theta$, so specific heat capacity has dimension $L^2T^{-2}\Theta^{-1}$.
• Latent heat ($L$): Defined as heat energy per unit mass: $L = \frac{Q}{m}$. Heat energy has dimension $ML^2T^{-2}$, mass has dimension $M$, so latent heat has dimension $L^2T^{-2}$.

We analyze the dimensions of each pair in the options:

1. Option A: Wave number and Rydberg’s constant
   Wave number: $L^{-1}$
   Rydberg’s constant: $L^{-1}$
   Both have the same dimension. Not the correct answer.
2. Option B: Stress and Coefficient of elasticity
   Stress: $ML^{-1}T^{-2}$
   Coefficient of elasticity: $ML^{-1}T^{-2}$
   Both have the same dimension. Not the correct answer.
3. Option C: Coercivity and Magnetisation
   Coercivity: $L^{-1}I$
   Magnetisation: $L^{-1}I$
   Both have the same dimension. Not the correct answer.
4. Option D: Specific heat capacity and Latent heat
   Specific heat capacity: $L^2T^{-2}\Theta^{-1}$
   Latent heat: $L^2T^{-2}$
   Specific heat capacity has an additional dimension of $\Theta^{-1}$ compared to latent heat. These have different dimensions.

Thus, the pair with different dimensions is Option D.

Students often confuse specific heat capacity and latent heat because both involve heat energy and mass. However, specific heat capacity includes temperature dependence, introducing the dimension of temperature ($\Theta$), while latent heat does not. This distinction is crucial for dimensional analysis.

Another common mistake is assuming that all constants related to waves (like wave number and Rydberg’s constant) might have different dimensions. In reality, both are inverse lengths and share the same dimension.

Always verify dimensions using the defining equations rather than relying on intuition or memorized units.