Ray Optics And Optical Instruments

Step 1: Condition for dispersion without deviation.
For dispersion without deviation, the total deviation produced by the two prisms must be zero. The deviation produced by a prism is given by: where is the refractive index and is the angle of the prism.
Step 2: Setting up the equation.
For dispersion without deviation, the deviation produced by should be equal and opposite to that produced by . Hence, we have: Substituting the known values , , and , we can solve for .
Step 3: Solving for .

Step 4: Conclusion.
Thus, the angle of prism is , which corresponds to option (D).

Step 1: Use the relationship between speed, distance, and time.
The speed of light in vacuum is , and the speed of light in the denser medium is . The critical angle is related to the refractive indices of the two media by the equation: Step 2: Substitute the values.
Substitute the expressions for the speeds into the equation: Thus, the critical angle is: Step 3: Conclusion.
Thus, the critical angle of the given pair of media is , which corresponds to option (B).

Step 1: Formula for frequency of a vibrating string.
The fundamental frequency of a vibrating string is given by: where is the length of the string, is the tension, and is the mass per unit length.
Step 2: Frequency ratio.
Since the strings are identical, we can relate the frequencies and tensions using the formula. The frequency is inversely proportional to the square root of the length and directly proportional to the square root of the tension. Therefore, the ratio of tensions is:
Step 3: Conclusion.
Thus, the ratio of tensions is , and the correct answer is (C).

Step 1: Energy conversion.
The total mechanical energy at the top of the inclined plane is potential energy , which is converted into rotational kinetic energy and translational kinetic energy at the bottom. The rotational kinetic energy for a rolling object is , where is the moment of inertia and is the angular velocity.
Step 2: Rotational kinetic energy of a solid cylinder.
For a solid cylinder rolling without slipping, the moment of inertia is , and the rotational kinetic energy at the bottom is:
Step 3: Conclusion.
Thus, the correct answer is (A) .

Step 1: Write the condition for critical angle.
For light going from a denser medium to a rarer medium,

Step 2: Substitute the given refractive indices.
Here, glass is denser and water is rarer:

Step 3: Write the critical angle.

Step 1: Understanding the angle of deviation.
The angle of deviation for a prism is given by the relation: where is the angle of incidence, is the angle of emergence, and is the angle of the prism.
Step 2: Using the given values.
It is given that , so we can substitute into the formula for : Step 3: Conclusion.
Thus, the angle of deviation is 30°, which corresponds to option (D).

Step 1: Using the lensmaker’s formula.
The lensmaker’s formula is given by: where is the focal length, is the refractive index, and and are the radii of curvature of the two faces of the lens. Since both faces have the same radius of curvature, , the formula simplifies to: Step 2: Substituting the given values.
Substitute and into the equation: Step 3: Conclusion.
The radius of curvature is , which corresponds to option (A).

Step 1: Relation between focal length and wavelength.
The focal length of a convex lens depends on the refractive index of the material.
Refractive index decreases as wavelength increases.

Step 2: Compare wavelengths of colors.
Among visible light, red light has the maximum wavelength.
Hence, refractive index is minimum for red light.

Step 3: Conclusion.
Since focal length is inversely related to refractive index, it is maximum for red light.

Step 1: Understanding the focal lengths for different instruments.
The focal length of the objective lens in optical instruments plays a crucial role in determining their magnification and resolving power. - For a compound microscope: The objective lens has a short focal length to allow for a larger magnification. This allows a small object to be magnified by a large amount, making it suitable for observing small details. - For a telescope: The objective lens has a large focal length to provide better resolution and the ability to view distant objects clearly. A larger focal length in a telescope means a higher resolving power and a clearer image over great distances. Step 2: Explanation of the options.
Let's analyze the given options: - (A) short and short: This is incorrect because both the objective lens of a compound microscope and a telescope do not have short focal lengths. - (B) large and large: This is incorrect because while the telescope requires a large focal length, the compound microscope needs a short focal length for high magnification. - (C) large and short: This is incorrect as it suggests that a telescope has a short focal length, which is false. - (D) short and large: Correct. The compound microscope requires a short focal length for the objective lens to magnify small objects, and the telescope requires a large focal length for better resolution over long distances. Step 3: Conclusion.
The correct answer is (D) short and large. The focal length of the objective in a compound microscope is short, and in a telescope, it is large.

Step 1: SHM formula.
In simple harmonic motion, the distance from the mean position at any time is given by: where is the amplitude and is the angular frequency, and is the period.
Step 2: Time for half the amplitude.
For half the amplitude, . So, we need to find such that: This gives:
Step 3: Conclusion.
Since , the time taken to cover half the amplitude is: Thus, the correct answer is (A).

Step 1: Understanding refractive index.
The refractive index of a medium is related to the speed of light in the medium. The refractive index decreases with the increase in wavelength.
Step 2: Correct proportionality relation.
For most materials, the refractive index is inversely proportional to the wavelength:
Step 3: Conclusion.
The correct relation is , so the correct answer is (C).

Step 1: Use of vector addition.
Let the components of be and , and the components of be and . The components of the resultant vector are given by: From these, we can find:
Step 2: Magnitude of .
The magnitude of is:
Step 3: Conclusion.
The magnitude of is 5, so the correct answer is (B).

Step 1: Use lens maker’s formula.
For a biconvex lens, , .

Step 2: Substitute values.


Step 3: Use mirror formula.
For a concave mirror, By sign convention for concave mirror, is negative.

Step 4: Final result.

Step 1: Condition for minimum deviation.
At minimum deviation, angle of incidence equals angle of emergence, and refraction angles are equal: Step 2: Apply Snell’s law at first surface.
Step 3: Calculate angle of refraction.
Step 4: Find angle of prism.

Step 1: Magnifying power of a telescope.
The magnifying power of a telescope is given by: where is the aperture of the objective lens and is the focal length of the eyepiece. If the focal length of the eyepiece is doubled, the magnifying power will be halved.
Step 2: Conclusion.
Thus, when the focal length of the eyepiece is doubled, the new magnifying power will be , which is option (B).

Step 1: Use definition of refractive index.
Refractive index of medium A with respect to medium B is given by
Step 2: Write given refractive indices.

Step 3: Multiply all three.

Step 4: Conclusion.
The correct relation is .

Step 1: Recall image formation in spherical mirrors.
In spherical mirrors, images can be formed either on the same side as the object or on the opposite side depending on the nature of the mirror and object position.
Step 2: Image on the same side as object.
When the image is formed on the same side as the object (as in concave mirror for certain positions), the image is real.
Step 3: Image on the opposite side of object.
When the image is formed behind the mirror (opposite side of object), the image is virtual.
Step 4: Conclusion.
Thus, images formed on the side of the object are real, and those formed on the opposite side are virtual.

Step 1: Using the relationship between refractive index and velocity.
The refractive index is related to the speed of light in a medium by the equation: Where is the speed of light in vacuum. Thus, for two media: This is the correct relationship. Therefore, the correct answer is (A) .

Step 1: Understanding the concept of invisibility.
For an object to become invisible when observed from outside, the light should not be refracted or reflected by the object. This can happen only when the refractive index of the object matches that of the surrounding medium. Thus, the object will become invisible when its refractive index is the same as that of the fluid, which is . Step 2: Conclusion.
Thus, the correct answer is (B) have refractive index same as surrounding fluid, that is, .

Step 1: Using the lens formula.
The lens formula is: Where: - is the focal length, - is the image distance, - is the object distance. Step 2: Relating the size of the image and object.
The magnification of the lens is given by: Given that the magnification is times the size of the object, we have: Step 3: Finding the image distance.
From the magnification equation, we can solve for : Now substitute this into the lens formula: Simplifying: Thus, the image distance is: Thus, the correct answer is (A) .

Step 1: Understanding the problem.
In a thin prism, when the ray emerges normally from the opposite surface, the angle of incidence is related to the angle of the prism and the refractive index . The relationship is given by the formula: Step 2: Conclusion.
Thus, the correct answer is (C) .

Snell's law:
n =
In this case, the refractive index of water with respect to air is given as . We are given that the fish appears to be at a depth of 30 cm from the surface (d'). We need to find the real depth (d) of the bird.
Applying Snell's law:
=
d = x 30
d = 40 cm
Therefore, the correct answer is (C) 40 cm.

Step 1: Use the formula for the power of a lens
The power of a lens is given by the formula: where is the focal length in meters.
Step 2: Convert the focal length to meters
Given: - Focal length .
Step 3: Calculate the power of the lens
Now, substitute the value of into the formula:
Answer:
Therefore, the power of the lens is . So, the correct answer is option (1).

Concept:
For equilibrium of proton: (Coulomb's law) Step 1: Assume position. Let distance of proton from be . Then distance from : (Proton is placed beyond )
Step 2: Apply force balance. Cancel common terms:
Step 3: Simplify.
Step 4: Take square root.
Step 5: Solve for .
Step 6: Rationalize.
Step 7: Conclusion.

Concept:
According to Stefan–Boltzmann law: where Step 1: Initial power.
Step 2: New conditions. Radius doubled: Temperature doubled:
Step 3: New power.
Step 4: Conclusion.

Step 1: Pressure-Radius Relation
Excess pressure in soap bubble: .
Given .
Step 2: Volume-Radius Relation
Volume .
.
Step 3: Calculation
.
Final Answer: (D)

Step 1: Displacement Equation
Since it starts from mean position: .
.
Step 2: Distances
Distance in 1st sec ( ): .
Displacement at 2nd sec ( ): .
Distance in 2nd sec ( ): .
Step 3: Ratio
Ratio .
Final Answer: (A)

To find the image distance for a convex lens, we use the lens formula: Where: - (focal length of the convex lens, positive for convex lens), - (object distance, negative as the object is on the side of incident light), - is the image distance (to be determined). Rearrange the lens formula to solve for : Substitute the values: Find a common denominator (LCM of 20 and 30 is 60): The positive value of indicates that the image is formed on the opposite side of the lens from the object, which is consistent for a real image formed by a convex lens when the object is beyond the focal point. Thus, the image distance from the lens is .

Using Snell's law: Where: - is the refractive index of air, - is the refractive index of water, - is the angle of incidence in air. Substituting the values: Thus, the angle of refraction in water is .