Which two of the following five physical parameters have the same dimensions? [PMT/NEET-2008]
1. energy density
2. refractive index
3. dielectric constant
4. Young’s modulus
5. magnetic field

a. 1 and 4
b. 1 and 5
c. 2 and 3
d. 2 and 4

Answer: a. 1 and 4

Explanation:
To solve this, we need to find the dimensions of each physical parameter and identify which pair has matching dimensions.

Dimensional Analysis

• Energy density = Energy per unit volume

• Energy: $ [ML^{2}T^{-2}] $
• Volume: $ [L^{3}] $
• Energy density: $ \frac{[ML^{2}T^{-2}]}{[L^{3}]} = [ML^{-1}T^{-2}] $

• Refractive index = Ratio of speed of light in vacuum to speed in medium

• Both speeds have dimensions $ [LT^{-1}] $
• Refractive index: $ \frac{[LT^{-1}]}{[LT^{-1}]} = \text{dimensionless} $

• Dielectric constant = Ratio of material’s permittivity to free space permittivity

• Since it’s a ratio of quantities with identical dimensions
• Dielectric constant: $ \text{dimensionless} $

• Young’s modulus = Stress ÷ Strain

• Stress = Force/Area = $ \frac{[MLT^{-2}]}{[L^{2}]} = [ML^{-1}T^{-2}] $
• Strain is dimensionless (ratio of lengths)
• Young’s modulus: $ [ML^{-1}T^{-2}] $

• Magnetic field = From $ F = BIl $

• Rearranging: $ B = \frac{F}{Il} = \frac{[MLT^{-2}]}{[I][L]} = [MT^{-2}I^{-1}] $

Comparison

• Energy density: $ [ML^{-1}T^{-2}] $
• Young’s modulus: $ [ML^{-1}T^{-2}] $

Both energy density and Young’s modulus have identical dimensions $ [ML^{-1}T^{-2}] $, which represents pressure or stress-like quantities. This makes physical sense since energy density can be thought of as energy stored per unit volume under stress.