Answer :
To solve this problem, we need to determine the wavelength of a quantum of light given certain parameters. We will use the relationship between the speed of light, frequency, and wavelength.
Here's the step-by-step solution:
1. Given Values:
- Planck's constant, [tex]\( h \)[/tex] = [tex]\( 6.63 \times 10^{-34} \)[/tex] Js (though this value isn't directly needed for calculating wavelength, it's given in the problem).
- Speed of light, [tex]\( c \)[/tex] = [tex]\( 3.0 \times 10^8 \)[/tex] m/s.
- Frequency, [tex]\( f \)[/tex] = [tex]\( 8 \times 10^{15} \)[/tex] Hz.
2. Formula for Wavelength:
The wavelength ([tex]\( \lambda \)[/tex]) of light can be calculated using the formula:
[tex]\[ \lambda = \frac{c}{f} \][/tex]
where:
- [tex]\( \lambda \)[/tex] is the wavelength,
- [tex]\( c \)[/tex] is the speed of light,
- [tex]\( f \)[/tex] is the frequency.
3. Calculate the Wavelength in Meters:
[tex]\[ \lambda = \frac{3.0 \times 10^8 \text{ m/s}}{8 \times 10^{15} \text{ Hz}} \][/tex]
[tex]\[ \lambda = \frac{3.0}{8} \times 10^{8-15} \text{ m} \][/tex]
[tex]\[ \lambda = 0.375 \times 10^{-7} \text{ m} \][/tex]
[tex]\[ \lambda = 3.75 \times 10^{-8} \text{ m} \][/tex]
4. Convert the Wavelength to Nanometers:
Since 1 meter = [tex]\( 1 \times 10^9 \)[/tex] nanometers:
[tex]\[ \lambda = 3.75 \times 10^{-8} \text{ m} \times 10^9 \text{ nm/m} \][/tex]
[tex]\[ \lambda = 37.5 \text{ nm} \][/tex]
5. Select the Closest Value from the Given Options:
The wavelengths provided in the choices are in different scales. Let's review them and see which one is closest to our calculated value of 37.5 nm.
- Option (a): [tex]\( 3 \times 10^7 \)[/tex] nm (this is [tex]\( 30,000,000 \)[/tex] nm - much too large).
- Option (b): [tex]\( 2 \times 10^{-25} \)[/tex] nm (this is [tex]\( 0.000...00002 \)[/tex] nm - much too small).
- Option (c): [tex]\( 5 \times 10^{-18} \)[/tex] nm (this is [tex]\( 0.000...00005 \)[/tex] nm - also too small).
- Option (d): [tex]\( 4 \times 10^1 \)[/tex] nm (this is [tex]\( 40 \)[/tex] nm - close to 37.5 nm).
Therefore, the value closest to the wavelength in nanometers of a quantum of light with the given frequency is:
[tex]\[ \boxed{4 \times 10^1} \][/tex]
So, the correct option is (d) [tex]\( 4 \times 10^1 \)[/tex] nm.
Here's the step-by-step solution:
1. Given Values:
- Planck's constant, [tex]\( h \)[/tex] = [tex]\( 6.63 \times 10^{-34} \)[/tex] Js (though this value isn't directly needed for calculating wavelength, it's given in the problem).
- Speed of light, [tex]\( c \)[/tex] = [tex]\( 3.0 \times 10^8 \)[/tex] m/s.
- Frequency, [tex]\( f \)[/tex] = [tex]\( 8 \times 10^{15} \)[/tex] Hz.
2. Formula for Wavelength:
The wavelength ([tex]\( \lambda \)[/tex]) of light can be calculated using the formula:
[tex]\[ \lambda = \frac{c}{f} \][/tex]
where:
- [tex]\( \lambda \)[/tex] is the wavelength,
- [tex]\( c \)[/tex] is the speed of light,
- [tex]\( f \)[/tex] is the frequency.
3. Calculate the Wavelength in Meters:
[tex]\[ \lambda = \frac{3.0 \times 10^8 \text{ m/s}}{8 \times 10^{15} \text{ Hz}} \][/tex]
[tex]\[ \lambda = \frac{3.0}{8} \times 10^{8-15} \text{ m} \][/tex]
[tex]\[ \lambda = 0.375 \times 10^{-7} \text{ m} \][/tex]
[tex]\[ \lambda = 3.75 \times 10^{-8} \text{ m} \][/tex]
4. Convert the Wavelength to Nanometers:
Since 1 meter = [tex]\( 1 \times 10^9 \)[/tex] nanometers:
[tex]\[ \lambda = 3.75 \times 10^{-8} \text{ m} \times 10^9 \text{ nm/m} \][/tex]
[tex]\[ \lambda = 37.5 \text{ nm} \][/tex]
5. Select the Closest Value from the Given Options:
The wavelengths provided in the choices are in different scales. Let's review them and see which one is closest to our calculated value of 37.5 nm.
- Option (a): [tex]\( 3 \times 10^7 \)[/tex] nm (this is [tex]\( 30,000,000 \)[/tex] nm - much too large).
- Option (b): [tex]\( 2 \times 10^{-25} \)[/tex] nm (this is [tex]\( 0.000...00002 \)[/tex] nm - much too small).
- Option (c): [tex]\( 5 \times 10^{-18} \)[/tex] nm (this is [tex]\( 0.000...00005 \)[/tex] nm - also too small).
- Option (d): [tex]\( 4 \times 10^1 \)[/tex] nm (this is [tex]\( 40 \)[/tex] nm - close to 37.5 nm).
Therefore, the value closest to the wavelength in nanometers of a quantum of light with the given frequency is:
[tex]\[ \boxed{4 \times 10^1} \][/tex]
So, the correct option is (d) [tex]\( 4 \times 10^1 \)[/tex] nm.
