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Sagot :
Certainly! Let's break down the problem step by step.
Firstly, let's discuss the concept of the critical angle and how it relates to the refractive index of a medium:
1. Critical Angle and Refractive Index:
The critical angle for light traveling from a medium (with refractive index [tex]\( n \)[/tex]) to air (refractive index = 1) is the angle of incidence at which light is refracted along the boundary, making the angle of refraction 90 degrees. This can be expressed mathematically by:
[tex]\[ \sin(\theta_c) = \frac{1}{n} \][/tex]
where [tex]\(\theta_c\)[/tex] is the critical angle.
2. Given Critical Angle:
The problem states that the critical angle ([tex]\(\theta_c\)[/tex]) is 30°.
3. Calculate the Refractive Index:
Using the formula:
[tex]\[ \sin(30°) = \frac{1}{2} \][/tex]
Therefore, the refractive index ([tex]\( n \)[/tex]) is calculated as:
[tex]\[ n = \frac{1}{\sin(30°)} = \frac{1}{0.5} = 2 \][/tex]
4. Velocity of Light in the Medium:
The speed of light in a vacuum ([tex]\( c \)[/tex]) is [tex]\( 3 \times 10^8 \)[/tex] m/s. The speed of light in the medium ([tex]\( v \)[/tex]) is related to the speed of light in vacuum ([tex]\( c \)[/tex]) and the refractive index ([tex]\( n \)[/tex]) by:
[tex]\[ v = \frac{c}{n} \][/tex]
Substituting the values:
[tex]\[ v = \frac{3 \times 10^8 \text{ m/s}}{2} = 1.5 \times 10^8 \text{ m/s} \][/tex]
Hence, the velocity of light in the medium is [tex]\( 1.5 \times 10^8 \)[/tex] m/s. So the correct option is:
[tex]\[ \text{d. } 1.5 \times 10^8 \text{ m/s} \][/tex]
5. Maximum Deviation in a Prism:
When light passes through a prism, the deviation depends on the wavelength of light. Shorter wavelengths (like blue light) are refracted more than longer wavelengths (like red light) due to dispersion.
- Among the given color options (Yellow, Blue, Green, Orange), the color with the shortest wavelength will suffer the maximum deviation.
6. Choosing the Maximum Deviation Color:
Blue light has the shortest wavelength among Yellow, Blue, Green, and Orange, and therefore suffers the maximum deviation.
So the correct answer for maximum deviation in a prism is:
[tex]\[ \text{b. Blue} \][/tex]
To summarize:
- Velocity of light in the medium: [tex]\( 1.5 \times 10^8 \)[/tex] m/s (Option d)
- Color suffering maximum deviation in a prism: Blue (Option b)
Firstly, let's discuss the concept of the critical angle and how it relates to the refractive index of a medium:
1. Critical Angle and Refractive Index:
The critical angle for light traveling from a medium (with refractive index [tex]\( n \)[/tex]) to air (refractive index = 1) is the angle of incidence at which light is refracted along the boundary, making the angle of refraction 90 degrees. This can be expressed mathematically by:
[tex]\[ \sin(\theta_c) = \frac{1}{n} \][/tex]
where [tex]\(\theta_c\)[/tex] is the critical angle.
2. Given Critical Angle:
The problem states that the critical angle ([tex]\(\theta_c\)[/tex]) is 30°.
3. Calculate the Refractive Index:
Using the formula:
[tex]\[ \sin(30°) = \frac{1}{2} \][/tex]
Therefore, the refractive index ([tex]\( n \)[/tex]) is calculated as:
[tex]\[ n = \frac{1}{\sin(30°)} = \frac{1}{0.5} = 2 \][/tex]
4. Velocity of Light in the Medium:
The speed of light in a vacuum ([tex]\( c \)[/tex]) is [tex]\( 3 \times 10^8 \)[/tex] m/s. The speed of light in the medium ([tex]\( v \)[/tex]) is related to the speed of light in vacuum ([tex]\( c \)[/tex]) and the refractive index ([tex]\( n \)[/tex]) by:
[tex]\[ v = \frac{c}{n} \][/tex]
Substituting the values:
[tex]\[ v = \frac{3 \times 10^8 \text{ m/s}}{2} = 1.5 \times 10^8 \text{ m/s} \][/tex]
Hence, the velocity of light in the medium is [tex]\( 1.5 \times 10^8 \)[/tex] m/s. So the correct option is:
[tex]\[ \text{d. } 1.5 \times 10^8 \text{ m/s} \][/tex]
5. Maximum Deviation in a Prism:
When light passes through a prism, the deviation depends on the wavelength of light. Shorter wavelengths (like blue light) are refracted more than longer wavelengths (like red light) due to dispersion.
- Among the given color options (Yellow, Blue, Green, Orange), the color with the shortest wavelength will suffer the maximum deviation.
6. Choosing the Maximum Deviation Color:
Blue light has the shortest wavelength among Yellow, Blue, Green, and Orange, and therefore suffers the maximum deviation.
So the correct answer for maximum deviation in a prism is:
[tex]\[ \text{b. Blue} \][/tex]
To summarize:
- Velocity of light in the medium: [tex]\( 1.5 \times 10^8 \)[/tex] m/s (Option d)
- Color suffering maximum deviation in a prism: Blue (Option b)
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