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) \(2.51 \times 10^{30}\) (b) \(2.27 \times 10^{28}\) (c) \(1.72 \times 10^{30}\) (d) \(1.77 \times 10^{27}\)

Step-by-Step Solution

Answer

Option (c): \(1.72 \times 10^{30}\) photons per second.

Step 1: Convert Frequency Units

The given frequency of the radio transmitter is \(880\, \text{kc/s}\) (kilocycles per second). To use it in calculations, we need to convert this to Hz.ewline \(1 \text{ kHz} = 10^3 \text{ Hz}\)ewlineThus, \(880 \text{ kc/s} = 880,000 \text{ Hz}\) or \(8.8 \times 10^5 \text{ Hz}\).

Step 2: Calculate Photon Energy

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 \(f = 8.8 \times 10^5 \text{ Hz}\).ewlineSubstitute the values:ewline\[ E = (6.626 \times 10^{-34}) \times (8.8 \times 10^5) \approx 5.833 \times 10^{-28} \text{ J} \]

Step 3: Calculate Number of Photons

The power output of the transmitter is given as \(1000 \text{ watts}\). The number of photons emitted per second, \( N \), can be found by \[ \text{Power} = N \times \text{Energy per photon} \].ewlineRearrange to find \( N \): ewline\( N = \frac{\text{Power}}{E} = \frac{1000 \text{ W}}{5.833 \times 10^{-28} \text{ J}} \)ewline\( N \approx 1.72 \times 10^{30} \text{ photons/second} \)

Step 4: Match to Answer Choices

The calculated number of photons per second is approximately \(1.72 \times 10^{30}\). Therefore, the correct answer is option (c): \(1.72 \times 10^{30}\).

Key Concepts

Radio Frequency

Radio frequency refers to the electromagnetic waves that are used in radio communications. These waves oscillate at frequencies between 3 Hz to 300 GHz. Radio frequency is represented in terms of cycles per second (Hz), kilocycles/sec (kc/s), or kilohertz (kHz).
In the provided exercise, the radio transmitter operates at a frequency of 880 kc/s, which can also be expressed as 880,000 Hz. This conversion to hertz is necessary in calculations to ensure consistency with standard physical equations, such as those involving photon energy. Understanding and converting frequencies is crucial for calculating the properties of electromagnetic waves, such as their energy.

Photon Energy

Photon energy is the energy carried by a single photon, the fundamental particle of light and all other forms of electromagnetic radiation. The energy of a photon depends on its frequency and can be calculated using the equation:

• \( E = h \times f \)
• Where \( E \) is the energy of the photon in joules, \( h \) is Planck's constant \((6.626 \times 10^{-34} \text{ Js})\), and \( f \) is the frequency in hertz.

In the exercise, with the frequency provided at 880,000 Hz, the photon energy becomes approximately \(5.833 \times 10^{-28} \text{ J}\). Each photon at this frequency carries this small amount of energy, which is a critical component in understanding how energy is transmitted by electromagnetic waves like those from a radio transmitter.

Power and Energy Conversion

Power is the rate at which energy is transferred or converted. In physics, power is generally measured in watts, where one watt equates to one joule per second. For example, in the problem, a 1000-watt transmitter means it transfers 1000 joules of energy per second.
This concept extends to converting energy into photons. If a transmitter radiates energy in the form of photons, we can find the number of photons per second by dividing the total power by the energy of an individual photon. The math involves rearranging the power formula:
\[ N = \frac{\text{Power}}{E} \]
Where \( N \) is the number of photons per second, power is in watts, and \( E \) is the energy per photon. By using this calculation, we determined that approximately \(1.72 \times 10^{30}\) photons are emitted every second by the transmitter.

Planck's Constant

Planck's constant is a fundamental quantity in quantum mechanics, serving as a bridge between the macroscopic and quantum scales. It is denoted as \( h \) and has a value of \(6.626 \times 10^{-34} \text{ Js}\).
This constant is crucial in the formula \( E = h \times f \), linking the energy of a photon to its frequency. Given its tiny magnitude, Planck's constant reflects the small scale at which quantum effects become significant, explaining phenomena like the discrete energy levels in atoms and photon energy calculations.
In contexts like radio transmission, understanding Planck's constant allows engineers and scientists to predict how much energy is carried by high-frequency waves, illustrating the quantum mechanical nature of electromagnetic radiation.