Class 10 Physics Chapter 6 Spectrum

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: 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.

Key Concept: Dependence on Wavelength

d) Red
[Solution Description]
Since the refractive index decreases with increasing wavelength, red light (with the longest wavelength among visible light) deviates the least compared to other colors like violet, which has a shorter wavelength.

Key Concept: Wavelength Range, Frequency-Wavelength Relationship

a) Both Assertion and Reason are true, and Reason is the correct explanation of Assertion.
[Solution Description]
First, calculate the wavelength using $c = f \times \lambda$ where $c = 3 \times 10^8 \, \text{m/s}$. Rearranging gives $\lambda = \frac{c}{f} = \frac{3 \times 10^8}{3 \times 10^{15}} = 10^{-7} \, \text{m} = 100 \, \text{nm}$. The assertion states the frequency corresponds to ultraviolet, which is correct as $3 \times 10^{15} \, \text{Hz}$ lies within the ultraviolet frequency range ($3 \times 10^{15} \, \text{Hz}$ to $7.5 \times 10^{14} \, \text{Hz}$). The reason correctly calculates the wavelength and confirms it lies in the ultraviolet range (10 nm to 400 nm). Hence, both are true, and the reason explains the assertion.

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: 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: 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 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: Speed of light in glass

b) Increases
[Solution Description]
The speed of light in a medium increases with an increase in its wavelength. This is because the refractive index $\mu$ decreases as the wavelength increases, leading to a higher speed of light in the medium. For example, red light (longer wavelength) travels faster in glass than violet light (shorter wavelength).

Key Concept: Blue Colour of Sky

c) Due to intense scattering of blue light by air molecules
[Solution Description]
The sky appears blue because sunlight is scattered by air molecules in the atmosphere. Blue light has a shorter wavelength ($\lambda \approx 4000 \, \text{\AA}$) and is scattered more intensely than red light due to the inverse fourth power dependence of scattering intensity on wavelength ($I \propto \frac{1}{\lambda^4}$). The scattered blue light reaches our eyes from all directions, making the sky appear blue.

Key Concept: Sky Color Explanation

c) Blue light is scattered most by air molecules
[Solution Description]
During the day, sunlight travels a shorter distance through the atmosphere, and due to Rayleigh scattering, blue light (shorter wavelength) is scattered more than other colors. This scattered blue light reaches our eyes from all directions, making the sky appear blue.

Key Concept: Scattering and Wavelength Dependence

d) All wavelengths are scattered equally
[Solution Description]
When the size of particles in the medium is larger than the wavelength of incident light, all wavelengths of light are scattered equally. This results in white light being scattered uniformly, which is why clouds appear white as they consist of water droplets much larger than visible light wavelengths.

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: Electromagnetic Spectrum - Speed

b) $3 \times 10^8 \, \text{m s}^{-1}$
[Solution Description]
All electromagnetic waves travel with the same speed in a vacuum, which is $3 \times 10^8 \, \text{m s}^{-1}$. This is a fundamental property of electromagnetic waves.

Key Concept: Speed of Electromagnetic Waves

c) $3 \times 10^8 \, \text{m/s}$
[Solution Description]
All electromagnetic waves travel at the same speed in a vacuum, which is $c = 3 \times 10^8 \, \text{m/s}$.

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: Rayleigh Scattering

c) Shorter wavelengths (blue/violet) are scattered away, leaving red/orange.
[Solution Description]
The Sun appears red at sunrise and sunset because the light has to travel through a larger portion of Earth's atmosphere, causing shorter wavelengths (blue, violet) to scatter more due to Rayleigh scattering. The intensity of scattered light follows $I \propto \frac{1}{\lambda^4}$, so longer wavelengths (red/orange) dominate the remaining unscattered light.

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: 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: Dispersion and Spectrum

d) Violet
[Solution Description]
The refractive index ($\mu$) of the prism material is higher for violet light compared to red light because violet light has a shorter wavelength. Since the angle of deviation depends on $\mu$, violet light will deviate the most.

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: Properties of Infrared Radiations

a) Both Assertion and Reason are true, and Reason is the correct explanation of Assertion.
[Solution Description]
The assertion states that infrared radiations are used for therapeutic purposes by doctors, which is correct as mentioned in the syllabus under "Uses of infrared radiations." The reason states that infrared radiations have a strong heating effect and can penetrate deeply due to less scattering, which is also true as per the syllabus (under "Properties of infrared radiations"). Furthermore, the heating effect and deep penetration directly explain why they are useful in therapy. Hence, both Assertion and Reason are true, and Reason correctly explains Assertion.

Key Concept: Refractive Index and Wavelength

d) Violet
[Solution Description]
The refractive index $\mu$ increases with a decrease in wavelength. Violet has the shortest wavelength ($4000 \, \text{\AA}$), so it has the highest refractive index in glass.

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: 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: Dispersion, Angle of deviation, Wavelength

b) $9.3^\circ$
[Solution Description]

Angular dispersion is the difference between deviation of violet and red light. Using the prism formula
$\sin r = \frac{\sin i}{\mu}$,
we calculate \(r_1\), \(r_2\), emerging angle \(e\), and deviation
$\delta = i + e - A$
for each color. Violet deviates more. The difference gives \(9.3^\circ\).

Key Concept: Dispersion Mechanism

c) First surface only
[Solution Description]
As per the syllabus, dispersion of white light occurs at the first surface of the prism due to the change in speed of light with wavelength. Refraction occurs at both surfaces, but dispersion specifically happens at the first surface.

Key Concept: Scattering Condition, Particle Size

b) When scattering by water droplets in fog
[Solution Description]
The condition for $I \propto \frac{1}{\lambda^4}$ relationship is that the size of scattering particles must be smaller than the wavelength of incident light.
Step 1: This dependence breaks down when particle size becomes comparable to or larger than the wavelength.
Step 2: Therefore, when particle size > wavelength, the scattering becomes independent of wavelength.

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: 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: Wavelength and Frequency Relationship

c) $6.25 \times 10^{14} Hz$
[Solution Description]
First, convert the wavelength from Angstroms to meters:
$4800 A = 4800 \times 10^{-10} m = 4.8 \times 10^{-7} m$
Using the formula $c = f \lambda$, rearrange to find frequency $f$:
$f = \frac{c}{\lambda} = \frac{3 \times 10^8}{4.8 \times 10^{-7}} = 6.25 \times 10^{14} Hz$

Key Concept: Sunrise/Sunset Phenomenon

c) Red light scatters less and dominates after long atmospheric travel
[Solution Description]
At sunrise/sunset, sunlight passes through a thicker layer of the atmosphere, causing most of the shorter-wavelength blue light to scatter out before reaching the observer. The remaining light, dominated by longer-wavelength red components, travels directly to the observer, making the Sun appear red.

Key Concept: Newton's Experiment

a) Both Assertion and Reason are true, and Reason is the correct explanation of Assertion.
[Solution Description] The assertion states that white light is polychromatic, which is correct as it contains multiple wavelengths (as concluded by Newton). The reason explains that dispersion occurs because different wavelengths travel at different speeds in the prism, causing them to bend at different angles. This is also correct and directly explains why the assertion is true. Thus, both the assertion and reason are true, and the reason correctly explains the assertion.

Key Concept: Concept: Wavelength Comparison

d) 16 times more
[Solution Description]
Using the relation $I_{\text{violet}} / I_{\text{red}} = (\lambda_{\text{red}} / \lambda_{\text{violet}})^4$, substitute $\lambda_{\text{red}} = 8000$ Å and $\lambda_{\text{violet}} = 4000$ Å.
$I_{\text{violet}} / I_{\text{red}} = (8000/4000)^4 = 2^4 = 16$
Hence, violet light is scattered 16 times more than red light.

Key Concept: Electromagnetic Spectrum - Properties

c) Ultraviolet radiations
[Solution Description]
Ultraviolet radiations cause fluorescence when they strike fluorescent materials such as zinc sulphide. Gamma rays also cause fluorescence, but ultraviolet rays are specifically mentioned for this property in the given syllabus.

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: Dependence on Angle of Incidence and Prism Angle, Refractive Index

a) Both Assertion and Reason are true, and Reason is the correct explanation of Assertion.
[Solution Description]
The assertion states that the angle of minimum deviation $\delta_m$ occurs when the light ray passes symmetrically through the prism, which is true. The reason states that this is because the angle of incidence equals the angle of emergence, ensuring symmetry in the path of light, which is also true. Moreover, the reason correctly explains why the assertion is true because symmetry in the path (equal angles of incidence and emergence) minimizes the deviation. Therefore, both Assertion and Reason are true, and Reason is the correct explanation of the Assertion.

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: 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: 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: Cause of Dispersion

c) Change in speed of light with wavelength
[Solution Description]
The dispersion of white light occurs due to the change in speed of light with wavelength (color). Different colors have different speeds in glass, causing them to refract at different angles. Violet light slows down the most and deviates the most, while red light slows down the least and deviates the least.

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: 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: Frequency range, wavelength conversion

b) Radio waves
[Solution Description]
Use $c = f \times \lambda$ to find $f$:
$f = \frac{c}{\lambda} = \frac{3 \times 10^8}{20} = 1.5 \times 10^7 \, \text{Hz}.$
From the syllabus, radio waves have frequencies below $3 \times 10^7 \, \text{Hz}$ and microwaves have frequencies from $3 \times 10^7 \, \text{Hz}$ to $3 \times 10^{11} \, \text{Hz}$. Since $1.5 \times 10^7 \, \text{Hz} < 3 \times 10^7 \, \text{Hz}$, the wave is a radio wave.

Key Concept: Rayleigh Scattering, 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 due to the higher scattering of shorter wavelengths (like blue light) compared to longer wavelengths (like red light). This is correct because blue light has a shorter wavelength and is scattered more according to Rayleigh scattering. The Reason correctly explains this phenomenon by stating Rayleigh's law, which mathematically describes the inverse proportionality between scattered light intensity and the fourth power of the wavelength. Therefore, both Assertion and Reason are true, and Reason is the correct explanation of Assertion.

Key Concept: Deviation in Prism

d) Violet
[Solution Description]
As per the syllabus, the deviation ($\delta$) is inversely related to the speed of light in the medium. Violet light has the least speed in glass, so it is deviated the most. Red light has the highest speed, so it is deviated the least.

Key Concept: Heating effect, Detection

d) Heating effect approximately doubles
[Solution Description]
The heating effect depends on the intensity of infrared radiation. From the given syllabus, infrared has strong heating effect. When intensity doubles (while wavelength remains same), the energy delivered per unit area per second doubles. Since the thermometer measures temperature change caused by absorbed energy, the heating effect will approximately double.
Calculation: Let original intensity = I, new intensity = 2I. Energy absorbed Q $\propto$ I. Temperature rise $\Delta T \propto Q$, so $\Delta T_{new} \approx 2\Delta T_{original}$.

Key Concept: Intensity of Scattering

c) Red, due to its long wavelength.
[Solution Description]
Red light has the longest wavelength ($\lambda = 8000 \, \text{A}$) among visible light colors. Since scattering intensity follows $I \propto \frac{1}{\lambda^4}$, red light is scattered the least because its longer wavelength leads to significantly lower scattering compared to shorter wavelengths like blue or violet.

Key Concept: Scattering and Cloud Color

c) All wavelengths of visible light are scattered equally by large particles
[Solution Description]
Clouds contain water droplets and dust particles larger than the wavelength of visible light. These particles scatter all wavelengths of sunlight equally (Mie scattering). As a result, the scattered light reaching our eyes retains the original white color of sunlight, making clouds appear white.

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: 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: 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: Refractive index, Speed of light, Deviation

b) Speed: $1.94 \times 10^8 \text{ m/s}, \delta = 42.7^\circ$
[Solution Description]

Speed of light inside prism: $v = \frac{c}{\mu} = \frac{3 \times 10^8}{1.55} \approx 1.935 \times 10^8 \text{ m/s}.$ For angle of deviation, first find refracted angle: $\sin r_1 = \frac{\sin 60^\circ}{1.55} \approx 0.5594 \implies r_1 \approx 34.03^\circ.$ Second refraction angle: $r_2 = A - r_1 = 60^\circ - 34.03^\circ = 25.97^\circ.$ Emergence angle: $\sin e = 1.55 \sin 25.97^\circ \approx 0.6786 \implies e \approx 42.74^\circ.$ Total deviation: $\delta = 60^\circ + 42.74^\circ - 60^\circ = 42.74^\circ.$

Key Concept: Scattering of Light

c) Blue
[Solution Description]
Rayleigh scattering states that the intensity of scattered light $I \propto \frac{1}{\lambda^4}$. Therefore, blue light (shorter wavelength than red) is scattered more than other colours.

Key Concept: Dispersion, Wavelength, Refractive Index

b) 1.067
[Solution Description]
The angle of deviation $\delta$ is given by $\delta = i - r$, where $i$ is the angle of incidence and $r$ is the angle of refraction. From Snell's law, $\mu = \frac{\sin i}{\sin r}$, so $\sin r = \frac{\sin i}{\mu}$. For violet light: $\sin r_v = \frac{\sin 45°}{1.53} = \frac{0.7071}{1.53} \approx 0.4622$, thus $r_v \approx \sin^{-1}(0.4622) \approx 27.5°$. For red light: $\sin r_r = \frac{\sin 45°}{1.48} = \frac{0.7071}{1.48} \approx 0.4778$, thus $r_r \approx \sin^{-1}(0.4778) \approx 28.6°$. Now, $\delta_v = 45° - 27.5° = 17.5°$ and $\delta_r = 45° - 28.6° = 16.4°$. The ratio $\frac{\delta_v}{\delta_r} = \frac{17.5}{16.4} \approx 1.067$.

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: Scattering of Light

d) Blue light is scattered more than other colors due to its shorter wavelength.
[Solution Description]
Blue light has a shorter wavelength and is scattered more by air molecules compared to other colors in sunlight. This scattered blue light reaches our eyes from all directions, making the sky appear blue.

Key Concept: Dispersion, Wavelength, Refraction

c) Violet light deviates more than red light.
[Solution Description]
The deviation of light depends on its wavelength and the refractive index of the prism material. Violet light has the shortest wavelength and highest refractive index in glass, causing it to deviate the most, while red light has the longest wavelength and lowest refractive index, deviating the least.

Key Concept: Atmospheric Scattering

c) Black, because no scattering occurs without an atmosphere.
[Solution Description]
If there were no atmosphere, there would be no scattering of sunlight. As a result, no scattered light would reach our eyes, making the sky appear black. This is observed on the Moon, where the sky is black due to the absence of an atmosphere.

Key Concept: Dispersion, Refractive Index, Angle of Deviation

b) 0.5°
[Solution Description]
First, calculate the angle of refraction for red light using Snell's law: $\sin r_{\text{red}} = \frac{\sin 45°}{\mu_{\text{red}}}$. So, $r_{\text{red}} = \sin^{-1}\left(\frac{\sin 45°}{1.52}\right) \approx 27.8°$. The deviation for red light is $\delta_{\text{red}} = i - r_{\text{red}} = 45° - 27.8° = 17.2°$. Similarly, for violet light: $r_{\text{violet}} = \sin^{-1}\left(\frac{\sin 45°}{1.54}\right) \approx 27.3°$. The deviation for violet light is $\delta_{\text{violet}} = 45° - 27.3° = 17.7°$. The difference in deviation is $17.7° - 17.2° = 0.5°$.

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: Frequency Calculation

b) $3.75 \times 10^{14}$ Hz
[Solution Description]
Frequency ($f$) and wavelength ($\lambda$) are related by $c = f \lambda$, where $c$ is the speed of light. Wavelength of red light ($\lambda_r$) = 2 $\times$ wavelength of violet light ($\lambda_v$). So, $\lambda_r = 2 \lambda_v$. Thus, $f_r = \frac{c}{\lambda_r} = \frac{c}{2 \lambda_v} = \frac{f_v}{2}$, since $f_v = \frac{c}{\lambda_v}$. Given $f_v = 7.5 \times 10^{14}$ Hz, $f_r = \frac{7.5 \times 10^{14}}{2} = 3.75 \times 10^{14}$ Hz.

Key Concept: Danger Signal and Scattering

c) Red light scatters the least due to its long wavelength
[Solution Description]
Red light has the longest wavelength in the visible spectrum, which means it is scattered the least by atmospheric particles compared to shorter wavelengths like blue or green. This allows red light to travel farther without significant attenuation, making it ideal for danger signals as it remains visible even from long distances or in foggy conditions.

Key Concept: Wavelength, Frequency, Refractive Index

a) Red
[Solution Description]
According to Snell's Law, $\mu = \frac{\sin i}{\sin r}$. Rearranging gives $\sin r = \frac{\sin i}{\mu}$. For a constant angle of incidence ($i = 60^\circ$), $\sin r$ is inversely proportional to $\mu$. Since $\mu_{\text{red}} = 1.52 < \mu_{\text{violet}} = 1.54$, $\sin r_{\text{red}}$ will be larger than $\sin r_{\text{violet}}$. Therefore, the angle of refraction $r$ will be largest for red light.

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: Electromagnetic Spectrum

a) Both Assertion and Reason are true, and Reason is the correct explanation of Assertion.
[Solution Description]
The energy $E$ of an electromagnetic wave is given by the equation $E = hf$, where $h$ is Planck''s constant and $f$ is the frequency. Since $c = f\lambda$, we can rewrite the energy as $E = \frac{hc}{\lambda}$. This shows that energy is inversely proportional to wavelength. Gamma rays have the shortest wavelength in the electromagnetic spectrum (below 0.01 nm), which implies they have the highest energy. Therefore, the Assertion is true, and the Reason correctly explains why gamma rays have the highest energy.

Key Concept: Angle of Deviation

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 for violet light is greater than for red light when passing through a prism. This is true because, according to the syllabus, the refractive index of the prism material depends on the wavelength (or colour) of light. Violet light has a shorter wavelength and thus experiences a higher refractive index compared to red light. A higher refractive index results in a larger angle of deviation.
The reason given is that the refractive index of the prism material is higher for violet light compared to red light. This is also correct and directly explains why violet light is deviated more than red light.
Therefore, both the assertion and reason are true, and the reason correctly explains the assertion.

Key Concept: Wavelength Range

a) Both Assertion and Reason are true, and Reason is the correct explanation of Assertion.
[Solution Description]
According to the syllabus, X-rays are defined as electromagnetic waves with wavelengths ranging from 0.1 Å to 100 Å (which is equivalent to 0.01 nm to 10 nm). The frequency range for X-rays is given as $3 \times 10^{16} \, \text{Hz}$ to $3 \times 10^{19} \, \text{Hz}$. Both the Assertion (A) and Reason (R) are true based on the syllabus. Additionally, the Reason correctly explains the Assertion because the frequency and wavelength ranges are directly related through the equation $c = f \times \lambda$.

Key Concept: Penetration through materials

b) Rock-salt
[Solution Description]
Infrared radiations can pass through rock-salt but are absorbed by glass, unlike visible light which passes through glass.

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: 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, 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: Visible Light Spectrum

d) Red
[Solution Description]
From the syllabus, red light has the longest wavelength in the visible spectrum (800 nm), while violet has the shortest (400 nm).

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: Role of Prism in Spectrum Formation

d) It splits the existing colours in the incident light
[Solution Description]
The syllabus states that the prism does not produce colours but only separates the existing colours present in the incident white light by refracting them at different angles based on their wavelengths.

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 sun appears red at sunrise and sunset due to the increased scattering of blue light compared to red light. This is a well-known phenomenon explained by Rayleigh scattering. The Reason provides the scientific basis for this observation: the intensity of scattered light follows $I \propto \frac{1}{\lambda^4}$, meaning shorter wavelengths (blue/violet) scatter much more intensely than longer wavelengths (red). Since the sunlight travels a longer path through the atmosphere during sunrise/sunset, most of the blue light is scattered away, leaving predominantly red light to reach the observer. Thus, the Reason correctly explains the Assertion.

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: Visible Spectrum - Wavelength and Frequency

a) Both Assertion and Reason are true, and Reason is the correct explanation of Assertion.
[Solution Description]
From the given syllabus, it is mentioned that red light has the longest wavelength (8000 Å or 800 nm) in the visible spectrum (VIBGYOR). This makes the Assertion (A) true.
The Reason (R) states that the speed of red light is the highest among all colours in glass, which is also correct as per the syllabus. Furthermore, the Reason correctly explains why red light has the least deviation when passing through a prism—because its speed is highest in glass, resulting in the least refraction. Therefore, the Reason is the correct explanation for the Assertion.

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: Scattering Intensity

c) Decreases by a factor of 16
[Solution Description]
The intensity of scattered light follows the relation $I \propto \frac{1}{\lambda^4}$. If $\lambda$ becomes $2\lambda$, the new intensity $I'$ will be:
$I' \propto \frac{1}{(2\lambda)^4} = \frac{1}{16\lambda^4} = \frac{I}{16}$
Hence, the intensity decreases by a factor of 16.

Key Concept: Concept: Size of particles in clouds

c) larger than the wavelength of visible light
[Solution Description]
The particles in clouds, including dust and aggregates of water molecules, are larger than the wavelength of visible light, which causes them to scatter all colors equally.

Key Concept: Angle of deviation, Refractive index, Wavelength dependence

b) $47.4^\circ$
[Solution Description]

The deviation is found using prism refraction steps. First, the refracted angle \(r_1\)
is obtained from
$\sin r_1 = \frac{\sin 45^\circ}{1.6}$
Then the second refracted angle is $r_2 = 60^\circ - r_1$
The emergent angle is calculated using $\sin e = 1.6 \sin r_2$
Finally, deviation is $\delta = i + e - A$
giving approximately 47.4∘47.4^\circ47.4∘.

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: Speed of light

c) Violet
[Solution Description]
The violet color has the shortest wavelength ($4000 \, \text{\AA}$) and hence the highest frequency ($7.5 \times 10^{14} \, \text{Hz}$) in the visible spectrum. From the table, red has the lowest frequency.

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: 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: 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: 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: Atmospheric scattering, sky color observation

b) Black due to lack of atmosphere
[Solution Description]
The moon has no atmosphere, so there are no air molecules to scatter sunlight. Therefore, no scattered light reaches the astronaut's eyes from directions other than the direct path of sunlight. As a result, the sky appears black in all directions except towards the sun. This is because scattering requires an atmosphere, which is absent on the moon.

Key Concept: Intensity of Scattering

c) Violet light is scattered approximately 16 times more than red light.
[Solution Description]
The intensity of scattered light is inversely proportional to the fourth power of the wavelength ($I \propto \frac{1}{\lambda^4}$). Since violet light has the shortest wavelength $(\approx 400 \, \text{nm})$ and red light has the longest wavelength $(\approx 700 \, \text{nm})$, violet light is scattered much more intensely $(\approx 16 \text{ times})$ more than red light.

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: Frequency Range

d) Violet
[Solution Description]
From the table, violet light has a frequency of $7.5 \times 10^{14} \, \text{Hz}$, which is the highest among all colours in white light. Frequency decreases as wavelength increases.

Key Concept: Scattering of light, Rayleigh scattering

c) 16 times
[Solution Description]
According to Rayleigh's scattering law, the intensity of scattered light is inversely proportional to the fourth power of the wavelength: $I \propto \frac{1}{\lambda^4}$.
Given $\lambda_{\text{violet}} = 4000 \, \text{Å}$ and $\lambda_{\text{red}} = 8000 \, \text{Å}$.
The ratio of intensities is calculated as:
$\frac{I_{\text{violet}}}{I_{\text{red}}} = \left( \frac{\lambda_{\text{red}}}{\lambda_{\text{violet}}} \right)^4 = \left( \frac{8000}{4000} \right)^4 = 2^4 = 16.$
Thus, violet light is scattered 16 times more than red light.

Key Concept: Scattering, Sky Color

d) Shorter path length at noon causes equal scattering
[Solution Description]
The sky appears blue due to the preferential scattering of shorter wavelengths (blue light) by atmospheric particles. At noon, sunlight travels the shortest distance through the atmosphere, resulting in nearly equal scattering of all visible wavelengths, which combines to appear white. Thus, the dominant effect changes based on the path length of sunlight through the atmosphere.

Key Concept: Deviation in Prism

d) Red
[Solution Description]
The angle of deviation depends on the wavelength of light. Red light has the longest wavelength (8000 Å), and its speed in glass is maximum, causing it to deviate the least compared to other colours. From the key points, $\delta_{red} < \delta_{violet}$.

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: Refractive index, Angle of deviation

d) Violet light (400 nm)
[Solution Description]
The angle of deviation $\delta$ depends on refractive index $\mu$, which varies with wavelength ($\lambda$).
1. For violet light ($\lambda \approx 4000$\AA), $\mu_{\text{v}}$ is highest because speed in glass is slowest.
2. Using Snell's law: $\sin i = \mu \sin r \Rightarrow r = \sin^{-1}(\frac{\sin i}{\mu})$
3. For higher $\mu$, $r$ becomes smaller, causing greater deviation $\delta = i - r$.
4. Thus, violet light (highest $\mu$) deviates most.

Key Concept: Cause of Dispersion

c) Violet
[Solution Description]
The deviation of light in a prism depends on its wavelength. According to the syllabus, violet light has the shortest wavelength and thus is deviated the most when passing through a prism, while red light is deviated the least.

Key Concept: Dependence on Angle of Incidence and Prism Angle

b) 37.2$^\circ$
[Solution Description]
The formula relating the refractive index $\mu$, prism angle $A$, and angle of minimum deviation $\delta_m$ is:
$\mu = \frac{\sin\left(\frac{A + \delta_m}{2}\right)}{\sin\left(\frac{A}{2}\right)}$
Given $A = 60^\circ$ and $\mu = 1.5$, plug in the values:
$1.5 = \frac{\sin\left(\frac{60^\circ + \delta_m}{2}\right)}{\sin(30^\circ)}$
Simplify $\sin(30^\circ) = 0.5$:
$1.5 = \frac{\sin\left(\frac{60^\circ + \delta_m}{2}\right)}{0.5}$
Multiply both sides by 0.5:
$0.75 = \sin\left(\frac{60^\circ + \delta_m}{2}\right)$
Take the inverse sine of both sides:
$\frac{60^\circ + \delta_m}{2} = \sin^{-1}(0.75) \approx 48.59^\circ$
Solve for $\delta_m$:
$60^\circ + \delta_m = 97.18^\circ$
$\delta_m = 37.18^\circ$

Key Concept: Spectrum Order

d) Violet
[Solution Description]
The order of colours in the spectrum (VIBGYOR) shows that violet light deviates the most and appears closest to the base of the prism, while red deviates the least and appears farthest.

Key Concept: Visible Spectrum, Frequency, Energy

d) Violet photons have higher energy than red photons.
[Solution Description]
The energy of a photon is given by $E = h\nu$, where $h$ is Planck's constant and $\nu$ is the frequency. Violet light has the highest frequency ($7.5 \times 10^{14}$ Hz) in the visible spectrum, so its photons have the highest energy. Red light has the lowest frequency ($4.6 \times 10^{14}$ Hz), so its photons have the lowest energy.

Key Concept: Rayleigh Scattering

a) Both Assertion and Reason are true, and Reason is the correct explanation of Assertion.
[Solution Description]
According to Rayleigh's law, the intensity of scattered light is inversely proportional to the fourth power of the wavelength ($I \propto \frac{1}{\lambda^4}$). Since blue light has a shorter wavelength compared to other visible colors, it is scattered more strongly in the atmosphere, making the sky appear blue. Therefore, both Assertion and Reason are correct, and the Reason properly explains the Assertion.

Key Concept: Atmospheric Scattering

c) It scatters blue light more than red light, allowing red light to reach us.
[Solution Description]
During sunrise and sunset, sunlight passes through a longer path in the atmosphere. Shorter wavelengths (blue light) are scattered away due to their higher scattering intensity ($\propto \frac{1}{\lambda^4}$), while longer wavelengths (red light) pass through with minimal scattering, resulting in the red color.

Key Concept: Color of Sky

d) Because blue light is scattered more than other colors
[Solution Description]
Blue light has a shorter wavelength compared to red light, so it is scattered more by air molecules in the atmosphere. This scattered blue light reaches our eyes from all directions, making the sky appear blue.

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: 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: 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: Scattering of Light

b) Blue light is scattered the most.
[Solution Description]
The blue color 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). Thus, blue light dominates the scattered light reaching our eyes from all directions except the direct path of the sun.

Key Concept: Gamma Rays Properties

a) Both Assertion and Reason are true, and Reason is the correct explanation of Assertion.
[Solution Description]
Gamma rays indeed have the shortest wavelength (less than $0.1 \, \text{A}$) among all electromagnetic waves, as mentioned in the syllabus. This property makes them highly energetic. Therefore, both Assertion and Reason are true, and the Reason correctly explains the Assertion.

Key Concept: Wavelength and Speed in Vacuum

d) All colours have the same speed in vacuum
[Solution Description] In vacuum, all colours of light travel at the same speed, which is the speed of light ($3 \times 10^8$ m/s). This is a fundamental property of electromagnetic waves. Therefore, no single colour has a greater speed than another in vacuum.

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: Relationship Between Colour, Wavelength, and Frequency

a) Both Assertion and Reason are true, and Reason is the correct explanation of Assertion.
[Solution Description]
The assertion states that violet light has a shorter wavelength than red light. According to the syllabus, violet light has a wavelength of $4000 \, \text{A}$ while red light has a wavelength of $8000 \, \text{A}$, so the assertion is true.
The reason states that the speed of light in glass is lower for violet light compared to red light. The syllabus explains that the speed of light in a medium decreases with decreasing wavelength, meaning violet light ($\lambda = 4000 \, \text{A}$) travels slower than red light ($\lambda = 8000 \, \text{A}$). Hence, the reason is also true.
Furthermore, the reason correctly explains why violet light is deviated more than red light when passing through a prism. Therefore, the reason is the correct explanation of the assertion.

Key Concept: Angle of Deviation, Dispersion, Wavelength dependence

a) Both Assertion and Reason are true, and Reason is the correct explanation of Assertion.
[Solution Description]
The assertion states that for a fixed angle of incidence and prism angle, violet light deviates more than red light. This is true because the refractive index $\mu$ of the prism material depends on the wavelength ($\lambda$) of light. Violet light has a shorter wavelength and higher refractive index ($\mu_{violet} > \mu_{red}$). According to the formula for deviation $\delta = i - r + e - A$, where $r$ decreases with increasing $\mu$, violet light bends more due to its higher $\mu$. Thus, $\delta_{violet} > \delta_{red}$. The reason correctly explains why this happens: $\mu_{violet} > \mu_{red}$ directly leads to greater deviation for violet light.

Key Concept: Nature of White Light

b) Polychromatic
[Solution Description]
As per the syllabus, white light is polychromatic, meaning it consists of multiple wavelengths (colors). The prism separates these colors but does not create them.

Key Concept: Scattering of light

d) Red light is scattered the least and blue light is scattered the most
[Solution Description]
The red appearance of the sun at sunrise and sunset is due to the scattering of sunlight by the Earth's atmosphere. During these times, sunlight travels a longer distance through the atmosphere, causing shorter wavelengths (blue light) to scatter more, while longer wavelengths (red light) scatter less. As a result, the remaining light that reaches our eyes is predominantly red.

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: Gamma Rays, X-rays

b) 248.5 keV
[Solution Description]
First, convert the wavelength from angstroms to meters: $0.05 \, \text{Å} = 0.05 \times 10^{-10} \, \text{m} = 5 \times 10^{-12} \, \text{m}$. The energy of a photon is given by $E = \frac{hc}{\lambda}$. Substituting the values:
$E = \frac{(6.626 \times 10^{-34})(3 \times 10^8)}{5 \times 10^{-12}} = \frac{1.9878 \times 10^{-25}}{5 \times 10^{-12}} = 3.9756 \times 10^{-14} \, \text{J}.$
Convert energy to eV: $\frac{3.9756 \times 10^{-14}}{1.6 \times 10^{-19}} = 2.48475 \times 10^5 \, \text{eV} \approx 248.5 \, \text{keV}$.

Key Concept: Dispersion and Spectrum

c) Assertion is true, but Reason is false.
[Solution Description]
The assertion is true because when white light passes through a prism, it indeed splits into seven colors (VIBGYOR) due to dispersion. However, the reason is false because different colors of light do not travel at the same speed in glass; violet travels slower than red, causing dispersion. Thus, the correct option is (c).

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: Wavelength, Frequency, Speed of Light

a) $2.7 \times 10^8$ m/s
[Solution Description]
The speed of light in a medium is given by $v = \lambda \nu$, where $\lambda$ is the wavelength in the medium and $\nu$ is the frequency. The frequency remains unchanged when light passes from one medium to another. Here, $\nu = 6 \times 10^{14}$ Hz and $\lambda = 450$ nm $= 450 \times 10^{-9}$ m. Thus, $v = (450 \times 10^{-9}) \times (6 \times 10^{14}) = 2.7 \times 10^8$ m/s.

Key Concept: Angle of Deviation Concept

c) Angle between the incident ray and the emergent ray
[Solution Description]
The angle of deviation ($\delta$) is the angle between the incident ray and the emergent ray after passing through the prism. It is the measure of how much the light bends when it travels through the prism.

Key Concept: Dependence on Angle of Incidence, Prism Angle, Refractive Index

b) 37.18$^\circ$
[Solution Description]
To find the minimum deviation $\delta_{min}$ for a prism, we use the formula:
$\mu = \frac{\sin\left(\frac{A + \delta_{min}}{2}\right)}{\sin\left(\frac{A}{2}\right)}$
Given $A = 60^\circ$, $\mu = 1.5$, we substitute these values into the equation:
$1.5 = \frac{\sin\left(\frac{60^\circ + \delta_{min}}{2}\right)}{\sin\left(30^\circ\right)}$
Since $\sin(30^\circ) = 0.5$, the equation simplifies to:
$1.5 = \frac{\sin\left(\frac{60^\circ + \delta_{min}}{2}\right)}{0.5}$
Multiply both sides by 0.5:
$0.75 = \sin\left(\frac{60^\circ + \delta_{min}}{2}\right)$
Taking the inverse sine (arcsin) of both sides:
$\frac{60^\circ + \delta_{min}}{2} = \arcsin(0.75) \approx 48.59^\circ$
Now solve for $\delta_{min}$:
$60^\circ + \delta_{min} = 97.18^\circ$
Subtract 60 from both sides:
$\delta_{min} = 37.18^\circ$

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: 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: Visible Spectrum, Dispersion of 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, splitting into its constituent colors (VIBGYOR). According to the syllabus, the refractive index of the prism material ($\mu$) is greater for violet light ($\mu_{\text{violet}}$) than for red light ($\mu_{\text{red}}$). Since $\mu = \frac{\text{Speed of light in air}}{\text{Speed of light in glass}}$, a higher refractive index implies slower speed and greater deviation. Thus, violet light ($7.5 \times 10^{14}$ Hz) has a higher refractive index and slower speed, causing more deviation ($\delta_{\text{violet}} > \delta_{\text{red}}$). Hence, both Assertion and Reason are true, and the Reason explains the Assertion correctly.

Key Concept: Refractive Index and Wavelength

d) Increases
[Solution Description]
The refractive index $\mu$ of glass increases as the wavelength of light decreases. This is because shorter wavelengths (like violet light) travel slower in glass compared to longer wavelengths (like red light), leading to higher refraction. The relationship can be expressed as:
$\mu = \frac{\text{Speed of light in air}}{\text{Speed of light in glass}}$
Since the speed of violet light in glass is the least, its refractive index is the highest.
Thus, the correct option states that the refractive index increases.

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: 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: Speed of infrared radiations

c) $3 \times 10^8 \, \text{m/s}$
[Solution Description]
Infrared radiations travel in straight lines with a speed equal to $3 \times 10^8 \, \text{m/s}$ in vacuum or air, similar to visible light.

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: 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: Electromagnetic Spectrum - Identifying Radiation Type

d) Microwaves
[Solution Description]
Microwaves are widely used in satellite communication and microwave ovens. Their wavelength range is from 1 mm to 10 m, and they fall between infrared and radio waves in the electromagnetic spectrum.

Key Concept: Electromagnetic Spectrum

a) Both Assertion and Reason are true, and Reason is the correct explanation of Assertion.
[Solution Description]
Gamma rays are known to have the shortest wavelength $\lambda < 0.1 \, \text{Å}$
among all electromagnetic waves, making the Assertion true. The Reason states that the energy ($E$) of electromagnetic waves is inversely proportional to their wavelength ($\lambda$), which is also true as per the relation $E = h\nu = \frac{hc}{\lambda}$, where $h$ is Planck's constant and $\nu$ is frequency. Since shorter wavelength corresponds to higher energy, the Reason correctly explains why gamma rays have the highest energy in the spectrum.

Key Concept: Refractive Index Relationship

c) $\mu_v > \mu_r$
[Solution Description]
Since violet light travels slower in glass than red light, its refractive index is higher. Therefore, $\mu_v > \mu_r$.

Key Concept: Wavelength Range

b) Both Assertion and Reason are true, but Reason is NOT the correct explanation of Assertion.
[Solution Description]
From the syllabus, infrared radiation has a wavelength range of 8000 Å to $10^7$ Å (or $\lambda = 800$ nm to 1 mm) and a corresponding frequency range of $3.75 \times 10^{14}$ Hz to $3 \times 10^{11}$ Hz.
Step 1: Check Assertion (A). The wavelength range given in (A) matches the syllabus ($800$ nm to $1$ mm), so (A) is true.
Step 2: Check Reason (R). The frequency range given in (R) matches the syllabus ($3.75 \times 10^{14}$ Hz to $3 \times 10^{11}$ Hz), so (R) is also true.
Step 3: Determine if (R) correctly explains (A). Since both wavelength and frequency are related by $c = f \lambda$, knowledge of one implies the other. However, (R) does not directly explain why the wavelength range is as stated in (A); it merely provides another correct but independent fact. Therefore, Reason is NOT the correct explanation of Assertion.
Conclusion: Both (A) and (R) are true, but (R) is not the correct explanation of (A).

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 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 ($\delta$) is greater for violet light than for red light when passed through a prism for a given angle of incidence. This is correct because the refractive index ($\mu$) of the prism material depends on the wavelength of light, being higher for shorter wavelengths (violet) and lower for longer wavelengths (red). A higher refractive index causes greater deviation.
The reason provided is that the refractive index of the prism material is higher for violet light compared to red light. This is also true since the refractive index indeed varies inversely with wavelength, making it higher for violet light.
Furthermore, the reason correctly explains the assertion because the greater refractive index for violet light leads to a larger angle of deviation compared to red light.

Key Concept: Spectrum, Deviation, Refractive Index

b) 1.02
[Solution Description]
The angle of deviation $\delta$ is related to the refractive index $\mu$ and the angle of the prism $A$ by the formula: $\delta = (\mu - 1)A$.
For green light: $38^\circ = (\mu_g - 1) \times 60^\circ \Rightarrow \mu_g - 1 = \frac{38}{60} \approx 0.633 \Rightarrow \mu_g \approx 1.633$.
For yellow light: $36^\circ = (\mu_y - 1) \times 60^\circ \Rightarrow \mu_y - 1 = \frac{36}{60} = 0.6 \Rightarrow \mu_y \approx 1.6$.
The ratio of refractive indices is:
$\frac{\mu_g}{\mu_y} \approx \frac{1.633}{1.6} \approx 1.02$

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: Frequency and Wavelength Relationship

c) It decreases
[Solution Description]
Frequency ($f$) and wavelength ($\lambda$) are inversely related by the equation $c = f\lambda$, where $c$ is the speed of light. Therefore, if wavelength increases, frequency decreases. The correct answer is "It decreases".

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: Visible Spectrum and Dispersion

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, splitting into its constituent colors due to different wavelengths. The deviation of light depends on its wavelength. Red light has the longest wavelength ($800 \, \text{nm}$) in the visible spectrum, resulting in the least deviation because the refractive index for red light is lower compared to shorter wavelengths like violet. The reason correctly explains why red light deviates the least.

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 and Frequency Relationship

c) Violet
[Solution Description]
The frequency of light is inversely proportional to its wavelength, meaning shorter wavelengths have higher frequencies. In the visible spectrum, violet light has the shortest wavelength ( $4000 \, \text{Å}$ or $4 \times 10^{-7}\, \text{m}$ ) and thus the highest frequency ( $7.5 \times 10^{14}\, \text{Hz}$ ).
Therefore, the correct answer corresponds to the option stating violet.

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: 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: 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

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 inversely proportional to the fourth power of the wavelength ($I \propto \frac{1}{\lambda^4}$). Blue light has a shorter wavelength (around $4000 \, \text{A}$) compared to other visible colors like red ($8000 \, \text{A}$), so it scatters more in the atmosphere. This scattered blue light reaches our eyes from all directions, making the sky appear blue. Therefore, both the Assertion and Reason are true, and the Reason correctly explains the Assertion.