Which of the following five physical parameters have the same dimensions? 1. energy density 2. refractive index 3. dielectric constant 4. Young's modulus 5. magnetic field

To determine which physical parameters share the same dimensions, we can derive the dimensional formula for each parameter step-by-step.

1. Energy density:
Energy density is defined as the energy stored per unit volume:
$\text{Energy density}=\frac{\text{Energy}}{\text{Volume}}$
The dimensions of energy are the same as work, which is force times distance:
$\left[\text{Energy}\right]=\left[{\text{ML}}^2{\text{T}}^{-2}\right]$
The dimensions of volume are:
$\left[\text{Volume}\right]=\left[{\text{L}}^3\right]$
Dividing the two, we get the dimensions of energy density:
$\left[\text{Energy density}\right]=\frac{\left[{\text{ML}}^2{\text{T}}^{-2}\right]}{\left[{\text{L}}^3\right]}=\left[{\text{ML}}^{-1}{\text{T}}^{-2}\right]$

2. Refractive index:
Refractive index is the ratio of the speed of light in a vacuum to the speed of light in a medium:
$\text{Refractive index}=\frac{\text{Speed of light in vacuum}}{\text{Speed of light in medium}}$
Since it is the ratio of two similar physical quantities (speeds), it is dimensionless:
$\left[\text{Refractive index}\right]=\left[{\text{M}}^0{\text{L}}^0{\text{T}}^0\right]$

3. Dielectric constant:
Dielectric constant is the ratio of the permittivity of a substance to the permittivity of free space:
$\text{Dielectric constant}=\frac{{\epsilon }_{\text{medium}}}{{\epsilon }_0}$
Being a ratio of two identical physical quantities, it is also dimensionless:
$\left[\text{Dielectric constant}\right]=\left[{\text{M}}^0{\text{L}}^0{\text{T}}^0\right]$

4. Young's modulus:
Young's modulus is the ratio of tensile stress to tensile strain:
$\text{Young's modulus}=\frac{\text{Stress}}{\text{Strain}}$
Strain is a ratio of change in length to original length, making it dimensionless:
$\left[\text{Strain}\right]=\left[{\text{M}}^0{\text{L}}^0{\text{T}}^0\right]$
Stress is defined as force per unit area:
$\left[\text{Stress}\right]=\frac{\left[\text{Force}\right]}{\left[\text{Area}\right]}=\frac{\left[{\text{MLT}}^{-2}\right]}{\left[{\text{L}}^2\right]}=\left[{\text{ML}}^{-1}{\text{T}}^{-2}\right]$
Thus, the dimensional formula for Young's modulus is:
$\left[\text{Young's modulus}\right]=\left[{\text{ML}}^{-1}{\text{T}}^{-2}\right]$

5. Magnetic field:
The force $F$ acting on a charge $q$ moving with velocity $v$ in a magnetic field $B$ is given by:
$F=qvB\Rightarrow B=\frac{F}{qv}$
The dimensions of charge $q$ are $\left[\text{AT}\right]$ (where $\text{A}$ represents electric current). Therefore:
$\left[\text{Magnetic field}\right]=\frac{\left[{\text{MLT}}^{-2}\right]}{\left[\text{AT}\right]·\left[{\text{LT}}^{-1}\right]}=\left[{\text{MT}}^{-2}{\text{A}}^{-1}\right]$

Comparing all the parameters, we see that energy density (1) and Young's modulus (4) share the same dimensional formula, which is $\left[{\text{ML}}^{-1}{\text{T}}^{-2}\right]$.

Therefore, the physical parameters with the same dimensions are 1 and 4.