Chapter 6: Problem 117

117\. An AM radio transmitter radiates 500 kW at a frequency of \(760 \mathrm{kHz}\). How many photons per second does the emitter emit?

Short Answer

Expert verified

The AM radio transmitter emits approximately \(9.937 \times 10^{29}\) photons per second.

Step by step solution

Convert given data to the standard units

Before we start the calculation, let's convert the given data into the standard units. The power of the transmitter is given as 500 kW, and we need it in watts (W). The frequency is given as 760 kHz, and we need it in hertz (Hz). Power (P) = 500 kW × 1000 = 500,000 W Frequency (f) = 760 kHz × 1000 = 760,000 Hz Now, we have the data in standard units: P = 500,000 W, and f = 760,000 Hz.

Calculate the energy per photon using the Planck's constant

The energy of a single photon (E) can be calculated using the formula E = hf, where h is the Planck's constant and f is the frequency of the radiation. The value of Planck's constant (h) is approximately \(6.63 \times 10^{-34}\) Js. Using the formula, E = hf, we'll now calculate the energy per photon. E = \( (6.63 \times 10^{-34}) \)^{Js} × (760,000 Hz) E = \(5.0388 \times 10^{-25}\) J So the energy of a single photon is approximately \(5.0388 \times 10^{-25}\) Joules.

Calculate the number of photons emitted per second

We know that the total power of the transmitter is P = 500,000 W. The power (P) can also be written as the total energy emitted per unit time, so P = nE/t, where n is the number of photons emitted, E is the energy per photon, and t is time in seconds. Now, we need to find the number of photons emitted per second, so we'll isolate 'n' from the formula: n = Pt/E Substitute the values P = 500,000 W, t = 1 s, and E = \(5.0388 \times 10^{-25}\) J: n = (500,000 W) × (1 s) / \(5.0388 \times 10^{-25}\) J n ≈ \(\frac{500,000}{5.0388 \times 10^{-25}}\) n ≈ \(9.937 \times 10^{29}\) Therefore, approximately \(9.937 \times 10^{29}\) photons are emitted by the AM radio transmitter per second.

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117\. An AM radio transmitter radiates 500 kW at a frequency of \(760 \mathrm{kHz}\). How many photons per second does the emitter emit?

Short Answer

Step by step solution

Convert given data to the standard units

Calculate the energy per photon using the Planck's constant

Calculate the number of photons emitted per second

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