Ray Optics And Optical Instruments

Step 1: Understanding the Concept:
A lens works by refracting light. Refraction, or the bending of light, occurs only when light passes from one medium to another with a different refractive index. The focal length of a lens is determined by its curvature and the difference in refractive index between the lens material and the surrounding medium.
Step 2: Key Formula or Approach:
The focal length ( ) of a lens is given by the Lens Maker's Formula:
where:
- is the refractive index of the lens material.
- is the refractive index of the surrounding medium.
- and are the radii of curvature of the lens surfaces.
Step 3: Detailed Explanation:
According to the problem statement:
- Refractive index of the lens, .
- Refractive index of the medium, .
Now, let's substitute these values into the Lens Maker's Formula:
If the reciprocal of the focal length is zero, the focal length itself must be infinite:
A lens with an infinite focal length has zero power ( ). This means it does not bend light at all and acts as a transparent, parallel-sided plate of glass. The light rays will pass through it undeviated.
Step 4: Final Answer:
The focal length of the lens will be infinite, and it will lose its converging property, effectively becoming invisible in the liquid.

Step 1: Understanding the Concept:
The critical angle is related to the refractive index of the medium. The refractive index, in turn, relates the speed of light in a vacuum to the speed of light in the medium.
Step 2: Key Formula or Approach:
1. The relationship between the refractive index ( ) of a medium and the critical angle ( ) for light going into a vacuum is:
2. The definition of refractive index is:
Step 3: Detailed Explanation:
Part 1: Calculate the refractive index (n)
Given the critical angle .
We know that or .
The refractive index of the medium is 2.
Part 2: Calculate the velocity of light in the medium (v)
We use the formula , where is the speed of light in vacuum, m/s.
Rearranging the formula to solve for :
Substituting the values:
Part 3: Compare with options
The calculated velocity is m/s.
Looking at the options:
(A) 3 x 10 m/s (This is the speed in vacuum)
(B) 1.5 x 10 m/s (The numerical value is correct, but the power of 10 is wrong, likely a typo)
(C) 6 x 10 m/s (Faster than light in vacuum, impossible)
(D) 4.5 x 10 m/s (Faster than light in vacuum, impossible)
The value m/s is the correct answer.
Step 4: Final Answer:
The calculated velocity of light in the medium is m/s. Option (B) is the intended answer, despite the typo.

Step 1: Understanding the Concept:
Total Internal Reflection (TIR) is an optical phenomenon that occurs when a ray of light traveling from a denser medium to a less dense medium strikes the boundary at an angle of incidence greater than the critical angle.
The coefficient of reflection (or reflectivity) is the ratio of the reflected power (or intensity) to the incident power (or intensity).
Step 2: Detailed Explanation:
When TIR occurs, the boundary between the two media acts like a perfect mirror. No light is transmitted or refracted into the rarer medium; instead, the entire incident light is reflected back into the denser medium.
Let be the intensity of the incident light and be the intensity of the reflected light.
The coefficient of reflection, , is defined as:
In the case of Total Internal Reflection, all the incident energy is reflected, so .
Therefore, the coefficient of reflection is:
This means 100% of the light is reflected.
Step 3: Final Answer:
For total internal reflection, the coefficient of reflection is 1. Therefore, option (B) is correct.

Step 1: Understanding the Concept:
The focal length of a spherical mirror is determined by the laws of reflection and the geometry of the mirror's surface.
Step 2: Key Formula or Approach:
The focal length ( ) of a spherical mirror is related to its radius of curvature ( ) by the formula:
Step 3: Detailed Explanation:
The laws of reflection state that the angle of incidence is equal to the angle of reflection, and these angles are measured with respect to the normal at the point of incidence. These laws are independent of the medium in which the reflection occurs.
The formula shows that the focal length of a mirror depends only on its radius of curvature ( ), which is a physical property of the mirror's shape.
Unlike a lens, whose focal length depends on the refractive index of its material and the surrounding medium (as described by the Lens Maker's Formula), a mirror's focal length is purely a geometric property.
Therefore, immersing a spherical mirror in water (or any other transparent medium) does not change its radius of curvature, and consequently, its focal length remains the same.
Step 4: Final Answer:
Since the focal length of a spherical mirror depends only on its radius of curvature and not on the surrounding medium, it will remain the same when immersed in water. Option (C) is correct.

Step 1: Understanding the Concept:
When thin lenses are placed in contact, the power of the combination is the algebraic sum of the powers of the individual lenses. The power of a lens is defined as the reciprocal of its focal length in meters.
Step 2: Key Formula or Approach:
Power of a lens, . The unit of power is the dioptre (D).
For a combination of lenses in contact: .
Sign Convention: The focal length of a convex lens is positive, and the focal length of a concave lens is negative.
Step 3: Detailed Explanation:
Lens 1 (Convex):
Focal length, cm = m.
Power, D.
Lens 2 (Concave):
Focal length, cm = m.
Power, D.
Power of the Combination:
Now, add the individual powers algebraically: Step 4: Final Answer:
The power of the combination is -2.5 dioptres. The negative sign indicates that the combination acts as a concave (diverging) lens. Option (B) is correct.

Step 1: Understanding the Concept:
Dispersive power ( ) is a property of the material of a prism that quantifies its ability to separate white light into its constituent colors (dispersion). It is defined as the ratio of the angular dispersion (the difference in deviation angles for two extreme colors, typically violet and red) to the deviation of a mean color (typically yellow).
Step 2: Key Formula or Approach:
The formula for dispersive power ( ) is: For a prism with a small angle , the angle of deviation is given by , where is the refractive index of the material.
Substituting this into the formula for : The prism angle cancels out from the numerator and the denominator: Step 3: Detailed Explanation:
The final expression, , shows that the dispersive power depends only on the refractive indices of the prism's material for different wavelengths of light ( ). The refractive index is an intrinsic property of the material itself.
Therefore, the dispersive power depends on the nature of the material of the prism. It does not depend on the geometric properties of the prism, such as its angle ( ), nor on how the light enters it, such as the angle of incidence.
Step 4: Final Answer:
Since dispersive power is determined solely by the refractive indices of the medium, it is a characteristic property of the material of the prism. Therefore, option (B) is correct.

Step 1: Understanding the Concept:
The problem requires finding the focal length of a biconvex lens given its radii of curvature and refractive index. This can be solved using the Lens Maker's Formula.
Step 2: Key Formula or Approach:
The Lens Maker's Formula is: where: = focal length of the lens = refractive index of the lens material with respect to the surrounding medium (air, in this case) = radius of curvature of the first surface (where light enters) = radius of curvature of the second surface
Step 3: Detailed Explanation:
Given data: - Type of lens: Biconvex - Refractive index, - Radius of curvature of each surface = 20 cm. We must apply the Cartesian sign convention. Assume light travels from left to right. - For the first surface (left), it is convex towards the incident light. Its center of curvature is on the right side. Thus, cm. - For the second surface (right), it is also convex, but its center of curvature is on the left side. Thus, cm. Now, substitute these values into the Lens Maker's Formula: Therefore, the focal length is: Step 4: Final Answer:
The focal length of the lens is 20 cm. This corresponds to option (C).

This formula is known as the Lens Maker's Formula. It relates the focal length of a lens to the refractive index of its material and the radii of curvature of its two surfaces.
Step 1: Assumptions:
1. The lens is thin, meaning its thickness is negligible compared to the radii of curvature.
2. The aperture of the lens is small.
3. The object is a point object placed on the principal axis.
4. The incident and refracted rays make small angles with the principal axis.
Step 2: Derivation using Refraction at Spherical Surfaces:
Let's consider a thin convex lens of refractive index placed in a medium of refractive index . Let and be the radii of curvature of the first and second surfaces, respectively.

The general formula for refraction at a single spherical surface is:
Refraction at the First Surface (surface 1 with radius ):
- An object O is placed in the medium ( ) at a distance from the lens.
- Light travels from the medium ( ) to the lens material ( ).
- A real image would be formed at a distance if the second surface were absent.
Applying the formula:
Refraction at the Second Surface (surface 2 with radius ):
- The image formed by the first surface acts as a virtual object for the second surface, at a distance .
- Light travels from the lens material ( ) back to the medium ( ).
- The final image is formed at a distance .
Applying the formula (note the swapping of and ):
Step 3: Combining the Equations:
Now, add equation (1) and equation (2):
The term cancels out.
Divide the entire equation by :
Step 4: Final Formula:
From the thin lens equation, we know that , where is the focal length of the lens.
Let be the refractive index of the lens material with respect to the surrounding medium.
Substituting these into the equation, we get the Lens Maker's Formula:
Meaning of Symbols:
- : Focal length of the thin lens.
- : Refractive index of the material of the lens relative to the surrounding medium.
- : Radius of curvature of the first surface where light enters.
- : Radius of curvature of the second surface where light exits.
(Note: A proper sign convention must be used for .)

Step 1: Understanding fibre optic communication.
Fibre optic communication relies on light signals transmitted through optical fibres. The medium used for transmitting data is electromagnetic waves, particularly light waves. These waves travel through the fibre core via total internal reflection. Step 2: Conclusion.
Hence, the type of wave used in fibre optic communication is electromagnetic waves.