Question

A beam of light is converging towards a point I on a screen. A plane parallel plate of glass whose thickness is in the direction of beam \(=\mathrm{t}\), refractive index \(=\mathrm{n}\) is introduced in the path of the beam. The convergence point is shifted by \(\ldots \ldots\) (A) \(t[1+(1 / n)]\) never (B) \(t[1-(1 / n)]\) nearer (C) \(\mathrm{t}[1+(1 / \mathrm{n})]\) away (D) \(t[1-(1 / n)]\) away

Short answer

The convergence point shifts closer to the screen by \(t[1 - (1/n)]\) when the glass plate is introduced in the path of the beam of light. The correct answer is (B) \(t[1 - (1/n)]\) nearer.

Step-by-step solution

Understanding the Optical Path Length concept

Optical path length is the product of the refractive index and the distance the light travels in a medium. It is a measure of how the light travels through a specific medium, considering the speed of light in the medium. When light travels through the glass plate, its optical path length will differ compared to its original path in the air.

Calculate Optical Path Length for air and glass

In this exercise, we have two optical paths: one in air and the other in the glass. First, let's calculate the optical path length in air: Optical Path Length in air: \(OPL_{air} = t\) Now let's calculate the optical path length when the light travels through the glass: Optical Path Length in glass: \(OPL_{glass} = nt\)

Find the difference in Optical Path Lengths

To find how the convergence point is shifted, we need to subtract the optical path length in air from the optical path length in glass: Difference in Optical Path Lengths: \(\Delta OPL = OPL_{glass} - OPL_{air} = nt - t\)

Simplify the expression for the shift in convergence point

Let's simplify the expression for the difference in optical path length: \(\Delta OPL = nt - t = t(n - 1)\) Now, let's rewrite the expression in terms of the thickness of the glass (t) and the refractive index (n): \(\Delta OPL = t[1 - (1/n)]\)

Compare the result to the given options and pick the right one

Finally, we can compare our result to the available options. The correct answer is: (B) \(t[1 - (1/n)]\) nearer So, the convergence point shifts closer to the screen by \(t[1 - (1/n)]\) when the glass plate is introduced in the path of the beam of light.

Key concepts

Refractive Index

The refractive index is a measurement that indicates how light or any other radiation propagates through a medium. It is represented by the letter \( n \). The refractive index describes how much the speed of light is reduced compared to its speed in a vacuum.
For instance, a refractive index of 1.5 means that light travels 1.5 times slower in the medium than it does in a vacuum.

Light slows down as it enters a medium with a higher refractive index. As a result, the light's path bends, a phenomenon called refraction. The bending of light depends on the angle of incidence and the refractive indices of the media involved.

• Higher refractive index - more bending of the light ray.
• Lower refractive index - less bending.

This concept is crucial in optical systems such as lenses and is essential in determining the convergence point of a light beam after passing through different mediums.

Convergence Point

A convergence point in optics refers to the point at which light rays intersect after passing through lenses or other optical components. This point is significant in determining the focus position in various optical instruments, like microscopes and cameras.

When a plane parallel glass plate is introduced, the convergence point can shift. The shift in the convergence point depends on the difference in optical path length through the medium compared to the surrounding air. The optical path length changes because the refractive index of the glass alters the speed and path of the light.

• Without the glass plate, light rays converge at the initial point on the screen.
• When the glass plate is added, light travels slower, causing a shift towards the glass.

Understanding how convergence points change helps solve complex optical problems, including lens design and corrective eyewear.

Plane Parallel Plate of Glass

A plane parallel plate of glass is essentially a flat slab of glass with two parallel faces. These types of glass plates are commonly used in optics to manipulate light paths without altering the overall focus or spreading of the beam too much.

When light enters such a plate, it experiences a shift, called optical path length change. The change arises because the refractive index of glass is greater than that of air. Thus, light within the plate travels slower, elongating the path length effectively.

• The thickness \( t \) of the plate dictates how far the convergence point shifts.
• The shift can be calculated using the difference in refractive indices.

These plates play a pivotal role in correcting alignment errors in optical devices and controlling beam paths precisely.