Problem 12 - Master Resource Book in JEE Main Physics

Question

What will be the number of photons emitted per second by a \(10 \mathrm{~W}\) sodium vapour lamp assuming that \(90 \%\) of the consumed energy is converted into light? [Wavelength of sodium light is \(590 \quad \mathrm{~nm}\), \(\left.h=6.63 \times 10^{-34} \mathrm{~J}_{-\mathrm{s}}\right]\) \(\begin{array}{ll}\text { (a) } 0.267 \times 10^{18} & \text { (b) } 0.267 \times 10^{19}\end{array}\) (c) \(0.267 \times 10^{20}\) (d) \(0.267 \times 10^{17}\)

Step-by-Step Solution

Answer

The number of photons emitted per second is approximately \(0.267 \times 10^{19}\) photons, which corresponds to option (b).

Step 1: Identify the total energy converted into light

First, we calculate the total energy that is converted to light. Given that the sodium vapor lamp has a power of 10 W and 90% of this power is converted into light, the energy converted into light per second is: \[ \text{Energy per second} = 0.9 \times 10 \text{ W} = 9 \text{ J} \]

Step 2: Calculate the energy of a single photon

The energy of a single photon can be calculated using the formula \( E = \frac{hc}{\lambda} \), where \( h \) is Planck's constant \(6.63 \times 10^{-34} \text{ J·s}\), \( c \) is the speed of light \(3 \times 10^8 \text{ m/s}\), and \( \lambda \) is the wavelength \(590 \text{ nm} = 590 \times 10^{-9} \text{ m} \). \[ E = \frac{6.63 \times 10^{-34} \times 3 \times 10^8}{590 \times 10^{-9}} \approx 3.37 \times 10^{-19} \text{ J} \]

Step 3: Calculate the number of photons emitted per second

Now we use the energy of a single photon to find the number of photons emitted per second. We divide the total energy per second converted into light, calculated in Step 1, by the energy of a single photon from Step 2:\[ \text{Number of photons per second} = \frac{9}{3.37 \times 10^{-19}} \approx 2.67 \times 10^{19} \]

Step 4: Compare calculations with provided options

The calculation shows that the number of photons emitted per second is approximately \(2.67 \times 10^{19}\), which matches option (b).

Key Concepts

Sodium Vapor Lamp

A sodium vapor lamp is a type of gas discharge lamp that uses sodium in an excited state to produce light. These lamps are commonly used for street lighting because they are highly efficient and produce a characteristic bright yellow color. The process begins when an electrical current is passed through the sodium vapor, which excites the sodium atoms, causing them to emit photons. These photons are part of the visible light spectrum, predominantly around the wavelength of 590 nm, which is the distinct yellow we see.

Sodium vapor lamps are highly energy-efficient, making them a favorable choice for many outdoor lighting scenarios. This efficiency is due to the substantial portion of electrical energy that is converted into visible light compared to other types of lamps. In our example, 90% of the lamp's energy is successfully converted to light, reducing waste in the form of heat.

Photon Energy Calculation

To understand photon energy, consider a photon as the smallest discrete amount of light energy. The energy of one photon depends on the light's wavelength. We can calculate the energy of a photon using the formula:

• E = \( \frac{hc}{\lambda} \)
• Where h is Planck's constant (\(6.63 \times 10^{-34} \text{ J·s}\))
• c is the speed of light (\(3 \times 10^8 \text{ m/s}\))
• \(\lambda\) is the wavelength of light (590 nm or\(590 \times 10^{-9}\) m)

Plugging these values into our formula gives:\[E = \frac{6.63 \times 10^{-34} \times 3 \times 10^8}{590 \times 10^{-9}} \approx 3.37 \times 10^{-19} \text{ J}\]Thus, each photon from the sodium vapor lamp has an energy of approximately\(3.37 \times 10^{-19} \text{ J}\).This calculated energy is crucial to determine how many photons are emitted per given power, as seen in the exercise.

Visible Light Wavelength

Wavelengths of visible light refer to the range of electromagnetic radiation that can be perceived by the human eye, typically between 400 nm to 700 nm. The sodium vapor lamp emits light at a wavelength of around 590 nm, which sits comfortably within the visible spectrum. This particular wavelength corresponds to a yellow light, which is highly recognizable.

Understanding light wavelength is essential not only for physics but also for practical applications like lighting design. Different wavelengths correspond to different colors, and each has unique properties. For instance, shorter wavelengths have higher energy and can appear more bluish, while longer wavelengths appear redder and have lower energy. The 590 nm wavelength strikes a balance that gives sodium vapor lamps their efficiency and their characteristic yellow glow.