Question 22

If a source of power $$4$$ kW produces $$10^{20}$$ photons/second, the radiation belong to a part of the spectrum called

Solution

Solution & Explanation

1. Establish the Energy-Power Relation

Power ($$P$$) is defined as the total energy emitted by a radiation source per unit time. If a source emits $$n$$ photons per second, and each individual photon carries an energy $$E$$, the total power is given by:

$$P = n \cdot E$$

We are given the following values from the system:

Source Power ($$P$$): $$4 \,\, \text{kW} = 4 \times 10^3 \,\, \text{W} = 4 \times 10^3 \,\, \text{J/s}$$
Photon Emission Rate ($$n$$): $$10^{20} \,\, \text{photons/second}$$

2. Calculate the Energy of a Single Photon ($$E$$)

Isolating the energy variable from the power equation:

$$E = \frac{P}{n}$$

Substituting our given parameters:

$$E = \frac{4 \times 10^3}{10^{20}} = 4 \times 10^{-17} \,\, \text{J}$$

3. Determine the Wavelength of the Radiation ($$\lambda$$)

According to Planck's quantum theory, the energy of a photon is related to its wavelength ($$\lambda$$) by the formula:

$$E = \frac{h \cdot c}{\lambda} \implies \lambda = \frac{h \cdot c}{E}$$

Where:

$$h$$ (Planck's constant) $$\approx 6.63 \times 10^{-34} \,\, \text{J}\cdot\text{s}$$
$$c$$ (Speed of light) $$\approx 3 \times 10^8 \,\, \text{m/s}$$

Substituting these universal constants into our wavelength equation:

$$\lambda = \frac{6.63 \times 10^{-34} \times 3 \times 10^8}{4 \times 10^{-17}}$$

$$\lambda = \frac{19.89 \times 10^{-26}}{4 \times 10^{-17}} \approx 4.97 \times 10^{-9} \,\, \text{m} = 4.97 \,\, \text{nm}$$

4. Match with the Electromagnetic Spectrum

Let us look at the standard wavelength thresholds defining parts of the electromagnetic spectrum:

Ultraviolet (UV) Rays: $$10 \,\, \text{nm}$$ to $$400 \,\, \text{nm}$$
X-rays: $$0.01 \,\, \text{nm}$$ to $$10 \,\, \text{nm}$$
Gamma ($$\gamma$$) Rays: Less than $$0.01 \,\, \text{nm}$$

Since our calculated wavelength of $$\lambda \approx 4.97 \,\, \text{nm}$$ sits cleanly within the range of $$0.01 \,\, \text{nm}$$ to $$10 \,\, \text{nm}$$, the radiation belongs to the X-rays region.