To determine the time taken by light to travel through a glass plate, we need to first understand how the speed of light changes when it passes from a vacuum (or air, approximately) into a medium like glass. This change in speed is characterized by the refractive index of the medium.

Calculating Light Travel Time in Glass

We are given the following information:

• Thickness of the glass plate, \(d = 1 \text{ mm}\).
• Refractive index of the glass, \(n = 1.5\).

We need to find the time \(t\) it takes for light to travel through this thickness normally (perpendicularly). The speed of light in a vacuum is a constant, approximately \(c = 3 \times 10^8 \text{ m/s}\).

Understanding Speed of Light in a Medium

The refractive index \(n\) of a medium is defined as the ratio of the speed of light in a vacuum (\(c\)) to the speed of light in the medium (\(v\)).

The formula for the speed of light in a medium is:

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

Let's calculate the speed of light in the glass plate using the given refractive index \(n = 1.5\) and the speed of light in vacuum \(c = 3 \times 10^8 \text{ m/s}\).

\(v = \frac{3 \times 10^8 \text{ m/s}}{1.5}\)

\(v = 2 \times 10^8 \text{ m/s}\)

So, the speed of light within the glass plate is \(2 \times 10^8 \text{ m/s}\).

Calculating Time Taken by Light

The time taken to travel a certain distance at a constant speed is given by the formula:

\(\text{Time} = \frac{\text{Distance}}{\text{Speed}}\)

In this case, the distance is the thickness of the glass plate, \(d\), and the speed is the speed of light in the glass, \(v\).

\(t = \frac{d}{v}\)

First, we need to convert the thickness from millimeters (mm) to meters (m):

\(d = 1 \text{ mm} = 1 \times 10^{-3} \text{ m}\)

Now, substitute the values of \(d\) and \(v\) into the formula for time:

\(t = \frac{1 \times 10^{-3} \text{ m}}{2 \times 10^8 \text{ m/s}}\)

\(t = \frac{1}{2} \times 10^{-3} \times 10^{-8} \text{ s}\)

\(t = 0.5 \times 10^{-11} \text{ s}\)

To express this in a standard scientific notation form, we can write \(0.5\) as \(5 \times 10^{-1}\):

\(t = (5 \times 10^{-1}) \times 10^{-11} \text{ s}\)

\(t = 5 \times 10^{(-1 - 11)} \text{ s}\)

\(t = 5 \times 10^{-12} \text{ s}\)

Thus, the time taken by light to travel through the glass plate is \(5 \times 10^{-12} \text{ s}\).

Analyzing the Options

Let's compare our calculated time with the given options:

• Option 1: \(0.1 \times 10^{-13} \text{ s} = 1 \times 10^{-14} \text{ s}\) - This is much smaller than our result.
• Option 2: \(12 \times 10^{-5} \text{ s}\) - This is much larger than our result.
• Option 3: \(5 \times 10^{-12} \text{ s}\) - This matches our calculated time.
• Option 4: \(1.5 \times 10^{-5} \text{ s}\) - This is much larger than our result.

Our calculation correctly gives \(5 \times 10^{-12} \text{ s}\).

Additional Information on Refractive Index and Light

The refractive index is a dimensionless quantity that tells us how much a medium slows down light. A higher refractive index means light travels slower in that medium. The refractive index is also responsible for phenomena like refraction (bending of light) when it passes from one medium to another.

Different wavelengths of light can have slightly different refractive indices in the same material; this effect is called dispersion and is why prisms can split white light into a spectrum.

In vacuum, the speed of light \(c\) is the maximum speed at which all conventional matter, energy, and information can travel. No object with rest mass can reach the speed of light, and light itself travels at this speed only in vacuum.

When light enters a material medium like glass or water, it interacts with the atoms and molecules of the medium, causing it to effectively slow down. The higher the density or optical density of the medium, generally the higher its refractive index and the slower the light travels.