## Energy of the quantum

### Task number: 4376

All electromagnetic radiation is mediated by photons which are characterized by wavelength, frequency, energy and momentum.

Calculate how big the quantum of energy is for the

- Signal from ČR1 Radiožurnál broadcasted from Prague on the frequency of 95 MHz,
- Wave from radio navigation to the southeast of the Ruzyně airport on the frequency of 127 MHz,
- Waves broadcasted/received by a mobile phone working on the frequency of 900 MHz,
- Waves broadcasted/received by a mobile phone working on the frequency of 1 800 MHz,
- Radiation in a microwave which has frequency of 2 450 MHz,
- Signal of satellite digital broadcasting channel ČT1 on the frequency of 12 382 GHz.

For all the cases determine the corresponding wavelength.

#### Notation

*f*_{1}= 95 MHzbroadcasting frequency ČR1 Radiožurnál *f*_{2}= 127 MHzradio navigation frequency near Prague airport *f*_{3}= 900 MHzsignal frequency from mobile phone *f*_{4}= 1 800 MHzsignal frequency from mobile phone *f*_{5}= 2 450 MHzradiation frequency in microwave *f*_{6}= 12 382 MHzfrequency of satellite digital broadcasting channel ČT1 *λ*_{i}= ? (m)wavelength of i-th radiation source *E*_{i}= ? (eV)energy of i-th radiation source

*From tables:**h*= 6.6·10^{−34}J sPlanck’s constant in SI units *h*= 4.1·10^{−15}eV sPlanck’s constant in eV s units *c*= 3.0·10^{8}m s^{−1}speed of light #### Hint

Energy of a photon is proportionate to the frequency of the corresponding wavelength.

Wavelength is inversely proportional to the photon’s frequency.

You can read more about properties of photon here .

#### Solution

Photon has zero rest mass and moves with speed of light. Photon’s overall energy is proportional to the frequency

\[E = hf ,\]*f*of given wavelengthwhere

*h*is the Planck’s constant.We can determine the corresponding wavelength of electromagnetic radiation with the following formula

\[\lambda = {c \over f},\]where

*c*is the speed of light in vacuum.Now we can easily determine the characteristics of given wavelengths:

*f*λ *E*1. 95 MHz 3.16 m 0.4 μeV 2. 127 MHz 2.36 m 0.5 μeV 3. 900 MHz 0.33 m 3.7 μeV 4. 1 800 MHz 0.17 m 7.4 μeV 5. 2 450 MHz 0.12 m 10.1 μeV 6. 12 382 MHz 0.02 m 51.2 μeV #### Answer

Wavelength and quantum of energy for given frequencies are shown in the following table:

*f*λ *E*a. 95 MHz 3.16 m 0.4 μeV b. 127 MHz 2.36 m 0.5 μeV c. 900 MHz 0.33 m 3.7 μeV d. 1 800 MHz 0.17 m 7.4 μeV e. 2 450 MHz 0.12 m 10.1 μeV f. 12 382 MHz 0.02 m 51.2 μeV