Theoretical Foundation: Refractive Index and Dispersion

The interaction between light and a dielectric medium, such as glass, is governed by the frequency-dependent nature of the material's permittivity. The speed of light $v$ in a medium is defined by the relation:

$v = \frac{c}{n(\lambda)}$

where $c$ is the speed of light in a vacuum ($3 \times 10^8 \text{ m/s}$) and $n(\lambda)$ is the refractive index of the medium, which is a function of the wavelength $\lambda$ of the incident light [Per Cauchy's Dispersion Formula: $n(\lambda) = A + \frac{B}{\lambda^2} + \dots$].

Step 1: Addressing Independence of Speed

The speed of light in glass is not independent of the color of light. Because the refractive index $n$ of glass varies with the wavelength (frequency) of the incident radiation, different colors experience different phase velocities within the medium. This phenomenon is known as chromatic dispersion.

Step 2: Comparative Analysis of Red and Violet Light

To determine which color travels slower, we must evaluate the relationship between wavelength and refractive index:

• Wavelengths: The wavelength of violet light ($\lambda_v \approx 400 \text{ nm}$) is shorter than the wavelength of red light ($\lambda_r \approx 700 \text{ nm}$).
• Cauchy's Relation: As $\lambda$ decreases, $\frac{1}{\lambda^2}$ increases, leading to a higher value for $n$. Consequently, $n_{violet} > n_{red}$.
• Velocity Relation: Since $v \propto \frac{1}{n}$, a higher refractive index results in a lower velocity.

Step 3: Visual Representation of Dispersion

Data Comparison Table

Final Solution: The speed of light in glass is dependent on the color of light. Violet light has a shorter wavelength, resulting in a higher refractive index for the glass medium, which causes violet light to travel slower than red light.

Step 1: Establishing Physical Parameters

To determine the speed of light in a medium, we utilize the relationship between the refractive index of the medium, the speed of light in a vacuum, and the speed of light in the specified medium.

• Refractive index of glass ($\mu$ or $n$): $1.5$
• Speed of light in vacuum ($c$): $3.0 \times 10^8 \, \text{m s}^{-1}$
• Speed of light in glass ($v$): Unknown

Step 2: Defining the Governing Relation

By definition, the absolute refractive index of a medium is the ratio of the speed of light in a vacuum ($c$) to the speed of light in the medium ($v$). [Per the Definition of Refractive Index]:

$n = \frac{c}{v}$

Step 3: Algebraic Rearrangement and Calculation

We rearrange the equation to solve for the target variable $v$:

$v = \frac{c}{n}$

Substituting the given numerical values:

$v = \frac{3.0 \times 10^8 \, \text{m s}^{-1}}{1.5}$

Perform the division:

$v = 2.0 \times 10^8 \, \text{m s}^{-1}$

Visual Representation of Wave Refraction

Summary of Analytical Results

The calculation confirms that as light enters a medium with a higher refractive index ($n > 1$), its velocity decreases proportionately to the index of the material. Given $n=1.5$, the velocity is reduced by a factor of exactly 1.5.

Final Solution: The speed of light in glass is $2.0 \times 10^8 \, \text{m s}^{-1}$.

Theoretical Foundation: Wavefront Geometry

A wavefront is defined as the locus of all points having the same phase of oscillation. The shape of a wavefront is determined by the geometry of the source and the path traversed by the light rays, which propagate perpendicularly to the wavefront [Per Huygens-Fresnel Principle].

Step 1: Identifying the Source Geometry

The source is identified as a point source. A point source radiates light energy symmetrically in all directions in a homogeneous, isotropic medium.

Step 2: Defining the Path of Rays

Let the point source be located at the origin $O(0,0,0)$ in a three-dimensional coordinate system. Light rays travel radially outward from this source along all possible directions in space. Consequently, the distance $r$ from the source at any time $t$ for a given wavefront is $r = ct$, where $c$ is the speed of light.

Step 3: Geometrical Analysis

Since the rays travel an equal distance $r$ from the point source $O$ in all directions, the set of all points $(x, y, z)$ having the same phase satisfies the equation of a sphere:

$x^2 + y^2 + z^2 = r^2$

[By the definition of a sphere in 3D Euclidean geometry, where the locus of points is equidistant from a fixed center].

Analytical Conclusion

Because the rays are radial vectors diverging from the point source, the wavefront, being a surface perpendicular to these rays at all points, must necessarily be spherical.

Final Solution: The shape of the wavefront for light diverging from a point source is spherical.