Problem 116 - A Complete Resource Book in Chemistry for JEE Main

Question

A 1000 watt radio transmitter operates at a frequen cy of \(880 \mathrm{kc} / \mathrm{sec}\). How many photons per sec does it emit? \(\left[\mathrm{h}=6.626 \times 10^{-34} \mathrm{Js}\right]\) (a) \(2.51 \times 10^{10}\) (b) \(2.27 \times 10^{2 n}\) (c) \(1.72 \times 10^{10}\) (d) \(1.77 \times 10^{27}\)

Step-by-Step Solution

Answer

The transmitter emits approximately \(1.72 \times 10^{30}\) photons per second, not matching given options.

Step 1: Understand the Relationship between Power and Photons

The power emitted by a transmitter is given as 1000 watts. This means 1000 joules of energy is emitted every second. Each photon will carry a specific energy that can be calculated using the frequency of the radiation and Planck's constant.

Step 2: Calculate the Energy per Photon

The energy of a single photon can be calculated using the formula \[ E = h \times f \] where \( h = 6.626 \times 10^{-34} \text{ Js} \) and \( f = 880 \times 10^3 \text{ Hz} \) (since 1 kc/sec = 1000 Hz). Substitute these values to find energy per photon.

Step 2 Calculation

Substituting the given values into the formula, we get: \[ E = 6.626 \times 10^{-34} \text{ Js} \times 880 \times 10^3 \text{ Hz} = 5.83288 \times 10^{-28} ext{ J/photon} \]

Step 3: Calculate Number of Photons Emitted Per Second

The power is the total energy emitted per second. Using the relationship: \[ \text{Number of photons per second} = \frac{\text{Total energy per second}}{\text{Energy per photon}} \] Substitute 1000 J for the total energy per second and the calculated energy per photon into this formula.

Step 3 Calculation

By substituting the known values:\[ \text{Number of photons per second} = \frac{1000 ext{ J/s}}{5.83288 \times 10^{-28} ext{ J/photon}} \approx 1.72 \times 10^{30} \text{ photons/s} \]

Step 4: Compare with Given Options

None of the choices seem to match the calculated answer. The calculated value does not match any of the answer options provided, suggesting a potential typing mistake in the options. Based on rigorous calculation, the correct answer should be approximately \(1.72 \times 10^{30}\).

Key Concepts

Planck's Constant

Planck's constant, represented as \( h \), plays an essential role in quantum mechanics. It's a very tiny but fundamental figure that relates the energy of a photon to the frequency of its associated electromagnetic wave. The importance of Planck's constant is often seen in the formula \( E = h \times f \), where \( E \) is the energy of the photon, and \( f \) stands for frequency.

Planck's constant is valued at approximately \( 6.626 \times 10^{-34} \text{ Js} \). This minuscule number tells us how little energy a single photon carries in typical interactions, which is why we need such significant calculations in physics to observe quantum effects.

• It bridges the gap between macroscopic observations and quantum mechanics.
• Understanding \( h \) allows us to calculate how particles like photons behave under different conditions.

Overall, Planck's constant is a stepping stone into the captivating world of quantum physics, helping us to comprehend the tiny scales at which quantum mechanics operates.

Frequency and Wavelength

Frequency and wavelength are pivotal characteristics of waves, particularly electromagnetic waves like radio waves or light. Frequency, denoted by \( f \), is the number of waves that pass a given point in one second. It's measured in Hertz (Hz). Wavelength is the distance between successive crests of a wave, usually measured in meters.

The relationship between these two properties is inversely proportional; as the frequency of a wave increases, its wavelength decreases. This can be mathematically represented by the equation \( c = \lambda \times f \), where \( c \) is the speed of light, \( \lambda \) is the wavelength, and \( f \) is the frequency.

• A high frequency means more wave crests pass a point in one second.
• A longer wavelength means fewer crests pass that point, indicating a lower frequency.

Understanding these properties helps us make sense of various phenomena, from explaining how radios work to why the sky appears blue. By connecting frequency and wavelength, we unlock a vast potential for applications in technology and science.

Energy of a Photon

The energy of a photon is a concept that connects the worlds of classical and quantum physics, revealing how energy is quantized in packets known as quanta or photons. A photon's energy can be measured using the formula \( E = h \times f \), with \( h \) representing Planck's constant and \( f \) the frequency of the light or radiation.

This equation tells us a few critical things:

• Higher frequency light (like ultraviolet or X-rays) has more energetic photons compared to lower frequency light (such as radio waves).
• Our understanding of a photon's energy enables technologies like solar cells and LEDs.

The quantization of energy revolutionized physics in the early 20th century, helping scientists understand complex phenomena like atomic structures and electromagnetic radiation. Recognizing how a photon carries and transfers energy aids in our grasp of the microscopic universe, paving the way for innovative advancements in science and technology.