Question

Radio signals from Voyager 1 in the 1970 s were broadcast at a frequency of 8.4 GHz. On Earth, this radiation was received by an antenna able to detect signals as weak as \(4 \times 10^{-21} \mathrm{W}\). How many photons per second does this detection limit represent?

Short answer

The detection limit represents approximately \(7.18 \times 10^{2}\) photons per second.

Step-by-step solution

Determine the frequency in Hz

First, we need to convert the frequency given in GHz to Hz, as the standard unit for frequency in the photon energy formula is Hz. As 1 GHz = \(1 \times 10^{9}\) Hz, this implies that 8.4 GHz = \(8.4 \times 10^{9}\) Hz.

Calculate the energy of a single photon

Next, we're going to compute the energy of a single photon by using Planck's equation, \(E = h \cdot f\), where \(E\) represents the photon's energy, \(f\) is its frequency, and \(h\) denotes the Planck constant (\(6.63 \times 10^{-34}\) joules-second). Substituting the given frequency, the energy of one photon becomes \(E = 6.63 \times 10^{-34} \cdot 8.4 \times 10^{9} = 5.57 \times 10^{-24} \) joules.

Compute the number of photons per second

Lastly, to calculate the number of photons per second, divide the power of the signal by the energy of a singular photon. Hence, using the signal power \(4 \times 10^{-21} W\) (where 1W = 1 joule/sec), the number of photons per second can be calculated as: \(4 \times 10^{-21}\) divided by \(5.57 \times 10^{-24}\) = \(7.18 \times 10^{2}\) photons per second.

Key concepts

Frequency Conversion

Frequency conversion is the process of changing the unit of frequency from one scale to another.
For radio signals, frequencies are often given in gigahertz (GHz) which is a convenient measurement for high frequency waves, such as microwaves and radio waves.
To make calculations involving energy, such as using Planck's Equation, we convert these frequencies to hertz (Hz) since it is the standard unit.

1 GHz is equal to \(1 \times 10^9\) Hz.
This conversion is essential because calculations in physics often require base units, like joules and seconds, to maintain consistency and accuracy.
By converting 8.4 GHz to Hz, we have \(8.4 \times 10^9\) Hz, which allows us to accurately compute photon energy using Planck's equation.
Keep in mind this conversion step any time you're dealing with frequency-related energy calculations.

Planck's Equation

Planck's Equation establishes a critical relationship between the frequency of a wave and its energy.
The formula is \(E = h \cdot f\), where \(E\) is the energy of a photon, \(h\) is Planck's constant, and \(f\) represents the frequency.
The value of Planck's constant \(h\) is approximately \(6.63 \times 10^{-34}\) joules-second.

Using Planck's Equation helps us move from the abstract concept of frequency to a measurable quantity of energy.
For example, with a frequency of \(8.4 \times 10^9\) Hz, the energy of a single photon is calculated as:\[E = 6.63 \times 10^{-34} \cdot 8.4 \times 10^9 = 5.57 \times 10^{-24} \text{ joules}\]
This equation is fundamental in quantum mechanics and underpins many technologies that rely on the interaction of photons and matter.
Understanding Planck's Equation allows us to compute the energy content of light sources.

Photon Detection Limit

The photon detection limit refers to the minimum energy level at which a detector can identify photons.
This is important in contexts such as space communications, where signals need to be detected over vast distances.

For instance, the detection equipment on Earth for Voyager 1 could sense signals as weak as \(4 \times 10^{-21}\) watts, allowing us to infer the number of photons arriving each second.
This detection limit is crucial because it determines the sensitivity of a system.
To compute the number of photons per second at this detection limit, we divide the signal power by the energy of a single photon:
\[\text{Number of photons per second} = \frac{4 \times 10^{-21}}{5.57 \times 10^{-24}}\approx 718\text{ photons per second}\]
Knowing the photon detection limit helps engineers and scientists design better communication systems to receive faint signals from distant sources.