Question

What is the time taken in seconds to cross a glass plate of thickness \(6 \mathrm{~mm}\) and \(\mu=2.0\) by light ? (A) \(8 \times 10^{-11}\) (B) \(4 \times 10^{-11}\) (C) \(2 \times 10^{11}\) (D) \(16 \times 10^{-11}\)

Short answer

The time taken by light to cross the glass plate is \(4 \times 10^{-11}\, \text{s}\).

Step-by-step solution

Convert the thickness of the glass plate into meters

We are given that the thickness of the glass plate is \(6\, \text{mm}\). We need to convert this into meters. Since 1 meter is equal to 1000 millimeters, we have : \[d = 6\, \text{mm} \times \frac{1\, \text{m}}{1000\, \text{mm}}\] \[d = 0.006\, \text{m}\]

Use the formula to calculate the time taken by light

Now, we have the distance \(d = 0.006\, \text{m}\) and the refractive index \(\mu = 2.0\). We also know that the speed of light in vacuum is \(c = 3\times 10^8 \,\text{m/s}\). We can use the formula \(t = \frac{d}{c/\mu}\) to find the time taken by light to cross the glass plate. \[t = \frac{0.006\, \text{m}}{(3 \times 10^8 \,\text{m/s}) / 2.0}\] \[t = \frac{0.006\, \text{m}}{(1.5 \times 10^8 \,\text{m/s})}\]

Calculate the time and express it in scientific notation

By performing the calculation, we can find the time taken by light in seconds: \[t = \frac{0.006\, \text{m}}{(1.5 \times 10^8\, \text{m/s})}\] \[t = 4 \times 10^{-11}\, \text{s}\] The time taken by light to cross the glass plate is \(4 \times 10^{-11}\, \text{s}\). Therefore, the correct answer is (B).

Key concepts

Speed of Light

When we talk about the speed of light, we refer to how fast light travels in a vacuum. Light is incredibly fast! It zooms through space at a speed of approximately \(3 \times 10^8\) meters per second (m/s). This is the universal speed limit. Nothing in the universe can travel faster than this speed.
Understanding the speed of light is important in optics. It helps us understand how light behaves when it moves through different materials. If light were slower, many of our everyday technologies like cameras and fiber optics wouldn't work the same way.
In our exercise, we use this speed to calculate how long it takes for light to travel through a glass plate. When light enters any material other than vacuum, it slows down. This is where the concept of refractive index comes into play.

Refractive Index

The refractive index, often symbolized by \(\mu\), measures how much the speed of light decreases in a material compared to its speed in a vacuum.

• An object with a higher refractive index slows light more than an object with a lower refractive index.
• For example, in our problem, glass has a refractive index of 2.0. This means light travels twice as slowly in this glass as it does in a vacuum.

When solving optical problems, the refractive index is crucial. It helps determine how light bends and changes speed as it enters materials. This change in speed can affect how we see objects through lenses or glass.
In the exercise, we use the refractive index to adjust the speed of light when calculating the time it takes light to travel through the glass. The refractive index, therefore, directly impacts our final result.

Scientific Notation

Scientific notation is a way to express very large or very small numbers conveniently. It uses powers of ten to simplify such numbers and makes them easier to read, write, and calculate.
For example:

• The number 300,000,000 can be written as \(3 \times 10^8\) in scientific notation.
• Similarly, \(0.000000001\) becomes \(1 \times 10^{-9}\).

In our optics exercise, time was expressed in scientific notation as \(4 \times 10^{-11}\) seconds. This format is practical when dealing with the minuscule scales and high speeds found in physics. Writing out these numbers in full would be cumbersome. Scientific notation offers a cleaner, more efficient way to handle calculations with precision and clarity.
Whether calculating speeds or expressing time taken for light to cross a glass plate, scientific notation ensures accuracy in the tiny details of the universe.