Q.The dimension of $\dfrac{1}{\mu_0\epsilon_0}$ is

Both statements are true, and the formula in R is exactly why speed is wavelength-independent.

Assertion is correct: in vacuum every electromagnetic wave, regardless of wavelength, travels at the same speed $c$ (vacuum is non-dispersive for EM waves). Reason is also correct: $c=\dfrac{1}{\sqrt{\mu_0\epsilon_0}}$, and since $\mu_0$ and $\epsilon_0$ are constants of free space (not functions of wavelength or frequency), this formula directly explains why $c$ is the same for all wavelengths. So R is the correct explanation of A.

The product $\mu\varepsilon$ has dimensions of $[T^2 L^{-2}]$, so $(\mu\varepsilon)^{-1}$ has dimensions of velocity squared: $[L^2 T^{-2}]$. The answer is (B).

Maxwell showed that combining the wave equations for E and B derived from his equations in free space gives a wave speed c = 1/sqrt(mu0 epsilon0), which matched the known speed of light and led him to conclude light itself is an electromagnetic wave.

From Maxwell's equations in vacuum, the electric and magnetic fields each satisfy a wave equation with speed:

c = 1 / sqrt(mu0 epsilon0)

Substituting mu0 = 4-pi x 10^-7 T.m/A and epsilon0 = 8.85 x 10^-12 C^2/(N.m^2) gives c approx 3 x 10^8 m/s, matching the measured speed of light - historically the strongest evidence that light is an electromagnetic wave.

Since $1/\sqrt{\mu_0\varepsilon_0}=c$ (the speed of light), $1/(\mu_0\varepsilon_0)=c^2$, whose dimension is $[L^2T^{-2}]$.

Working

Maxwell's relation gives the speed of electromagnetic waves in vacuum as

$c = \dfrac{1}{\sqrt{\mu_0\varepsilon_0}}$

Squaring both sides:

$c^2 = \dfrac{1}{\mu_0\varepsilon_0}$

Since $c$ has dimension of velocity, $[c]=[LT^{-1}]$, so

$\left[\dfrac{1}{\mu_0\varepsilon_0}\right] = [c^2] = [L^2T^{-2}]$

1 light year = c × (1 year) ≈ 3×10⁸ m/s × 3.15×10⁷ s ≈ 9.46 × 10¹⁵ m.

A light year is the distance travelled by light (an electromagnetic wave) in vacuum in one year.

$1\ \text{ly} = c \times t = (3\times10^8\ \text{m/s}) \times (365.25 \times 24 \times 3600\ \text{s})$

$= (3\times10^8) \times (3.156\times10^7) \approx 9.46\times10^{15}\ \text{m}$

Maxwell's equations predict that electromagnetic waves travel in free space at a speed fixed by the permeability and permittivity of free space.

From Maxwell's electromagnetic wave equation, the speed of an electromagnetic wave in free space (vacuum) is

$c = \frac{1}{\sqrt{\mu_0 \varepsilon_0}}$

where $\mu_0$ is the permeability of free space and $\varepsilon_0$ is the permittivity of free space. Substituting their values gives $c \approx 3 \times 10^8\ \text{m/s}$, matching the measured speed of light.

Speed of light in a medium is $v=c/n$; water has $n>1$, so speed decreases relative to air.

The speed of an electromagnetic wave in a medium is $v = \dfrac{c}{n}$, where $n$ is the refractive index of that medium. Water has a refractive index greater than that of air ($n_{water}\approx1.33 > n_{air}\approx1$), so the speed of the electromagnetic wave in water is less than in air — the speed decreases as the wave moves from air into water.

Maxwell showed that the speed of light equals $1/\sqrt{\mu_0\varepsilon_0}$, one of the great unifications of physics.

From Maxwell's equations, an electromagnetic wave in free space travels with speed $c = 1/\sqrt{\mu_0\varepsilon_0}$, where μ₀ is the permeability of free space and ε₀ is the permittivity of free space. Substituting $\mu_0 = 4\pi \times 10^{-7} \text{ T m/A}$ and $\varepsilon_0 = 8.85 \times 10^{-12} \text{ C}^2 \text{N}^{-1}\text{m}^{-2}$ gives $c \approx 3 \times 10^8$ m/s, matching the measured speed of light — this agreement was the key evidence that light itself is an electromagnetic wave.

A light year is a unit of distance, not time.

Despite the word 'year', a light year is the distance travelled by light (speed c ≈ 3 × 10⁸ m/s) in one year:

$1\ \text{light year} = c \times (1\ \text{year}) \approx 9.46 \times 10^{15}\ \text{m}$

It is an astronomical unit of length used to express the vast distances between stars.

The speed of an EM wave in free space comes directly from Maxwell's equations and equals $c = 1/\sqrt{\varepsilon_0\mu_0}$.

Maxwell's equations predict electromagnetic waves travelling in vacuum with speed:

$v = \frac{1}{\sqrt{\varepsilon_0 \mu_0}}$

Substituting $\varepsilon_0 = 8.85\times10^{-12}\,\text{C}^2\text{N}^{-1}\text{m}^{-2}$ and $\mu_0 = 4\pi\times10^{-7}\,\text{Tm/A}$ gives $v \approx 3\times10^8$ m/s, matching the measured speed of light — this agreement is what led Maxwell to conclude light itself is an electromagnetic wave.

Gamma rays, X-rays and microwaves are all part of the same electromagnetic spectrum, and every electromagnetic wave travels at the speed of light c in vacuum, independent of its frequency.

Gamma rays, X-rays, and microwaves differ enormously in frequency and wavelength (gamma rays have the shortest wavelength/highest frequency, microwaves the longest wavelength/lowest frequency among these three), but they are all transverse electromagnetic waves consisting of oscillating electric and magnetic fields. A defining property of electromagnetic waves derived from Maxwell's equations is that in vacuum they all propagate at the same universal speed,

$c = \frac{1}{\sqrt{\mu_0\varepsilon_0}} \approx 3\times10^{8}\text{ m/s}$

This speed does not depend on frequency or wavelength in vacuum (that dependence only appears inside a dispersive medium). So $V_g = V_x = V_m = c$.