Which two of the following five physical parameters have the same dimension?
(1) Energy density
(2) refractive index
(3) dielectric constant
(4) Young's modulus
(5) magnitic field

Options

A. (ii) and (iv)
B. (iii) and (v)
C. (i) and (iv)
D. (i) and (v)

Explanation

To determine which two of the five physical parameters have the same dimension, we will analyze each parameter one by one.

1. **Energy Density (U)**:
- Energy density is defined as energy per unit volume.
- The dimension of energy (E) is given by \( [E] = ML^2T^{-2} \).
- Therefore, the dimension of energy density is:
\[ [U] = \frac{[E]}{[L^3]} = \frac{ML^2T^{-2}}{L^3} = ML^{-1}T^{-2} \]

2. **Refractive Index (n)**:
- The refractive index is defined as the ratio of the speed of light in vacuum to the speed of light in a medium.
- It is a dimensionless quantity, so:
\[ [n] = 1 \]

3. **Dielectric Constant (ε)**:
- The dielectric constant is also defined as the ratio of the capacitance of a capacitor with a dielectric to the capacitance without it.
- Like the refractive index, it is dimensionless, so:
\[ [ε] = 1 \]

4. **Young's Modulus (Y)**:
- Young's modulus is defined as the ratio of stress to strain.
- The dimension of stress (force per unit area) is:
\[ [\text{Stress}] = \frac{[F]}{[A]} = \frac{MLT^{-2}}{L^2} = ML^{-1}T^{-2} \]
- Strain is dimensionless, so:
\[ [Y] = \frac{[\text{Stress}]}{[\text{Strain}]} = ML^{-1}T^{-2} \]

5. **Magnetic Field (B)**:
- The magnetic field can be defined in terms of force per unit charge per unit velocity.
- The dimension of the magnetic field is:
\[ [B] = \frac{[F]}{[q][v]} = \frac{MLT^{-2}}{IT} = MA^{-1}T^{-2} \]

Now, we summarize the dimensions:
- Energy Density: \( ML^{-1}T^{-2} \)
- Refractive Index: \( 1 \) (dimensionless)
- Dielectric Constant: \( 1 \) (dimensionless)
- Young's Modulus: \( ML^{-1}T^{-2} \)
- Magnetic Field: \( MA^{-1}T^{-2} \)

From the above analysis, we can see that:
- Energy Density and Young's Modulus both have the dimension \( ML^{-1}T^{-2} \).

Thus, the two parameters with the same dimension are:
**Energy Density (1)** and **Young's Modulus (4)**.

### Final Answer:
The two parameters with the same dimension are (1) Energy Density and (4) Young's Modulus.

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