Answer :
Certainly! Let's solve the problem of finding the wavelength and energy of a photon with the given frequencies of electromagnetic radiation for both FM and AM radio.
### Step-by-Step Solution:
Constants Needed:
1. Speed of Light (c): Approximately [tex]\(3.00 \times 10^8\)[/tex] meters per second (m/s).
2. Planck's Constant (h): Approximately [tex]\(6.626 \times 10^{-34}\)[/tex] joules-seconds (J·s).
### Part a: FM Radio (Frequency = 100.2 MHz)
1. Convert Frequency to Hertz:
- The frequency of 100.2 MHz is equivalent to [tex]\(100.2 \times 10^6\)[/tex] Hz.
2. Calculate the Wavelength:
- Wavelength ([tex]\(\lambda\)[/tex]) is given by the formula:
[tex]\[
\lambda = \frac{c}{f}
\][/tex]
- Substituting the values:
- [tex]\(c = 3.00 \times 10^8\)[/tex] m/s
- [tex]\(f = 100.2 \times 10^6\)[/tex] Hz
- Calculate:
[tex]\[
\lambda \approx 2.994 \, \text{meters}
\][/tex]
3. Calculate the Energy of the Photon:
- Energy (E) is given by:
[tex]\[
E = h \times f
\][/tex]
- Substituting the values:
- [tex]\(h = 6.626 \times 10^{-34}\)[/tex] J·s
- [tex]\(f = 100.2 \times 10^6\)[/tex] Hz
- Calculate:
[tex]\[
E \approx 6.639252 \times 10^{-26} \, \text{joules}
\][/tex]
### Part b: AM Radio (Frequency = 1070 kHz)
1. Convert Frequency to Hertz:
- The frequency of 1070 kHz is equivalent to [tex]\(1070 \times 10^3\)[/tex] Hz.
2. Calculate the Wavelength:
- Using the same formula for wavelength:
[tex]\[
\lambda = \frac{c}{f}
\][/tex]
- Substituting the values:
- [tex]\(c = 3.00 \times 10^8\)[/tex] m/s
- [tex]\(f = 1070 \times 10^3\)[/tex] Hz
- Calculate:
[tex]\[
\lambda \approx 280.374 \, \text{meters}
\][/tex]
3. Calculate the Energy of the Photon:
- Using the formula for energy:
[tex]\[
E = h \times f
\][/tex]
- Substituting the values:
- [tex]\(h = 6.626 \times 10^{-34}\)[/tex] J·s
- [tex]\(f = 1070 \times 10^3\)[/tex] Hz
- Calculate:
[tex]\[
E \approx 7.08982 \times 10^{-28} \, \text{joules}
\][/tex]
This provides the wavelength and energy for both FM and AM radio frequencies.
### Step-by-Step Solution:
Constants Needed:
1. Speed of Light (c): Approximately [tex]\(3.00 \times 10^8\)[/tex] meters per second (m/s).
2. Planck's Constant (h): Approximately [tex]\(6.626 \times 10^{-34}\)[/tex] joules-seconds (J·s).
### Part a: FM Radio (Frequency = 100.2 MHz)
1. Convert Frequency to Hertz:
- The frequency of 100.2 MHz is equivalent to [tex]\(100.2 \times 10^6\)[/tex] Hz.
2. Calculate the Wavelength:
- Wavelength ([tex]\(\lambda\)[/tex]) is given by the formula:
[tex]\[
\lambda = \frac{c}{f}
\][/tex]
- Substituting the values:
- [tex]\(c = 3.00 \times 10^8\)[/tex] m/s
- [tex]\(f = 100.2 \times 10^6\)[/tex] Hz
- Calculate:
[tex]\[
\lambda \approx 2.994 \, \text{meters}
\][/tex]
3. Calculate the Energy of the Photon:
- Energy (E) is given by:
[tex]\[
E = h \times f
\][/tex]
- Substituting the values:
- [tex]\(h = 6.626 \times 10^{-34}\)[/tex] J·s
- [tex]\(f = 100.2 \times 10^6\)[/tex] Hz
- Calculate:
[tex]\[
E \approx 6.639252 \times 10^{-26} \, \text{joules}
\][/tex]
### Part b: AM Radio (Frequency = 1070 kHz)
1. Convert Frequency to Hertz:
- The frequency of 1070 kHz is equivalent to [tex]\(1070 \times 10^3\)[/tex] Hz.
2. Calculate the Wavelength:
- Using the same formula for wavelength:
[tex]\[
\lambda = \frac{c}{f}
\][/tex]
- Substituting the values:
- [tex]\(c = 3.00 \times 10^8\)[/tex] m/s
- [tex]\(f = 1070 \times 10^3\)[/tex] Hz
- Calculate:
[tex]\[
\lambda \approx 280.374 \, \text{meters}
\][/tex]
3. Calculate the Energy of the Photon:
- Using the formula for energy:
[tex]\[
E = h \times f
\][/tex]
- Substituting the values:
- [tex]\(h = 6.626 \times 10^{-34}\)[/tex] J·s
- [tex]\(f = 1070 \times 10^3\)[/tex] Hz
- Calculate:
[tex]\[
E \approx 7.08982 \times 10^{-28} \, \text{joules}
\][/tex]
This provides the wavelength and energy for both FM and AM radio frequencies.