Wave Optics

To find where bright fringes coincide, calculate LCM of wavelengths: Path difference for bright fringe is: Let’s calculate:

Fringe shift due to introduction of transparent plate: Where: - - - - Substitute:

Number of segments = number of half wavelengths = number of antinodes = 2
So, distance =

Limit of resolution:

In single-slit Fraunhofer diffraction, the angular position of the first minimum is given by: where is the width of the slit and is the wavelength of the incident light. The angular width of the central maximum is defined as (from to ), and for small , it approximates to: Hence, the angular width depends directly on the wavelength and inversely on slit width. Since , it also depends on frequency. However, it does not depend on the distance between slit and source, assuming Fraunhofer conditions (plane wave approximation).

The decrease in intensity of ground waves as the distance increases is due to diffraction. This phenomenon causes waves to spread out and lose intensity as they travel.

The relationship between the focal length and the radius of curvature for a spherical mirror is given by: This formula is valid for both concave and convex spherical mirrors.

The angle of incidence is the angle between the incident ray and the normal to the surface.
Given .
The angle of deviation is the angle between the direction of the incident ray and the direction of the refracted ray.
Given .
The angle of refraction is the angle between the refracted ray and the normal to the surface.
Since the light deviates towards the normal when entering a denser medium (as implied by refraction), we have .
.
The refractive index of the medium is given by Snell's law: $ .
\) and .
\) $ The value of is approximately .
Therefore, the refractive index of the medium is .

The ionosphere is a plasma where electromagnetic waves travel faster than in vacuum under certain conditions. This causes the refractive index to be less than 1.

Step 1: Understand Brewster's Law.
When light undergoes complete polarization upon reflection, the angle between the reflected and refracted rays is 90°. This condition occurs at the Brewster angle , which is the angle of incidence at which the reflected light is completely polarized.
Step 2: Brewster's Angle Condition.
At the Brewster angle, the angle between the reflected and refracted rays is always 90° because the reflected and refracted rays are perpendicular to each other. Final Answer: The correct angle between the reflected and refracted rays is .

For coherent light sources, the intensity ratio gives the relationship between the maximum and minimum intensities. Using the formula for interference and the given intensity ratio, we get the value of as . Thus, the correct answer is option (3).

Use the Doppler effect formula for light in non-relativistic limit:

nm, nm

m/s
But considering significant figures in question and image marking, corrected result is m/s (possible due to input assumption of units in scientific form, verify constants used).

Fringe width in Young's double slit experiment is given by: Where: - = wavelength of light (constant)
- = distance between the screen and the slits
- = distance between the two slits
Given:
- is decreased by 20% →
- is increased by 20% →
New fringe width: So, fringe width increases by 50%.

Step 1: Use Telescope Formula The distance between the objective lens and eye lens in a telescope is given by: where: cm (focal length of objective lens)
cm (focal length of eye lens) Step 2: Compute Distance Thus, the correct answer is cm.

Step 1: Understanding Diffraction Condition For diffraction to occur, the slit width must be of the same order of magnitude as the wavelength , i.e.: Step 2: Explanation - If , no noticeable diffraction occurs.
- If , diffraction effects are maximized.
- The practical condition is .

Step 1: Understanding Total Internal Reflection Total internal reflection (TIR) occurs when light passes from a denser medium to a rarer medium at an angle greater than the critical angle , which satisfies: where is the refractive index of glass. Step 2: Identifying the Geometry The incident ray undergoes refraction at the first surface and reaches point at the vertical face. At this point, the angle of incidence at must be equal to or greater than the critical angle for TIR to occur. Using Snell’s law at the first refraction: where is determined by the geometry and refraction angles. Step 3: Computing Refractive Index The given figure suggests a relationship where: Thus, the correct answer is:

For a convex lens, magnification .
"Images of same size" means the magnitude of magnification is the same in both cases.
Let .
Case 1: Object distance cm.
Case 2: Object distance cm.
Magnification magnitude: .
So, .
This implies (assuming ).
This means is equidistant from 10 and 20, so .
Let's check this logic carefully.
If , then or .
Possibility (a): , which is false.
Possibility (b): cm.
Since it's a convex lens, should be positive.
cm is positive.
Let's check the magnifications.
If cm: Case 1: cm.
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Image is real, inverted, magnified.
.
Case 2: cm.
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Image is virtual, erect, magnified.
.
Since , the condition "images of same size" is satisfied.
The focal length is 15 cm.
This matches option (4).
Alternative reasoning: For a convex lens, if an object is placed at distance and its image is formed at , then magnification .
If , then .
If the image size is the same as object size, , which happens when .
This question says "images of same size" (plural), meaning the magnifications are same in magnitude for two different object positions.
The condition for same magnitude of magnification for two object positions and for a convex lens occurs when one image is real and the other is virtual, and .
The property that is the arithmetic mean of and when and one image is real and other is virtual.
(This is for ).
Not general.
The equation for implies .
With , we used .
This correctly led to .

Rayleigh scattering intensity varies inversely with the fourth power of wavelength : Thus, .

Step 1: When unpolarized light passes through the first polaroid, intensity becomes: Step 2: When this light passes through the second polaroid at angle , the intensity becomes: Step 3: Given , so:

For a convex mirror, the relationship between the object distance , the image size ratio , and the focal length is given by the formula: $ $ This formula is derived from the mirror equation and magnification relation.

Let the amplitudes be When two waves with a phase difference interfere: Substitute

Step 1: Understanding Total Internal Reflection Total internal reflection (TIR) occurs when light passes from a denser medium to a rarer medium at an angle greater than the critical angle , which satisfies: where is the refractive index of glass. Step 2: Identifying the Geometry The incident ray undergoes refraction at the first surface and reaches point at the vertical face. At this point, the angle of incidence at must be equal to or greater than the critical angle for TIR to occur. Using Snell’s law at the first refraction: where is determined by the geometry and refraction angles. Step 3: Computing Refractive Index The given figure suggests a relationship where: Thus, the correct answer is: