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Step-by-Step Solution
Step 1: Understand the Problem
We are given five physical quantities and asked to determine which two have the same dimensions. The quantities are:
Energy density
Refractive index
Dielectric constant
Young's modulus
Magnetic field
The correct answer states that Energy Density (1) and Young's Modulus (4) share the same dimensions. We will verify this by performing dimensional analysis.
Step 2: Recall Dimension Basics
In dimensional analysis, the fundamental dimensions are typically expressed as:
$M$ for mass
$L$ for length
$T$ for time
We will use these to express each quantity.
Step 3: Dimension of Energy Density
Energy density is defined as energy per unit volume. Symbolically,
Energy density = $ \frac{\text{Energy}}{\text{Volume}} $
• The dimension of Energy (e.g., joule) is $ML^2T^{-2}$.
• The dimension of Volume is $L^3$.
Thus, the dimension of energy density is:
$$
\frac{ML^2T^{-2}}{L^3} \;=\; M\,L^{-1}\,T^{-2}.
$$
Step 4: Dimension of Young's Modulus
Young's modulus (Y) is defined as the ratio of stress to strain:
$$
Y = \frac{\text{Stress}}{\text{Strain}}.
$$
• Stress has the same dimensions as pressure: $ \frac{\text{Force}}{\text{Area}} \Rightarrow \frac{MLT^{-2}}{L^2} = M\,L^{-1}\,T^{-2}$.
• Strain is dimensionless (ratio of lengths).
Therefore, the dimension of Young's modulus (Y) is:
$$
M\,L^{-1}\,T^{-2}.
$$
Step 5: Conclusion on Matching Dimensions
Energy density and Young's modulus both have the dimension $M\,L^{-1}\,T^{-2}$. Hence, they share the same dimensions, confirming that the correct pair is “1 and 4”.
Answer: 1 and 4.
