Class 10 Physics Chapter 6 Spectrum

Key Concept: Frequency Calculation

c) 500 nm
[Solution Description]
Using the formula $c = f \times \lambda$, we can find the wavelength $\lambda$:
$\lambda = \frac{c}{f} = \frac{3 \times 10^8 \, \text{m/s}}{6 \times 10^{14} \, \text{Hz}} = 5 \times 10^{-7} \, \text{m} = 500 \, \text{nm}$
From the syllabus, the wavelength range for visible light is 400 nm to 800 nm. Since 500 nm falls within this range, the wave is visible light.

Key Concept: Scattering of light, Rayleigh scattering

a) Both Assertion and Reason are true, and Reason is the correct explanation of Assertion.
[Solution Description]
According to Rayleigh scattering, the intensity of scattered light is given by $I \propto \frac{1}{\lambda^4}$. This means shorter wavelengths (like blue light) are scattered much more than longer wavelengths (like red light). At sunrise and sunset, sunlight travels a longer distance through the atmosphere, causing most of the blue light to be scattered away, leaving primarily red light to reach the observer. Thus, both the Assertion and Reason are true, and the Reason correctly explains the Assertion.

Key Concept: Electromagnetic spectrum applications, wavelength ranges

c) Both are microwaves
[Solution Description]
Referring to the electromagnetic spectrum: Microwaves have wavelengths from 1 mm to 10 m. The first wave at 2 mm (0.002 m) falls into the lower end of microwave range, while the second wave at 20 cm (0.2 m) is also in the microwave range but with longer wavelength. Both are microwaves.

Key Concept: Scattering applications, Atmospheric phenomena

c) Sunlight travels a longer path, scattering blue light more
[Solution Description] At sunrise and sunset, sunlight travels a longer distance through the atmosphere. Shorter wavelengths (blue/violet) are scattered more and removed from the direct path, leaving the longer wavelengths (red/orange) to dominate the observed color. At noon, the path length is shorter, allowing all wavelengths to reach the observer with minimal scattering, making the sun appear white.

Key Concept: Refractive index, Angle of deviation

b) $2^\circ$
[Solution Description]
The angle of deviation ($\delta$) is given by $\delta = i - r$.
For violet light: $\delta_{\text{violet}} = 40^\circ - 26^\circ = 14^\circ$.
For red light: $\delta_{\text{red}} = 40^\circ - 28^\circ = 12^\circ$.
The difference is $14^\circ - 12^\circ = 2^\circ$.

Key Concept: Concept: Rayleigh Scattering

d) Blue light scatters more, so red dominates the remaining light.
[Solution Description]
Rayleigh scattering explains why shorter wavelengths (e.g., blue) scatter more than longer wavelengths (e.g., red). Since the sun appears near the horizon during sunrise/sunset, sunlight travels a longer distance through the atmosphere, causing most of the blue light to scatter away, leaving predominantly red light to reach the observer.

Key Concept: Dispersion, Scattering

c) Selective scattering of shorter wavelengths
[Solution Description]
This involves both dispersion and scattering concepts:
1. Near horizon, starlight passes through thicker atmospheric layers.
2. Shorter wavelengths (blue/violet) scatter more (Rayleigh scattering: $I \propto 1/\lambda^4$).
3. Longer wavelengths (red) scatter less and dominate the transmitted light.
4. Prismatic dispersion also separates colors but is secondary to scattering in this context.

Key Concept: Wavelength Range

c) Green
[Solution Description]
According to the syllabus, the wavelength ranges for colours are: Green (500 nm to 570 nm), Yellow (570 nm to 590 nm). Since 550 nm lies between 500 nm and 570 nm, it corresponds to the green colour.

Key Concept: Relationship Between Wavelength and Speed in Glass

a) Both Assertion and Reason are true, and Reason is the correct explanation of Assertion.
[Solution Description]
From the syllabus, we know that:
- The speed of light in glass decreases with decreasing wavelength.
- Violet light has the shortest wavelength ($400 \, \text{nm}$), while red light has the longest wavelength ($800 \, \text{nm}$).
- The refractive index $\mu_{\text{glass}}$ increases as wavelength decreases because $\mu_{\text{glass}} = \frac{\text{Speed of light in air}}{\text{Speed of light in glass}}$.
Step 1: Assertion (A) states that violet light travels slower than red light in glass.
This is true because shorter wavelengths (violet) experience more refraction due to higher $\mu_{\text{glass}}$, reducing their speed more than longer wavelengths (red).
Step 2: Reason (R) states that the refractive index of glass increases with a decrease in wavelength.
This is also true based on the given formula for $\mu_{\text{glass}}$.
Step 3: The reason correctly explains why violet light travels slower—because its shorter wavelength results in a higher $\mu_{\text{glass}}$, leading to reduced speed.
Hence, both Assertion and Reason are true, and Reason is the correct explanation of Assertion.

Key Concept: Wavelength and frequency relationship

b) Frequency decreases
[Solution Description]
From the table provided, as we move from violet to red, the wavelength increases while the frequency decreases. This is because frequency ($\nu$) and wavelength ($\lambda$) are inversely related by the equation:
$\nu = \frac{c}{\lambda}$
where $c$ is the speed of light. Thus, an increase in wavelength results in a decrease in frequency.

Key Concept: Scattering of light

a) Both Assertion and Reason are true, and Reason is the correct explanation of Assertion.
[Solution Description]
According to the given syllabus, the intensity of scattered light depends on the inverse fourth power of the wavelength ($I \propto \frac{1}{\lambda^4}$). This means shorter wavelengths (like blue light) scatter much more than longer wavelengths (like red light). At sunrise and sunset, sunlight travels a longer distance through the atmosphere, causing most of the blue light to be scattered away, leaving predominantly red light reaching the observer. Thus, the assertion that the sun appears red due to more scattering of blue light is correct. The reason correctly explains this phenomenon by stating the inverse relationship between scattering intensity and wavelength raised to the fourth power. Therefore, both the Assertion and Reason are true, and the Reason correctly explains the Assertion.

Key Concept: Concept: Scattering of light in clouds

b) they scatter all colors of light equally
[Solution Description]
Clouds contain dust particles and water molecules larger than the wavelength of visible light. These particles scatter all colors of sunlight equally, resulting in the white appearance of clouds.

Key Concept: Refractive Index and Wavelength

d) Violet
[Solution Description]
The refractive index $\mu$ is inversely proportional to the speed of light in the medium. Violet light has the shortest wavelength and highest frequency, so its speed in glass is the lowest, leading to the highest refractive index.

Key Concept: Frequency Range of Radio Waves

b) Below $3 \times 10^7 \, \text{Hz}$
[Solution Description]
The frequency range for radio waves is below $3 \times 10^7 \, \text{Hz}$.

Key Concept: Electromagnetic Spectrum - Harmful Effects

c) Ultraviolet rays
[Solution Description]
Ultraviolet (UV) rays are known to cause skin cancer with prolonged exposure. They have higher energy than visible light and can damage DNA in skin cells, leading to mutations and cancer.

Key Concept: Refractive Index and Wavelength

d) Violet
[Solution Description]
The refractive index $\mu$ of a medium depends on the wavelength of light. Since $\mu$ increases with decreasing wavelength, violet light (with the smallest wavelength) has the highest refractive index.

Key Concept: Danger Signal

c) Red light does not scatter as much and can be seen from a greater distance
[Solution Description]
Red light has the longest wavelength in the visible spectrum, and according to Rayleigh scattering, the intensity of scattered light is inversely proportional to $\lambda^4$. Since red light scatters the least, it can travel longer distances without losing much intensity. This makes red light highly visible from far away, even in foggy conditions, hence its use in danger signals.

Key Concept: Earth's appearance from space, scattering phenomenon

b) Scattering of blue light by the atmosphere
[Solution Description]
From space, the blue color of Earth is due to Rayleigh scattering of sunlight by the Earth's atmosphere. Shorter wavelengths (blue/violet) are scattered more than longer wavelengths (red/orange). While violet is scattered even more, our eyes perceive blue better than violet. Hence, scattered blue light dominates the reflected sunlight reaching the astronaut's eyes, making Earth appear blue.

Key Concept: Frequency-wavelength relationship, speed of light

c) 4000-7000 Å
[Solution Description]
Using the formula $\lambda = \frac{c}{\nu}$, where $\nu$ is frequency. For $\nu_1 = 4.3 \times 10^{14} \, \text{Hz}$: $\lambda_1 = \frac{3 \times 10^8}{4.3 \times 10^{14}} = 6.98 \times 10^{-7} \, \text{m} = 6980 \, \text{Å}$. For $\nu_2 = 7.5 \times 10^{14} \, \text{Hz}$: $\lambda_2 = \frac{3 \times 10^8}{7.5 \times 10^{14}} = 4 \times 10^{-7} \, \text{m} = 4000 \, \text{Å}$. So the range is approximately 4000-7000 Å.

Key Concept: Dispersion and Spectrum Formation

a) Both Assertion and Reason are true, and Reason is the correct explanation of Assertion.
[Solution Description]
The assertion states that dispersion occurs because the speed of light changes with wavelength in the prism. This is correct since different colors (wavelengths) of light travel at different speeds in the prism (due to varying refractive indices), leading to dispersion. The reason explains that the refractive index of the prism material varies with wavelength, which is also true and directly explains why the speed of light changes with wavelength. Thus, both Assertion and Reason are true, and Reason correctly explains the Assertion.

Key Concept: Dependence on Wavelength

c) $I \propto \frac{1}{\lambda^4}$
[Solution Description]
The intensity of scattered light follows Rayleigh's law: $I \propto \frac{1}{\lambda^4}$. This means shorter wavelengths (e.g., blue light) scatter much more intensely than longer wavelengths (e.g., red light), given that the scattering particles are smaller than the wavelength of light.

Key Concept: Electromagnetic spectrum, wavelength-frequency relation

c) Ultraviolet rays
[Solution Description]
To determine the region of the electromagnetic spectrum, we first calculate the wavelength using the formula:
$c = f \lambda \Rightarrow \lambda = \frac{c}{f}$.
Given $c = 3 \times 10^8 \, \text{m/s}$ and $f = 1.5 \times 10^{15} \, \text{Hz}$,
$\lambda = \frac{3 \times 10^8}{1.5 \times 10^{15}} = 2 \times 10^{-7} \, \text{m} = 200 \, \text{nm}$.
From the syllabus, ultraviolet rays range from 10 nm to 400 nm, so 200 nm falls in this range.

Key Concept: Scattering, Wavelength dependence

b) 5.8 times
[Solution Description]
According to Rayleigh's law, the intensity of scattered light $I$ is inversely proportional to the fourth power of the wavelength ($I \propto \frac{1}{\lambda^4}$).
Given:
Wavelength of blue light ($\lambda_b$) = $4500 \, \text{Å}$
Wavelength of red light ($\lambda_r$) = $7000 \, \text{Å}$
The ratio of scattering intensities is given by:
$\frac{I_b}{I_r} = \left( \frac{\lambda_r}{\lambda_b} \right)^4 = \left( \frac{7000}{4500} \right)^4$
First, simplify the fraction:
$\frac{7000}{4500} = \frac{7}{4.5} \approx 1.5556$
Now, raise it to the fourth power:
$(1.5556)^4 \approx 5.8$
Thus, blue light is scattered approximately 5.8 times more intensely than red light.

Key Concept: Concept: Scattering Intensity and Wavelength Relation

d) Shorter wavelengths scatter more, making the sun appear red.
[Solution Description]
The intensity of scattered light is inversely proportional to the fourth power of the wavelength ($I \propto \frac{1}{\lambda^4}$). At sunrise and sunset, sunlight passes through a thicker layer of the atmosphere. Shorter wavelengths (blue/violet) are scattered more due to their higher frequency, while longer wavelengths (red/orange) are scattered less. Thus, red light dominates the observed color of the sun.

Key Concept: Electromagnetic spectrum classification

b) X-rays
[Solution Description]
An electromagnetic wave with a frequency of $2 \times 10^{17}$ Hz lies in the high-energy region of the electromagnetic spectrum. Such extremely high frequencies correspond to X-rays. Therefore, this wave belongs to the X-ray category, which is known for its strong penetrating power and medical imaging applications

Key Concept: Dispersion, Angle of Deviation

b) 1.2 degrees
[Solution Description]
The angle of deviation for a prism is given by $\delta = i + e - A$, where $A = 60^\circ$. First, find the angles of emergence $e$ for red and violet using $\mu = \frac{\sin \left(\frac{A + \delta}{2}\right)}{\sin \left(\frac{A}{2}\right)}$. However, for small angles, we can approximate $\delta = (\mu - 1)A$. Thus, $\delta_{\text{red}} = (1.51 - 1) \times 60^\circ = 30.6^\circ$ and $\delta_{\text{violet}} = (1.53 - 1) \times 60^\circ = 31.8^\circ$. The difference is $31.8^\circ - 30.6^\circ = 1.2^\circ$.

Key Concept: Scattering, Size of particles

c) The large particles in clouds scatter all colours equally
[Solution Description]
The white appearance of clouds is due to Mie scattering, where the size of water droplets and dust particles in clouds is larger than the wavelength of visible light. These particles scatter all wavelengths of sunlight equally, resulting in the combined white light reaching our eyes. During midday, the sun's light is most intense and directly incident, enhancing this effect.

Key Concept: Dispersion and Deviation in a Prism

a) Both Assertion and Reason are true, and Reason is the correct explanation of Assertion.
[Solution Description]
The assertion states that violet light undergoes greater deviation than red light when white light passes through a prism. This is true because the refractive index ($\mu$) of the prism material is wavelength-dependent, being higher for shorter wavelengths (violet light) and lower for longer wavelengths (red light). A higher refractive index causes greater bending (deviation) of light.
The reason correctly explains the assertion since the difference in deviation arises due to the variation in the refractive index with wavelength. Therefore, both the assertion and reason are true, and the reason is the correct explanation of the assertion.

Key Concept: Dependence of deviation on wavelength

a) Both Assertion and Reason are true, and Reason is the correct explanation of Assertion.
[Solution Description]
According to the given syllabus, the speed of light in glass depends on its wavelength. Violet light has a shorter wavelength \(\lambda = 4000 \, \text{Å}\) compared to red light \(\lambda = 8000 \, \text{Å}\)
The refractive index $\mu$ is inversely proportional to the speed of light in the medium, so $\mu$ is higher for violet light and lower for red light. This leads to a greater angle of refraction $r$ for red light compared to violet light, resulting in a larger deviation $\delta$ for violet light \(\delta = i - r\)
. Therefore, both Assertion (A) and Reason (R) are true, and Reason correctly explains Assertion.

Key Concept: Dispersion of Light

b) Difference in the speed of light for different colours
[Solution Description]
Dispersion occurs because different colours of light travel at different speeds in glass, causing them to refract (bend) by different amounts. Violet light slows down the most and bends the most, while red light slows down the least and bends the least.

Key Concept: Speed in Glass

d) Red
[Solution Description] In glass, the speed of light varies with wavelength. Red light has the longest wavelength and thus travels faster compared to other colours in glass. From the given syllabus, we know $\mu_{\text{red}} > \mu_{\text{violet}}$, so red light has the maximum speed in glass.

Key Concept: White Colour of Clouds

a) Both Assertion and Reason are true, and Reason is the correct explanation of Assertion.
[Solution Description]
The assertion states that clouds appear white, which is true. The reason explains that dust particles and water molecules in clouds are larger than the wavelength of visible light, causing them to scatter all colours of incident white light equally. This is also true and correctly explains why clouds appear white. Hence, both the assertion and reason are true, and the reason is the correct explanation of the assertion.

Key Concept: Speed of light in medium

c) Red light travels faster
[Solution Description]
The refractive index $\mu$ is inversely proportional to the speed of light in the medium: $\mu = c/v$. Since $\mu_v > \mu_r$, it implies that the speed of violet light $v_v$ is less than the speed of red light $v_r$ ($v_v < v_r$). Thus, red light travels faster than violet light in glass.

Key Concept: Cause of Dispersion, Refractive Index

d) Assertion is false, but Reason is true.
[Solution Description]
The assertion states that the angular dispersion $\theta$ depends only on the material of the prism and not on the angle of incidence. However, this is false because the angular dispersion also depends on the angle of incidence, as it affects the deviation angles $\delta_v$ and $\delta_r$.
The reason states that the refractive index $\mu$ varies with wavelength, causing different colors to deviate at different angles. This is true, as the variation in $\mu$ with wavelength (dispersion) is the fundamental cause of the separation of colors.
Therefore, the Assertion is false, but the Reason is true.

Key Concept: Gamma Rays

b) Gamma rays are chemically more active than ultraviolet radiations.
[Solution Description]
Gamma rays are highly energetic electromagnetic waves with wavelengths less than $0.1 \, \text{\AA}$. They can penetrate thick metallic sheets, cause fluorescence in materials like zinc sulphide, and are used in medical treatments such as radiotherapy. However, they are not chemically more active than ultraviolet radiations, which is a property specific to X-rays.

Key Concept: Electromagnetic Spectrum, Harmful Effects

c) Ultraviolet rays
[Solution Description]
Ultraviolet (UV) radiation is known for its ionizing properties and is a leading cause of skin cancer due to its ability to damage DNA in skin cells.

Key Concept: Deviation Produced by a Triangular Prism

a) Both Assertion and Reason are true, and Reason is the correct explanation of Assertion.
[Solution Description]
The assertion states that the angle of deviation ($\delta$) depends on the wavelength of light, which is true because different wavelengths (colors) of light experience different deviations due to dispersion. The reason states that the refractive index of the prism material varies with the wavelength of light, which is also true as refractive index is inversely proportional to wavelength (Cauchy's formula). Moreover, the reason correctly explains why the angle of deviation depends on the wavelength—it is directly due to the variation in refractive index with wavelength.

Key Concept: Properties and Applications of Infrared Radiations

b) Both Assertion and Reason are true, but Reason is NOT the correct explanation of Assertion.
[Solution Description]
The assertion states that infrared radiations are used in remote controls because they are not visible to the human eye, which is correct as per the syllabus. The reason given is also true as infrared radiations indeed have wavelengths in the specified range and do not affect ordinary photographic films. However, the reason does not explain why infrared radiations are used in remote controls; it merely states additional properties. Thus, both statements are true, but the reason does not correctly explain the assertion.

Key Concept: Sunrise/Sunset Color

d) Because blue light is scattered away, leaving red light
[Solution Description]
During sunrise and sunset, sunlight travels a longer distance through the atmosphere. Shorter wavelengths (blue/violet) are scattered away, leaving mostly longer wavelengths (red/orange) to reach our eyes, making the sun appear red.

Key Concept: Scattering Intensity, Inverse Fourth Power Law

d) Decreases by a factor of 16
[Solution Description]
According to Rayleigh's law, the intensity of scattered light $I$ is inversely proportional to the fourth power of the wavelength $\lambda$ ($I \propto \frac{1}{\lambda^4}$). If the wavelength $\lambda$ is doubled, the intensity $I$ decreases by a factor of 16 because $(2\lambda)^4 = 16\lambda^4$.

Key Concept: Use of red light for danger signal

c) Red light is least scattered by the atmosphere
[Solution Description]
Red light has the longest wavelength in the visible spectrum and hence is scattered the least by atmospheric particles. As a result, red light can travel longer distances without losing intensity, making it visible even in foggy conditions or from far away.

Key Concept: Colours and Wavelength

d) Red
[Solution Description]
According to the given information, red light has the longest wavelength of $8000 A$ (or $800 nm$) in the spectrum of white light. The order of wavelengths from longest to shortest is Red, Orange, Yellow, Green, Blue, Indigo, Violet.

Key Concept: Application of Scattering

b) Red light can penetrate dust and fog better than other colors
[Solution Description]
Red light has the longest wavelength in the visible spectrum, so it is scattered the least by atmospheric particles. This allows red light to travel farther without losing intensity, making it ideal for danger signals as it remains visible from long distances even in foggy conditions.

Key Concept: Scattering of Light, Red Sun at Sunrise/Sunset

d) Blue light is scattered more, leaving mostly red light to reach the observer
[Solution Description]
During sunrise or sunset, sunlight travels a longer distance through the Earth's atmosphere. Since blue light (shorter wavelength) is scattered much more than red light (longer wavelength), most of the blue light is scattered away, leaving mainly red light to reach the observer's eyes. This makes the sun appear red.

Key Concept: Dispersion of White Light through a Prism

c) Violet
[Solution Description]
According to the syllabus, violet light has the shortest wavelength ($\lambda = 400\, \text{nm}$) and slows down the most in the prism, causing it to deviate the most. The order of deviation from most to least is VIBGYOR (Violet, Indigo, Blue, Green, Yellow, Orange, Red).

Key Concept: Order of Colours

a) Both Assertion and Reason are true, and Reason is the correct explanation of Assertion.
[Solution Description]
The assertion states that violet light is deviated the most when white light passes through a prism, which is true because the angle of deviation ($\delta$) is greater for shorter wavelengths. The reason given is that violet light has the shortest wavelength among the visible spectrum, which is also true and directly explains why violet light is deviated the most. Therefore, both the assertion and reason are true, and the reason correctly explains the assertion.

Key Concept: Electromagnetic Spectrum

a) Gamma rays
[Solution Description]
From the syllabus:
- Gamma rays have the highest frequency ($> 10^{19} \, \text{Hz}$).
- X-rays have frequencies from $3 \times 10^{16} \, \text{Hz}$ to $3 \times 10^{19} \, \text{Hz}$.
- Ultraviolet rays have lower frequencies than X-rays.
- Infrared radiations have even lower frequencies.
Therefore, gamma rays have the highest frequency.

Key Concept: Spectrum Formation

c) Violet, Indigo, Blue, Green, Yellow, Orange, Red
[Solution Description]
The order of colours in the spectrum is VIBGYOR: Violet, Indigo, Blue, Green, Yellow, Orange, and Red. Starting from the base of the prism, the first colour is violet, followed by indigo, blue, green, yellow, orange, and red.

Key Concept: Scattering of light, Wavelength dependence

c) Violet
[Solution Description] According to Rayleigh scattering, the intensity of scattered light is inversely proportional to the fourth power of the wavelength ($I \propto \frac{1}{\lambda^4}$). Among the given wavelengths, violet has the smallest wavelength ($400 \, \text{nm}$), so it is scattered the most. Red has the largest wavelength ($650 \, \text{nm}$), so it is scattered the least.

Key Concept: Properties, Comparison with other radiations

d) Different wavelength-dependent absorption mechanisms in materials
[Solution Description]
The difference arises from how these radiations interact with atomic structures. Infrared has longer wavelengths $ (8000\, \text{Å} \text{ to } 10^7\, \text{Å}) $
which match vibrational modes in glass molecules, leading to absorption. Rock-salt's crystalline structure doesn't have matching vibrational modes for IR. Ultraviolet has much shorter wavelengths

$ (100\, \text{Å} \text{ to } 4000\, \text{Å}) $
that can be absorbed by electron transitions in rock-salt but pass through glass.

Key Concept: Electromagnetic Spectrum, Wavelength-Frequency Relationship

c) 200 nm
[Solution Description]
Given the frequency $f = 1.5 \times 10^{15}$ Hz, and the speed of light $c = 3 \times 10^8$ m/s, we can use the formula:
$c = f \lambda$
Solving for $\lambda$:
$\lambda = \frac{c}{f} = \frac{3 \times 10^8}{1.5 \times 10^{15}} = 2 \times 10^{-7}$ meters
Convert meters to nanometers:
$2 \times 10^{-7} \, \text{m} = 200 \, \text{nm}$

Key Concept: Applications of scattering, Danger signals

c) Red light is scattered much less than violet light
[Solution Description]
Red light has the longest wavelength in the visible spectrum, so it is scattered the least according to Rayleigh's scattering law ($I \propto \frac{1}{\lambda^4}$). This makes red light more visible over long distances and in poor weather conditions like fog, where scattering is significant. Violet light, having a shorter wavelength, would scatter too much and not be as clearly visible.

Key Concept: Color Scattering

c) Because blue light is scattered much more than red light
[Solution Description]
The sky appears blue because shorter wavelengths of light (blue/violet) are scattered more strongly by atmospheric molecules compared to longer wavelengths (red). When sunlight passes through the atmosphere, blue light is scattered in all directions due to Rayleigh scattering and reaches our eyes from various parts of the sky, making it appear blue.

Key Concept: Intensity of Scattering, Wavelength Dependence

b) About 9 times more scattering for violet
[Solution Description]
According to Rayleigh's scattering law, $I \propto \frac{1}{\lambda^4}$.
Step 1: Given wavelengths - $\lambda_v = 400$ nm (violet), $\lambda_r = 700$ nm (red).
Step 2: Calculate the ratio: $\frac{I_v}{I_r} = \left(\frac{\lambda_r}{\lambda_v}\right)^4 = \left(\frac{700}{400}\right)^4$.
Step 3: Compute the value: $\left(1.75\right)^4 \approx 9.38$.

Key Concept: Red colour of sun at sunrise and sunset

a) Both Assertion and Reason are true, and Reason is the correct explanation of Assertion.
[Solution Description]
Both Assertion and Reason are true, and Reason correctly explains why the sun appears red. During sunrise/sunset, sunlight travels a longer path through the atmosphere, causing more scattering of shorter wavelengths (blue light). Red light scatters less due to its longer wavelength, making the sun appear red. The inverse relationship $I \propto \frac{1}{\lambda^4}$ supports this explanation.

Key Concept: Dispersion and Wavelength

a) Violet
[Solution Description]
According to the given data, violet light has the shortest wavelength (4000 Å) and highest frequency ($7.5 \times 10^{14}$ Hz). Frequency is inversely proportional to wavelength, so shorter wavelengths correspond to higher frequencies.

Key Concept: Particle size effects, Scattering in clouds

d) The particles are too large for wavelength-dependent scattering
[Solution Description] When the scattering particles (e.g., water droplets in clouds) are much larger than the wavelength of light, they scatter all wavelengths almost equally (Mie scattering). This results in the incident white light being scattered uniformly, making clouds appear white.

Key Concept: Factors affecting deviation

b) It increases
[Solution Description]
The total angle of deviation $\delta = \delta_1 + \delta_2$, where $\delta_1$ and $\delta_2$ are deviations at the first and second surfaces respectively. As the angle of the prism $A$ increases, the angle of incidence at the second surface also increases, leading to an increase in $\delta_2$. Thus, the total deviation increases with an increase in the angle of the prism.

Key Concept: Sunrise/Sunset Color

d) Because blue light is scattered away, leaving mostly red light
[Solution Description]
At sunrise and sunset, sunlight travels a longer distance through the atmosphere. Since blue light is scattered away more due to its short wavelength, mostly red light (which scatters the least) reaches our eyes, making the sun appear red.

Key Concept: Electromagnetic Spectrum - Applications

d) Ultraviolet rays
[Solution Description]
Ultraviolet rays are commonly used for sterilization because they can destroy bacteria and viruses by damaging their DNA. Other radiations like infrared, microwaves, or radio waves do not possess this property.

Key Concept: Intensity of Scattering

a) Both Assertion and Reason are true, and Reason is the correct explanation of Assertion.
[Solution Description]
According to Rayleigh scattering, the intensity of scattered light follows $I \propto \frac{1}{\lambda^4}$. Since violet light has a shorter wavelength ($4000 \, \text{A}$) compared to red light ($8000 \, \text{A}$), it scatters more strongly. Thus, the Assertion (A) is true. The Reason (R) correctly states the dependence of scattering intensity on wavelength and explains why violet light scatters more. Therefore, both Assertion and Reason are true, and Reason is the correct explanation of Assertion.

Key Concept: Dispersion of White Light

d) Violet
[Solution Description]
Dispersion occurs because different colors travel at different speeds in the prism. Violet light has the shortest wavelength and highest frequency, causing it to deviate the most. Red light deviates the least. Thus, the correct answer is violet.

Key Concept: Rayleigh Scattering, Wavelength Dependence

c) Shorter wavelengths of light are scattered more by atmospheric particles
[Solution Description]
The blue color of the sky is due to Rayleigh scattering, where shorter wavelengths (blue light) are scattered more than longer wavelengths (red light). When sunlight passes through the atmosphere, the air molecules scatter blue light more effectively in all directions, making the sky appear blue to an observer on Earth.

Key Concept: Order of Colours, Wavelength and Frequency

a) Both Assertion and Reason are true, and Reason is the correct explanation of Assertion.
[Solution Description]
The assertion states that green light has a higher frequency than red light in the visible spectrum. This is correct because from the syllabus, we know the frequency decreases from violet to red. The reason states that green has a shorter wavelength than red, which is true, and mentions the inverse relationship between frequency and wavelength ($f \propto \frac{1}{\lambda}$), which correctly explains why green's frequency is higher. Hence, both are true, and the reason correctly explains the assertion.

Key Concept: Minimum Deviation Condition

c) Light travels parallel to the base of the prism
[Solution Description]
When a light ray passes through a prism under the condition of minimum deviation, the path of light inside the prism becomes symmetric. This means the angle of incidence equals the angle of emergence, and the light travels parallel to the base of an equilateral prism.

Key Concept: Deviation of Light Colors

d) Violet
[Solution Description]
The deviation of light depends on its speed in the glass. Violet light has the minimum speed in glass and hence the highest refractive index, resulting in the greatest angle of deviation.

Key Concept: Refractive Index and Wavelength

c) $\mu_v > \mu_r$
[Solution Description]
The refractive index $\mu$ is inversely related to the speed of light in the medium. Since violet light has the lowest speed in glass, its refractive index ($\mu_v$) is the highest. Conversely, red light has the highest speed, so its refractive index ($\mu_r$) is the lowest. Therefore, the statement that $\mu_v > \mu_r$ is correct.

Key Concept: Scattering of Light

d) All wavelengths of visible light are scattered equally by large particles in clouds.
[Solution Description]
The white colour of clouds is due to the scattering of sunlight by dust particles and water droplets in the clouds. These particles have sizes larger than the wavelength of visible light, causing all colours of the incident white light to be scattered equally. This uniform scattering results in the combined light reaching our eyes appearing white.

Key Concept: Wavelength Range

c) 400 nm to 800 nm
[Solution Description] According to the syllabus, the spectrum of white light ranges from $4000 \, \text{A}$ (violet) to $8000 \, \text{A}$ (red). Since $1 \, \text{nm} = 10 \, \text{A}$, converting gives $400 \, \text{nm}$ to $800 \, \text{nm}$.

Key Concept: Wavelength Range of Visible Light

c) 400 nm to 800 nm
[Solution Description]
According to the syllabus, the wavelength range for visible light is 4000 A to 8000 A (or $\lambda = 400 \, \text{nm} \, \text{to} \, 800 \, \text{nm}$).

Key Concept: Scattering of Light and Red Colour of Sun at Sunrise/Sunset

a) Both Assertion and Reason are true, and Reason is the correct explanation of Assertion.
[Solution Description]
The assertion states that the sun appears red during sunrise and sunset due to more scattering of blue light compared to red light. This is correct because sunlight travels a longer distance through the atmosphere at these times, causing shorter-wavelength blue light to scatter more and longer-wavelength red light to dominate. The reason correctly explains this by stating the inverse fourth power dependence of scattering on wavelength, which means shorter wavelengths (blue) scatter much more than longer wavelengths (red). Therefore, both the assertion and reason are true, and the reason correctly explains the assertion.

Key Concept: Composition of White Light, Wavelength and Frequency of Colors

a) Both Assertion and Reason are true, and Reason is the correct explanation of Assertion.
[Solution Description]
According to the given syllabus, violet light has the shortest wavelength (400 nm) and highest frequency ($7.5 \times 10^{14}$ Hz), while red light has the longest wavelength (800 nm) and lowest frequency ($3.75 \times 10^{14}$ Hz). This confirms that Assertion (A) is true. The Reason (R) correctly explains why violet light has a higher frequency because frequency ($f$) and wavelength ($\lambda$) are inversely related by the speed of light ($c$): $c = f \lambda$. Thus, both Assertion and Reason are true, and Reason is the correct explanation of Assertion.

Key Concept: Path length, Scattering phenomena

b) Shorter atmospheric path reduces selective scattering
[Solution Description]
At noon, sunlight travels a shorter path through the atmosphere compared to sunrise/sunset. This means less scattering occurs, allowing all wavelengths (including blue/violet) to reach the observer almost equally. Thus, the Sun appears white due to minimal wavelength-selective scattering.

Key Concept: Scattering of Light

a) Both Assertion and Reason are true, and Reason is the correct explanation of Assertion.
[Solution Description]
The Assertion states that the sky appears black in the absence of an atmosphere due to no scattering of sunlight. This is true because, without an atmosphere, there are no air molecules or particles to scatter sunlight, so no scattered light reaches our eyes, making the sky appear black.
The Reason states that scattering requires the presence of particles or molecules in the atmosphere to deflect light, which is also true and directly explains why the sky would be black without an atmosphere—no scattering occurs without these particles.
Therefore, both Assertion and Reason are true, and Reason correctly explains the Assertion.

Key Concept: Scattering of Light, Wavelength Dependence

a) Both Assertion and Reason are true, and Reason is the correct explanation of Assertion.
[Solution Description]
The Assertion states that the sky appears blue because blue light is scattered more than other colors due to its shorter wavelength. This is correct because blue light has a shorter wavelength ($\lambda_{\text{blue}} \approx 450 \, \text{nm}$) compared to red light ($\lambda_{\text{red}} \approx 700 \, \text{nm}$). According to Rayleigh's scattering law, $I \propto \frac{1}{\lambda^4}$, shorter wavelengths are scattered much more intensely. Thus, blue light dominates the scattered light reaching our eyes.
The Reason correctly explains Rayleigh's scattering law, which quantitatively describes why shorter wavelengths like blue light are scattered more. Therefore, both Assertion and Reason are true, and the Reason correctly explains the Assertion.

Key Concept: Scattering of Light

b) Red light has a shorter wavelength and scatters less than blue light.
[Solution Description]
During sunrise and sunset, sunlight travels through a longer distance in the atmosphere. Blue light is scattered more due to its shorter wavelength, leaving predominantly red light to reach the observer. This makes the sun appear red.

Key Concept: Properties of Electromagnetic Waves

d) They are transverse waves
[Solution Description]
Electromagnetic waves are transverse in nature, meaning their oscillations are perpendicular to the direction of propagation. They are not deflected by electric or magnetic fields.

Key Concept: Gamma Radiation

d) Gamma radiations
[Solution Description]
Gamma radiations have the shortest wavelength, less than $0.1 \, \text{A}$ ($0.01 \, \text{nm}$). They are highly energetic and emitted during nuclear transitions. Other radiations like X-rays, ultraviolet, and infrared have longer wavelengths.

Key Concept: Scattering of Light

a) Both Assertion and Reason are true, and Reason is the correct explanation of Assertion.
[Solution Description]
The Assertion states that the sky appears blue due to greater scattering of blue light, which is true according to Rayleigh's scattering law. The Reason states the mathematical relationship between scattering intensity and wavelength ($I \propto \frac{1}{\lambda^4}$), which is also correct. Moreover, the Reason correctly explains why blue light (with shorter wavelength $\approx 4000 \, \text{Å}$) scatters more than other colors, leading to the blue appearance of the sky.

Key Concept: Cloud Color, Scattering by Large Particles

b) All wavelengths are scattered equally by large particles in clouds.
[Solution Description]
Clouds contain water droplets and dust particles larger than the wavelength of visible light, causing all wavelengths to scatter equally (Mie scattering), resulting in white light.

Key Concept: Dispersion and Angle of Deviation

d) Violet
[Solution Description]
The angle of deviation $\delta = i - r$. Since violet light has the smallest angle of refraction $r_v$ due to its highest refractive index ($\mu_v > \mu_r$), it experiences the maximum angle of deviation, i.e., $\delta_v > \delta_r$.

Key Concept: Wavelength and Deviation

a) Both Assertion and Reason are true, and Reason is the correct explanation of Assertion.
[Solution Description]
The assertion states that violet light is deviated more than red light when white light passes through a prism. This is true because violet light has the shortest wavelength ($\lambda = 400 \, \text{nm}$) among visible colors, and according to the given syllabus, shorter wavelengths experience greater deviation due to higher refractive index.
The reason states that the refractive index of glass increases with a decrease in wavelength. This is also true, as mentioned in the syllabus. Moreover, the reason correctly explains why violet light is deviated more—because its shorter wavelength results in a higher refractive index, causing greater bending.
Therefore, both assertion and reason are true, and the reason correctly explains the assertion.

Key Concept: Wavelength dependence, Intensity comparison

c) 16 times
[Solution Description]
Using Rayleigh's scattering formula, the intensity $I$ is inversely proportional to $\lambda^4$.
Given:
Wavelength of violet light ($\lambda_v$) = $4000 \, \text{Å}$
Wavelength of red light ($\lambda_r$) = $8000 \, \text{Å}$
The ratio of scattered intensities is:
$\frac{I_v}{I_r} = \left( \frac{\lambda_r}{\lambda_v} \right)^4 = \left( \frac{8000}{4000} \right)^4 = (2)^4 = 16$
Therefore, violet light is scattered 16 times more intensely than red light.

Key Concept: Intensity of Scattering

c) Inversely proportional to the fourth power of the wavelength
[Solution Description]
As per Rayleigh's law of scattering, the intensity $I$ of scattered light is inversely proportional to the fourth power of the wavelength $\lambda$. Mathematically, $I \propto \frac{1}{\lambda^4}$. Therefore, shorter wavelengths (like violet) are scattered more than longer wavelengths (like red).

Key Concept: Wavelength calculation

c) 1 cm
[Solution Description]
Using the formula \(c = f \times \lambda\), rearrange to find \(\lambda\):
$\lambda = \frac{c}{f} = \frac{3 \times 10^8}{3 \times 10^{10}} = 0.01 \,\text{m} = 1 \,\text{cm}$.
From the syllabus, microwaves have wavelengths ranging from 1 mm to 10 m, which includes 1 cm.

Key Concept: Scattering of Light, Wavelength Dependence

c) Shorter wavelengths (blue light) scatter more in the atmosphere.
[Solution Description]
The sky appears blue due to Rayleigh scattering, where shorter wavelengths (blue/violet) are scattered more than longer wavelengths (red/orange). Since violet light is scattered even more than blue, but our eyes are more sensitive to blue and the sun emits less violet light, the sky appears blue.

Key Concept: Blue colour of sky

c) Blue light is scattered more than other colors
[Solution Description]
The sky appears blue because sunlight is scattered by air molecules in the atmosphere. Blue light (shorter wavelength) is scattered much more than red light (longer wavelength). This scattered blue light reaches our eyes from all directions, making the sky appear blue.

Key Concept: Wavelength dependence, Critical angle

[Solution Description]

c) It refracts into air at angle $ \mu_{\text{red}}$ (from $\mu \propto 1/\lambda$ approximation).
2. Thus, $\theta_{c,\text{red}} = \sin^{-1}(1/\mu_{\text{red}}) > \theta_{c,\text{blue}}$.
3. At $\theta_c$ for blue light ($=\theta_{c,\text{blue}}$):
- Blue light grazes the interface.
- Red light has $\theta_i = \theta_{c,\text{blue}} < \theta_{c,\text{red}}$, so it refracts out.

Key Concept: Relationship Between Wavelength and Frequency

c) $5 \times 10^{14} \, \text{Hz}$
[Solution Description]
Given: $\lambda = 600 \, \text{nm} = 600 \times 10^{-9} \, \text{m}$ and $c = 3 \times 10^8 \, \text{m/s}$. Using $c = f \times \lambda$, we can find $f$ as follows:
$f = \frac{c}{\lambda} = \frac{3 \times 10^8}{600 \times 10^{-9}} = 5 \times 10^{14} \, \text{Hz}.$

Key Concept: Electromagnetic Spectrum and Wavelength

a) Both Assertion and Reason are true, and Reason is the correct explanation of Assertion.
[Solution Description]
According to the given syllabus, radio waves have wavelengths above $10^{11}$ A ($\lambda > 10 \, \text{m}$), whereas microwaves have wavelengths in the range $10^7$ A to $10^{11}$ A ($1 \, \text{mm} \, \text{to} \, 10 \, \text{m}$). Therefore, the Assertion that radio waves have longer wavelengths than microwaves is true. The Reason provides the correct numerical range for both types of waves, which justifies the Assertion. Hence, both Assertion and Reason are true, and Reason correctly explains the Assertion.

Key Concept: Scattering Phenomenon

c) Rayleigh scattering
[Solution Description]
The red appearance of the Sun at sunrise and sunset is explained by Rayleigh scattering, which causes shorter wavelengths (blue/violet) to be preferentially scattered, leaving longer wavelengths (red/orange).

Key Concept: Red Colour of the Sun at Sunrise and Sunset

a) Both Assertion and Reason are true, and Reason is the correct explanation of Assertion.
[Solution Description]
The assertion states that the sun appears red at sunrise and sunset because blue light is scattered more than red light. This is true because during these times, sunlight passes through a thicker layer of Earth's atmosphere, leading to increased scattering of shorter wavelengths (blue light). The reason states that blue light has a shorter wavelength than red light, causing it to scatter more due to Rayleigh scattering, which follows $I \propto \frac{1}{\lambda^4}$. Both statements are correct, and the Reason correctly explains the Assertion based on the inverse fourth power law of wavelength dependency in scattering.

Key Concept: Deviation and Wavelength

d) Violet
[Solution Description]
The deviation $\delta$ depends on the refractive index $\mu$. Since $\mu_{\text{violet}} > \mu_{\text{red}}$, violet light is refracted more, resulting in greater deviation.

Key Concept: Scattering Applications, Danger Signal

b) Red light is scattered the least and can penetrate further
[Solution Description]
Red light has the longest wavelength in the visible spectrum. According to Rayleigh scattering, the intensity of scattered light is inversely proportional to the fourth power of the wavelength, so red light is scattered the least. This allows red light to travel farther without significant attenuation, making it ideal for danger signals.

Key Concept: Dependence of deviation on wavelength

d) Violet
[Solution Description]
The angle of deviation $\delta$ for a prism is given by $\delta = i - r$. Since refractive index $\mu$ is higher for violet light ($\mu_v > \mu_r$), the angle of refraction $r$ is smaller for violet light ($r_v < r_r$) for a given angle of incidence $i$. Therefore, the deviation is larger for violet light. Hence, violet light experiences the maximum deviation.

Key Concept: Wavelength Range

b) 400 nm to 700 nm
[Solution Description]
The visible light spectrum ranges from approximately 400 nm (violet) to 700 nm (red). This range corresponds to the colors observed when white light is dispersed by a prism.

Key Concept: Wavelength dependence of scattering

d) Shorter wavelengths are scattered more than longer wavelengths
[Solution Description]
Scattering of light depends on its wavelength, with shorter wavelengths being scattered more than longer wavelengths. This is described by the relationship $I \propto \frac{1}{\lambda^4}$, where $I$ is the intensity of scattered light and $\lambda$ is the wavelength. Since red light has a longer wavelength, it is scattered much less than blue light.

Key Concept: Dispersion cause

b) Change in speed of light with wavelength
[Solution Description]
Dispersion occurs because different colors of light have different speeds in glass, causing them to refract at different angles. Violet deviates the most while red deviates the least due to their respective wavelengths.

Key Concept: Scattering, Danger Signal

c) Red light scatters less and travels farther
[Solution Description]
Despite violet light scattering more due to its shorter wavelength, red light is preferred for danger signals because it is scattered the least ($I \propto \frac{1}{\lambda^4}$). Red light's longer wavelength allows it to travel farther with minimal scattering, making it clearly visible even in fog or low-visibility conditions.

Key Concept: Spectrum Formation, Prism Angle

c) The spectrum widens.
[Solution Description]
Increasing the prism angle increases the angle between the two refracting surfaces, which enhances the dispersion of light. This results in a wider spectrum with greater separation between the colors.

Key Concept: Wavelength and Frequency of Colours

d) Red
[Solution Description]
According to the given table, red light has a wavelength of $8000 \, \text{\AA}$, which is the longest among all colours in white light. The wavelength decreases from red to violet.

Key Concept: Electromagnetic Spectrum Classification

b) Between 400 nm and 800 nm
[Solution Description]
The visible spectrum lies between the red and violet regions, with a wavelength range of 400 nm to 800 nm.

Key Concept: Wavelength dependence, Scattering efficiency

b) They would appear blue
[Solution Description]
If the size of water droplets were smaller than the wavelength of visible light, Rayleigh scattering would dominate instead of Mie scattering. Rayleigh scattering is more efficient for shorter wavelengths (blue light), so the clouds would appear bluish rather than white. This is because smaller particles scatter blue light more than red light.

Key Concept: Properties of infrared radiations, wavelength comparison

b) Both Assertion and Reason are true, but Reason is NOT the correct explanation of Assertion.
[Solution Description]
Assertion (A) states that infrared radiations are used for therapeutic purposes due to their strong heating effect, which is correct as per the syllabus. Reason (R) correctly mentions the wavelength range of infrared radiations ($8000 \, \text{\AA}$ to $10^7 \, \text{\AA}$) and compares it with visible light. However, the reason does not directly explain why infrared radiations are used for therapeutic purposes; it merely describes a property of infrared radiations. Thus, both statements are true, but Reason is not the correct explanation of Assertion.

Key Concept: Wavelength and Dispersion in White Light

a) Both Assertion and Reason are true, and Reason is the correct explanation of Assertion.
[Solution Description]
From the syllabus, we know that the wavelength of violet light is $4000\,Å$, which is the shortest in the visible spectrum. When white light passes through a prism, the speed of light decreases more for shorter wavelengths (violet) compared to longer wavelengths (red). This means the refractive index ($\mu$) of glass is higher for violet light than for red light ($\mu_{violet} > \mu_{red}$). The angle of deviation $\delta$ depends on the refractive index, with higher refractive indices causing greater deviation. Therefore, violet light, having the highest refractive index, is deviated the most. The reason correctly explains why violet light is deviated the most, as it directly links the shorter wavelength to the higher refractive index and thus the greater deviation.

Key Concept: Intensity of Scattering

b) It is scattered the least and travels far.
[Solution Description]
Red light has the longest wavelength in the visible spectrum ($\lambda = 8000 \, \text{A}$), and scattering intensity follows $I \propto \frac{1}{\lambda^4}$. Therefore, red light is scattered the least, allowing it to travel farther without significant attenuation, making it ideal for danger signals that need high visibility over long distances.

Key Concept: Dispersion of White Light

a) Both Assertion and Reason are true, and Reason is the correct explanation of Assertion.
[Solution Description]
When white light passes through a prism, it undergoes dispersion due to the variation in refractive index with wavelength. Violet light has the shortest wavelength ($4000$ Å) and highest frequency ($7.5 \times 10^{14}$ Hz), which results in the highest deviation as $\delta = i - r$ and $r_{\text{violet}} < r_{\text{red}}$ (minimum angle of refraction for violet). Thus, both Assertion and Reason are true, and Reason correctly explains why violet light deviates the most.

Key Concept: Relation between Angles

b) $\delta = i + e - A$
[Solution Description]
The relationship between the angle of incidence $i$, angle of emergence $e$, angle of prism $A$, and angle of deviation $\delta$ is given by:
$\delta = i + e - A$
This formula shows that the total deviation is the sum of the angle of incidence and angle of emergence minus the angle of the prism.

Key Concept: Intensity of Scattering

a) Both Assertion and Reason are true, and Reason is the correct explanation of Assertion.
[Solution Description]
According to Rayleigh scattering, the intensity of scattered light ($I$) is given by $I \propto \frac{1}{\lambda^4}$. Since violet light has a shorter wavelength ($4000$ Å) compared to red light ($8000$ Å), it is scattered more. Therefore, both Assertion and Reason are true, and Reason correctly explains the Assertion.

Key Concept: Nature of White Light

b) It is polychromatic
[Solution Description]
Dispersion occurs because white light consists of multiple colours (polychromatic light). The prism separates these colours as they travel at different speeds and refract at different angles within the material.

Key Concept: Composition of Clouds

d) Equal scattering of all visible wavelengths by large cloud particles.
[Solution Description]
The primary reason for the white appearance of clouds is the presence of water droplets and ice crystals that are much larger than the wavelengths of visible light. These particles scatter all colours of sunlight equally (Mie scattering), resulting in the combined light appearing white to an observer.

Key Concept: Spectrum Formation

b) Violet, Indigo, Blue, Green, Yellow, Orange, Red
[Solution Description]
The spectrum order from the base of the prism is Violet, Indigo, Blue, Green, Yellow, Orange, Red (VIBGYOR). Therefore, the correct sequence is red appearing opposite to the base.

Key Concept: Scattering angle, Particle composition

c) They enhance equal scattering of all colours
[Solution Description]
Dust particles in clouds contribute to the Mie scattering process. Their size, being larger than the wavelength of visible light, ensures that all colours are scattered equally in all directions. Without these particles, the scattering efficiency would decrease, but as long as water droplets are of sufficient size, the clouds would still appear white.

Key Concept: Scattering and Wavelength Dependence

c) Shorter wavelengths (blue) are scattered more by air molecules
[Solution Description]
The blue color of the sky is due to Rayleigh scattering, where shorter wavelengths (blue/violet) are scattered more than longer wavelengths (red). Since sunlight passes through a greater thickness of the atmosphere when it travels obliquely (not directly overhead), blue light is scattered in all directions, making the sky appear blue.

Key Concept: Wavelength-Frequency Relation

d) 600 nm
[Solution Description]
Using the relation $c = f \times \lambda$, we can calculate the wavelength as follows:
$\lambda = \frac{c}{f} = \frac{3 \times 10^8 \, \text{m/s}}{5 \times 10^{14} \, \text{Hz}} = 6 \times 10^{-7} \, \text{m} = 600 \, \text{nm}$
This falls within the visible light range.

Key Concept: Refractive index and deviation

c) Violet
[Solution Description]
The refractive index ($\mu$) of glass depends on the wavelength of light. Since $\mu = \frac{\text{Speed of light in air}}{\text{Speed of light in glass}}$, and violet light travels the slowest in glass (shortest wavelength), it has the highest refractive index. Therefore, violet undergoes the most deviation.

Key Concept: Danger Signal

d) Red light scatters the least and can be seen from far away.
[Solution Description]
Red light has the longest wavelength in the visible spectrum, so it is scattered the least by atmospheric particles. This allows it to travel longer distances without losing intensity, making it ideal for danger signals.

Key Concept: Rayleigh Scattering Formula

c) 16 times
[Solution Description]
According to Rayleigh scattering, the intensity of scattered light ($I$) is inversely proportional to the fourth power of the wavelength ($\lambda$), i.e., $I \propto \frac{1}{\lambda^4}$. Given $\lambda_{violet} = 400$ nm and $\lambda_{red} = 800$ nm, we can compute the ratio of scattering as follows:
$\frac{I_{violet}}{I_{red}} = \left( \frac{\lambda_{red}}{\lambda_{violet}} \right)^4 = \left( \frac{800}{400} \right)^4 = 2^4 = 16$
Thus, violet light is scattered 16 times more than red light.

Key Concept: Electromagnetic Spectrum, Gamma Rays, X-rays

a) Both Assertion and Reason are true, and Reason is the correct explanation of Assertion.
[Solution Description]
Gamma rays and X-rays are both high-energy electromagnetic waves, but gamma rays have shorter wavelengths ($ 10^{19}$ Hz) compared to X-rays (0.1 Å to 100 Å). Due to their extremely short wavelength and high energy, gamma rays can penetrate dense materials like lead and are thus effective for sterilization. The reason correctly explains why gamma rays are used for sterilization by linking their high energy (due to highest frequency/shortest wavelength) to their application.

Key Concept: Dependence on Wavelength

d) Inversely proportional to the fourth power of the wavelength
[Solution Description]
The intensity of scattered light is inversely proportional to the fourth power of the wavelength of light ($I \propto \frac{1}{\lambda^4}$). This means shorter wavelengths (like blue or violet) are scattered more than longer wavelengths (like red).

Key Concept: Frequency-wavelength relation, overlapping regions

b) 300 nm to 700 nm (Visible light)
[Solution Description]
Calculate the wavelength: $\lambda = \frac{c}{f} = \frac{3 \times 10^8}{5 \times 10^{14}} = 6 \times 10^{-7} \, \text{m} = 600 \, \text{nm}$.
From the syllabus:
- Visible light ranges from 400 nm to 800 nm.
- Infrared starts beyond 800 nm but overlaps near the boundary.
Thus, 600 nm is in the visible light range with no overlap here.

Key Concept: Red colour of sun at sunrise and sunset

d) Blue light is scattered away, leaving mostly red light
[Solution Description]
During sunrise and sunset, sunlight travels the longest distance through the atmosphere. Blue light (short wavelength) is scattered more than red light (long wavelength). Most of the blue light is lost due to scattering, while the red light reaches our eyes almost unchanged. Thus, the sun appears red.

Key Concept: Speed of Light in Medium, Wavelength

b) 450 nm
[Solution Description]
The speed of light in the medium is 75% of its speed in air, so $\mu = \frac{c}{v} = \frac{1}{0.75} = \frac{4}{3}$. The frequency remains unchanged, so $\lambda_{\text{medium}} = \frac{\lambda_{\text{air}}}{\mu} = \frac{600 \, \text{nm}}{4/3} = 450 \, \text{nm}$.

Key Concept: Wavelength and Frequency Relationship, Refractive Index

a) Both Assertion and Reason are true, and Reason is the correct explanation of Assertion.
[Solution Description]
The assertion states that violet light is deviated more than red light when passing through a glass prism due to the refractive index of glass being higher for violet light compared to red light. This is indeed correct because, as per the syllabus, the refractive index $\mu_{\text{violet}} > \mu_{\text{red}}$ because the speed of violet light in glass is lower than that of red light, given by the formula:
$\mu = \frac{\text{Speed of light in air}}{\text{Speed of light in glass}}$
The reason provided is also true and correctly explains the assertion. According to the key points mentioned in the syllabus, the speed of light in glass decreases with decreasing wavelength, making violet light travel slower than red light in glass, thus increasing its refractive index. This causes violet light to deviate more than red light when passing through a prism.
Therefore, both the assertion and reason are true, and the reason correctly explains the assertion.

Key Concept: Ultraviolet Radiations, Visible Light

b) Visible, 428.57 nm
[Solution Description]
The frequency $7 \times 10^{14} \, \text{Hz}$ falls within the range of visible light ($4.3 \times 10^{14} \, \text{Hz}$ to $7.5 \times 10^{14} \, \text{Hz}$). To find the wavelength, use $c = \lambda f$. Rearranged, $\lambda = \frac{c}{f} = \frac{3 \times 10^8}{7 \times 10^{14}} = 4.2857 \times 10^{-7} \, \text{m} = 428.57 \, \text{nm}$, which is in the violet region of visible light.

Key Concept: Concept: Source of incident light on clouds

d) from the Sun
[Solution Description]
The white light scattered by clouds originates from the Sun, as the clouds scatter the incident white light from the Sun.

Key Concept: Refractive Index, Speed of Light

d) Red
[Solution Description]
The speed of light in a medium is inversely related to its refractive index. Red light has the lowest refractive index in glass, meaning it travels the fastest compared to other colors.

Key Concept: Prism Dispersion, Speed of Light

b) 1.015
[Solution Description]
The refractive index ($\mu$) is given by $\mu = \frac{c}{v}$, where $c$ is the speed of light in vacuum and $v$ is the speed in the medium. The ratio of refractive indices is $\frac{\mu_{\text{violet}}}{\mu_{\text{red}}} = \frac{c / v_{\text{violet}}}{c / v_{\text{red}}} = \frac{v_{\text{red}}}{v_{\text{violet}}} = \frac{1.97 \times 10^8}{1.94 \times 10^8} \approx 1.015$.

Key Concept: Deviation and Wavelength

d) Violet
[Solution Description]
The angle of deviation $\delta$ in a prism depends on the refractive index $(\mu)$
of the material for different colours. Since violet light has the shortest wavelength and highest refractive index in glass, it bends the most. The deviation is given by:
$\delta = i - r$
where r (angle of refraction) is smallest for violet light, resulting in the largest deviation $(\delta_{violet} > \delta_{red})$
Therefore, the correct answer is the option mentioning violet.

Key Concept: Scattering of Light, Applications

c) Because blue light is scattered more by air molecules
[Solution Description]
The blue colour of the sky is due to the scattering of sunlight by air molecules in the atmosphere. According to Rayleigh's law, shorter wavelengths (blue light) are scattered more than longer wavelengths (red light). When sunlight passes through the atmosphere, the blue light is scattered in all directions, making the sky appear blue to an observer looking away from the sun.

Key Concept: Refractive Index Dependence

c) Speed increases
[Solution Description]
The speed of light in a medium is inversely related to the refractive index ($v = c/\mu$). Since the refractive index decreases with increasing wavelength (as longer wavelengths like red light bend less), the speed of light in glass increases with increasing wavelength.

Key Concept: Wavelength and frequency relationship

b) 6 $\times$ 10$^{14}$ Hz
[Solution Description]
The relationship between speed (c), frequency (f), and wavelength \(\lambda\)
is given by:
\(c = f \times \lambda\)
Given:
Wavelength \(\lambda = 500 \text{ nm} = 500 \times 10^{-9} \text{ m}\).
Speed of light \(c = 3 \times 10^8 \,\text{m/s}\).
Rearranging the formula to find frequency:
$f = \frac{c}{\lambda} = \frac{3 \times 10^8}{500 \times 10^{-9}} = 6 \times 10^{14} \,\text{Hz}$

Key Concept: Refractive Index and Wavelength

d) Violet
[Solution Description]
The refractive index ($\mu$) of a medium is inversely proportional to the wavelength ($\lambda$) of light. From VIBGYOR, violet has the shortest wavelength (4000 $\text{A}$), so it has the highest refractive index. Hence, the correct answer is violet.

Key Concept: Assertion-Reason

a) Both Assertion and Reason are true, and Reason is the correct explanation of Assertion.
[Solution Description]
According to the given data, violet light has a wavelength of $4000 Å$ (shortest in the visible spectrum). This makes the Assertion true. The Reason states that violet light has the minimum speed in glass, leading to maximum deviation, which is also correct and directly explains why violet light has the shortest wavelength (since $\mu_{\text{violet}} > \mu_{\text{red}}$ due to higher refractive index for shorter wavelengths). Thus, both Assertion and Reason are true, and Reason correctly explains the Assertion.

Key Concept: Wavelength Range, Electromagnetic Spectrum

b) Both Assertion and Reason are true, but Reason is NOT the correct explanation of Assertion.
[Solution Description]
From the given syllabus, radio waves have wavelengths above $10^{11}$ Å (or $\lambda > 10 \, \text{m}$), which is indeed the longest among all electromagnetic waves. The assertion is true. The reason states that the frequency of radio waves is lower than microwaves, which is correct because radio waves have frequencies below $3 \times 10^7 \, \text{Hz}$, while microwaves range from $3 \times 10^{11} \, \text{Hz}$ to $3 \times 10^7 \, \text{Hz}$. However, the reason does not explain why radio waves have the longest wavelength; it only describes the relationship between frequency and wavelength. Thus, both Assertion and Reason are true, but Reason is NOT the correct explanation of Assertion.

Key Concept: Scattering intensity, Mathematical relationship

d) $I_2$ is 16 times $I_1$
[Solution Description]
The intensity of scattered light follows the relation $I \propto \frac{1}{\lambda^4}$. If $\lambda_1 = 2\lambda_2$, then the ratio of intensities is $\frac{I_2}{I_1} = \left(\frac{\lambda_1}{\lambda_2}\right)^4 = 16$. Thus, light with $\lambda_2$ is scattered 16 times more intensely than light with $\lambda_1$.

Key Concept: Scattering of Light

a) Both Assertion and Reason are true, and Reason is the correct explanation of Assertion.
[Solution Description]
The scattering of light by air molecules in the atmosphere causes the blue appearance of the sky. According to Rayleigh's Law, $I \propto \frac{1}{\lambda^4}$, which means shorter wavelengths (blue light) are scattered more than longer wavelengths (red light). This is why the sky appears blue, and the reason correctly explains the assertion.

Key Concept: Scattering of Light

c) Blue light is scattered more than other colours due to its shorter wavelength
[Solution Description]
The blue colour of the sky is due to Rayleigh scattering. According to Rayleigh's law, the intensity of scattered light is inversely proportional to the fourth power of its wavelength ($I \propto \frac{1}{\lambda^4}$). Blue light has a shorter wavelength compared to red light, so it is scattered more in all directions by the atmospheric molecules. When we look at the sky, we see this scattered blue light, making the sky appear blue.

Key Concept: Order of colors in VIBGYOR

d) Violet
[Solution Description]
According to the given information, violet has the shortest wavelength ($400 \, \text{nm}$ or $4000 \, \text{\AA}$). The order is VIBGYOR, where violet is at one end with the shortest wavelength and red is at the other end with the longest wavelength.

Key Concept: Wavelength Dependency

d) $I \propto \frac{1}{\lambda^4}$
[Solution Description]
The intensity of scattered light follows Rayleigh's law: $I \propto \frac{1}{\lambda^4}$. This means shorter wavelengths (like blue/violet) scatter much more than longer wavelengths (like red).

Key Concept: Red Sun at Sunset

d) Most of the blue light is scattered away, leaving red light dominant
[Solution Description]
At sunrise and sunset, sunlight has to travel a longer distance through the atmosphere before reaching our eyes. During this long journey, most of the shorter-wavelength blue light is scattered away by the atmosphere, leaving behind the longer-wavelength red light which is scattered much less. Thus, the direct sunlight that reaches us appears red.

Key Concept: Refractive Index and Dispersion

d) Violet
[Solution Description]
The refractive index $\mu$ determines the extent of bending. Since $\mu_v > \mu_r$, violet light bends more than red light towards the base of the prism during dispersion.

Key Concept: Atmospheric path length, Scattering

c) The atmospheric path length is shorter, reducing scattering of shorter wavelengths
[Solution Description]
At sunrise or sunset, sunlight travels the longest distance through the Earth's atmosphere. During this extended path, most of the shorter wavelengths (blue, violet) are scattered away, leaving predominantly longer wavelengths (red, orange) to reach the observer. When the sun is higher in the sky, the path length through the atmosphere is shorter, resulting in less scattering of shorter wavelengths. As a result, the sun appears less red because more of the blue component reaches the observer, making the sunlight closer to white.

Key Concept: White Colour of Clouds

a) Both Assertion and Reason are true, and Reason is the correct explanation of Assertion.
[Solution Description]
The assertion states that clouds appear white due to equal scattering of all colors of incident white light. This is correct because the large size of particles in clouds causes all wavelengths of visible light to be scattered uniformly. The reason correctly explains this by mentioning that the particles are larger than the wavelength of visible light, which leads to uniform scattering. Therefore, both the assertion and reason are true, and the reason provides the correct explanation for the assertion.

Key Concept: Spectrum formation, Speed of light in prism

b) Red light travels slower than violet light inside the prism.
[Solution Description]
When white light enters the prism, each color slows down differently due to their different wavelengths. Shorter wavelengths (violet) slow down more and bend more, while longer wavelengths (red) slow down less and bend less. This causes dispersion.

Key Concept: Scattering, Wavelength Dependency

b) Decreases by a factor of 16
[Solution Description]
According to Rayleigh scattering, the intensity of scattered light ($I$) is inversely proportional to the fourth power of the wavelength ($\lambda$) of light, given by $I \propto \frac{1}{\lambda^4}$. If the wavelength is doubled ($\lambda' = 2\lambda$), the new intensity ($I'$) will be $I' \propto \frac{1}{(2\lambda)^4} = \frac{1}{16\lambda^4} = \frac{I}{16}$. Therefore, the intensity reduces to $\frac{1}{16}$th of its original value.

Key Concept: Scattering of Light, Wavelength Dependence

c) 16
[Solution Description]
Given:
$I \propto \frac{1}{\lambda^4}$
For violet light ($\lambda_v = 4000 \, \text{A}$):
$I_v \propto \frac{1}{(4000)^4}$
For red light ($\lambda_r = 8000 \, \text{A}$):
$I_r \propto \frac{1}{(8000)^4}$
The ratio $\frac{I_v}{I_r}$ is:
$\frac{I_v}{I_r} = \left( \frac{\lambda_r}{\lambda_v} \right)^4 = \left( \frac{8000}{4000} \right)^4 = 2^4 = 16$

Key Concept: Deviation of colors, Dispersion of light

b) Violet light has a higher refractive index in glass than red light.
[Solution Description]
The deviation of light depends on its wavelength. Violet light has a shorter wavelength and higher frequency, causing it to slow down more in the prism and deviate more. Red light has a longer wavelength and lower frequency, so it deviates less.

Key Concept: Scattering of Light

a) Both Assertion and Reason are true, and Reason is the correct explanation of Assertion.
[Solution Description]
The Assertion states that the sky appears blue due to the scattering of blue light, which is correct as shorter wavelengths (blue light) scatter more efficiently according to Rayleigh scattering. The Reason provides the scientific basis for this phenomenon: the intensity of scattered light $I \propto \frac{1}{\lambda^4}$. Since both the Assertion and Reason are true and the Reason correctly explains the Assertion, the correct answer is option a).

Key Concept: Scattering of Light and Sky Color

a) Both Assertion and Reason are true, and Reason is the correct explanation of Assertion.
[Solution Description]
The assertion states that the sky appears blue because blue light is scattered more than red light by the Earth's atmosphere, which is true. This phenomenon is due to Rayleigh scattering, where shorter wavelengths (blue/violet) scatter more effectively than longer wavelengths (red).
The reason states that the intensity of scattered light is inversely proportional to the fourth power of the wavelength of light, given by $I \propto \frac{1}{\lambda^4}$. This is also true and explains why blue light (shorter wavelength) scatters much more than red light (longer wavelength).
Therefore, both the assertion and reason are correct, and the reason correctly explains the assertion.