Optics

An astronomical telescope uses two lenses: the objective lens and the eyepiece lens. The tube length of the telescope is the sum of the focal lengths of these two lenses. Given:
Focal length of the objective lens,
Focal length of the eyepiece lens,
Tube length, , is given by:

Substituting the values:

Now, to find the magnification , we use the formula:

Substituting the given values:

Thus, the tube length is 1010 cm, and the magnification is 100. Therefore, the correct answer is: 1010 cm, 100

To determine the correct condition for total internal reflection at an interface between two media, we need to consider the refractive indices and the direction of light travel. The phenomenon of total internal reflection occurs when light travels from a medium with a higher refractive index to a medium with a lower refractive index, and the incidence angle exceeds the critical angle.

For the given problem:

1. Refractive index of water, .

2. Refractive index of glass, .

Since glass has a higher refractive index than water ( ), total internal reflection can only occur when light travels from glass to water.

The condition for total internal reflection is that the angle of incidence must be greater than the critical angle . The critical angle can be calculated from:

Therefore, is the angle whose sine is , and total internal reflection occurs when:

Matching List-I (Optical Instruments) with List-II (Nature of Lens/Mirror Used):

Final Matching:

• A → III: Human eye uses a lens with adjustable focal length (crystalline lens).
• B → IV: Microscope uses an objective lens with small aperture and small focal length for high magnification.
• C → I: Reflecting telescope uses a concave mirror with large aperture and large focal length to collect more light.
• D → II: Refracting telescope uses an objective lens with large aperture and large focal length for better resolution.

To address the given question about the optical properties of a light pipe, we will evaluate each statement based on known principles of optics:

(A) Optical density of core should be greater than the optical density of cladding.
This statement is true. For a light pipe (optical fiber) to function correctly, total internal reflection must occur. This requires that light travel from a medium of higher optical density (the core) to one of lower optical density (the cladding).

(B) r and θ will always be equal.
This statement is false. In the context of refraction and total internal reflection, angles r (refraction angle) and θ (incidence or reflection angle) are not necessarily equal. They are determined by Snell's law which typically results in different values unless specified conditions are met.

(C) Optical density of cladding is .
This statement is incorrect. The expression does not follow the general principles of optics for determining optical densities.

(D) Optical density of cladding is .
This statement is also incorrect based on the same reasoning as in statement (C).

Given this analysis, the correct answer is (A) only, as it is the only true statement from the options provided.

An equilateral prism has an angle of . The refractive index . For an equilateral prism, the angle of minimum deviation is determined using the formula:

Substituting the values:

We know:

Thus,

Hence,

We know:

Therefore,

Solving gives:

Thus,

However, the equation was incorrectly set, reevaluating gives:

Resulting in:

Thus,

The correct angle of minimum deviation is .

To determine how the power of a lens is affected by the refractive index , we need to use the lens maker's formula:

where is the focal length of the lens, is the refractive index of the lens material, and and are the radii of curvature of the lens surfaces. The power of the lens is defined as the reciprocal of the focal length:

Upon substituting the expression for from the lens maker's formula, we get:

This equation shows that for fixed values of the radii of curvature and , the power is directly proportional to .

Thus, the correct answer is .

To determine the correct order of focal lengths for different colors in a convex lens, we need to understand how colors are refracted differently. In a convex lens, the refractive index affects the focal length for different colors in the visible spectrum.

The refractive index is higher for violet light and lower for red light, as violet light has a shorter wavelength and red has a longer wavelength.

The formula for the focal length of a lens is related to the refractive index by .
Hence, greater implies a smaller .

The order of decreasing refractive indices and hence increasing focal lengths is:

• Violet
• Blue
• Yellow
• Red

Thus, the increasing order of focal lengths is: (A), (C), (D), (B).

In single-slit diffraction, the width of any band (central or bright) is given by:

where:
λ = wavelength of light
D = distance between slit and screen
d = slit width

Analysis of each case:

A. Central band width (original): (central band is twice as wide as others)

B. First bright band width (original):

C. Central band width with doubled D:

D. First bright band width with tripled D:

Width Comparison:

1. Original first bright band (B):

2. Original central band (A):

3. First bright band with tripled D (D):

4. Central band with doubled D (C):

Increasing Order: B < A < D < C

The question involves a convex lens with its lower half made opaque. We need to understand how this affects the image of an object positioned in front of the lens.

Let's analyze each option:

1. No change in image: For a lens, light that passes through any part contributes to the formation of the complete image. Hence, even if half the lens is covered, the entire image still forms, though with reduced brightness. Therefore, this option is incorrect.
2. Image will show only half of the object: Since the lens still focuses the light from all parts of the object, the entire image is formed but with less intensity. Thus, the option stating that only half of the object is shown is incorrect.
3. Intensity of image gets reduced: By blocking half of the lens, we indeed reduce the amount of light that can pass through and form the image. This results in an image with reduced intensity. Therefore, this option aligns with the physical principle.

Based on the analysis, the correct solution is: (C) only

The problem involves determining the distance of the image formed by a spherical refracting surface using the formula for refraction at a spherical surface. The formula is given by:
, where:

• (refractive index of air)
• (refractive index of the material)
• \) (object distance, negative according to sign convention)
• \) (radius of curvature, positive as it is measured from air to the convex surface)

Substitute these values into the formula:

Solving step by step:

• Finding common denominator for right side:
  and , so
• Thus,
• Cross-multiplying gives:

The positive value of v indicates that the image is formed in air on the same side as the object, hence 16 cm left of P in air.

The angle of minimum deviation ( ) for a prism can be found using the prism formula:

where is the refractive index and is the prism angle.
Given: and for an equilateral prism, .
We want to find .
Given the equation:

Calculating gives .
So, .
This implies:
.
Thus:
.
Solving for :
.
.
.
Thus, the angle of minimum deviation is 30°.

The problem involves understanding the interference pattern created by a double-slit experiment. The formula for fringe separation in a double-slit experiment is given by:

where:

• is the wavelength of the light used
• is the distance between the screen and the slits
• is the distance between the slits

Given:

Substitute these values into the formula:

Calculate:

Therefore, the fringe separation is 0.1 cm.