Problem 62 - University Physics with Modern Physics

Question

The Hydrogen Maser. You can use the radio waves generated by a hydrogen maser as a standard of frequency. The frequency of these waves is \(1,420,405,751.786\) hertz. (A hertz is another name for one cycle per second) A clock controlled by a hydrogen maser is off by only 1 sin \(100,000\) years. For the following questions, use only three significant figures. (The large number of significant figures given for the frequency simply illustrates the remarkable accuracy to which it has been measured.) (a) What is the time for one cycle of the radio wave? (b) How many cycles occur in 1 \(\mathrm{h} 7\) (c) How many cycles would have occurred during the age of the earth, which is estimated to be \(4.6 \times 10^{9}\) years? (d) By how many seconds would a hydrogen maser clock be off after a time interval equal to the age of the earth?

Step-by-Step Solution

Answer

(a) 7.04 × 10^{-10} s, (b) 5.11 × 10^{12} cycles, (c) 2.06 × 10^{26} cycles, (d) 4.60 × 10^4 s off.

Step 1: Determine time for one cycle

The time for one cycle is the reciprocal of the frequency. Given the frequency \( f = 1,420,405,751.786 \) Hz, the time \( T \) for one cycle is \( T = \frac{1}{f} \). Calculate this to three significant figures:\[ T = \frac{1}{1,420,405,751.786} \approx 7.04 \times 10^{-10} \text{ seconds}. \]

Step 2: Calculate cycles in 1 hour

Convert 1 hour to seconds (1 hour = 3600 seconds). Using the frequency \( f = 1,420,405,751.786 \) Hz, calculate the number of cycles in 1 hour:\[ \text{Number of cycles} = f \times 3600 \approx 5.11 \times 10^{12} \text{ cycles}. \]

Step 3: Calculate cycles in the age of Earth

Convert the age of the Earth from years to seconds. One year is approximately \(3.154 \times 10^7\) seconds, thus:\[ \text{Age of Earth in seconds} = 4.6 \times 10^9 \times 3.154 \times 10^7 \approx 1.45 \times 10^{17} \text{ seconds}. \]Then calculate the number of cycles in this time period using the frequency:\[ \text{Number of cycles} = f \times 1.45 \times 10^{17} \approx 2.06 \times 10^{26} \text{ cycles}. \]

Step 4: Calculate time deviation in the age of Earth

A hydrogen maser clock is off by 1 second every \(100,000\) years. First, convert \(100,000\) years to seconds:\[ 100,000 \text{ years} = 100,000 \times 3.154 \times 10^7 \approx 3.15 \times 10^{12} \text{ seconds}. \]Now, determine how many such intervals are there in the age of Earth:\[ \frac{1.45 \times 10^{17}}{3.15 \times 10^{12}} \approx 4.60 \times 10^4. \]Finally, the time deviation in seconds would be the same as the number of these intervals:\[ \approx 4.60 \times 10^4 \text{ seconds}. \]

Key Concepts

Frequency

Frequency is a fundamental concept in physics and is an essential part of understanding how waves behave. It refers to how often an event, such as a wave cycle, repeats over a specific period of time. For waves, like the radio waves produced by a hydrogen maser, frequency is measured in hertz (Hz), which represents the number of cycles per second.
The higher the frequency, the more cycles occur in each second. For example, the frequency of radio waves generated by a hydrogen maser is extremely high, at approximately 1,420,405,751.786 Hz. This means that over a billion cycles happen every second!
When dealing with frequency, one can also discuss the reciprocal, which involves calculating the time interval for one complete cycle to occur. This is known as the period and is represented by the letter T, given by the equation \( T = \frac{1}{f} \). Using the hydrogen maser frequency, the period is very short, around \( 7.04 \times 10^{-10} \) seconds per cycle.

• Frequency is measured in Hertz (Hz).
• The hydrogen maser frequency is over a billion cycles per second.
• Period is the reciprocal of frequency.

Radio Waves

Radio waves are a type of electromagnetic wave used for communication and various technologies. They are part of the electromagnetic spectrum and have the longest wavelengths, ranging from about 1 millimeter to 100 kilometers.
These waves are characterized by their frequency. The hydrogen maser, for instance, emits radio waves at an astonishingly precise frequency of about 1.42 GHz. This precise frequency allows hydrogen masers to be used as frequency standards for scientific research and technology.

• Radio waves travel at the speed of light.
• They are used in broadcasting, radar, and other communications.
• The frequency of radio waves determines their propagation characteristics, like how far they can travel and how they can be received.

Hydrogen masers are used as a frequency standard because their precise radio waves are stable and reliable. This stability is crucial for applications requiring high precision, such as in radio astronomy and timekeeping.

Time Deviation

Time deviation refers to the degree to which a timekeeping device, like a clock or watch, strays from accurate time measurement. With high-precision devices like the hydrogen maser clock, the time deviation is exceptionally low.
A hydrogen maser clock is off by only one second every 100,000 years. This remarkable precision allows hydrogen masers to maintain a consistent and reliable measurement of time over long periods. This feature is highly beneficial in scientific research that requires precise timekeeping, like tracking the movements of celestial bodies or synchronizing networks.

• Time deviation is critical for applications like GPS, communication technologies, and scientific experiments.
• Low time deviation means higher accuracy in timing mechanisms.
• The hydrogen maser's incredibly stable frequency contributes to its minimal time deviation.

In the calculation presented, the time deviation for a hydrogen maser over the age of the Earth was found to be about 46,000 seconds, illustrating its reliability over even geological timescales.