Problem 52 - University Physics with Modern Physics

Question

A maser is a laser-type device that produces electromagnetic waves with frequencies in the microwave and radio-wave bands of the electromagnetic spectrum. 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 s in 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 h? (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

The time for one cycle is approximately \(7.042 \times 10^{-10}\) s. 5.113 trillion cycles occur in 1 hour. \(2.06 \times 10^{26}\) cycles occurred over the Earth's age. The clock would be off by \(4.6 \times 10^4\) seconds.

Step 1: Determine the time for one cycle of the radio wave

The time for one cycle of a wave is the reciprocal of its frequency. Given the frequency \( f = 1,420,405,751.786 \) Hz, we calculate the period \( T \) as follows:\[T = \frac{1}{f} = \frac{1}{1,420,405,751.786} \approx 7.042 \times 10^{-10} \text{ s}\]

Step 2: Calculate cycles in one hour

To find how many cycles occur in 1 hour, we multiply the frequency by the number of seconds in an hour. There are 3600 seconds (60 minutes \( \times \) 60 seconds each) in an hour: \[\text{Cycles in 1 hour} = 1,420,405,751.786 \times 3,600 = 5.113 \times 10^{12} \text{ cycles}\]

Step 3: Determine cycles in 4.6 billion years

First, convert 4.6 billion years into seconds:1 year has about \(3.1536 \times 10^{7}\) seconds. Therefore,\(\ 4.6 \times 10^{9}\times 3.1536 \times 10^{7} \approx 1.45 \times 10^{17}\) seconds. The total number of cycles is:\[\text{Number of cycles} = 1,420,405,751.786 \times 1.45 \times 10^{17} \approx 2.06 \times 10^{26} \text{ cycles}\]

Step 4: Calculate error in time over the age of the Earth

The clock is off by 1 second in 100,000 years. Therefore, during \(4.6 \times 10^9\) years, the error (\(e\)) is:\[e = \frac{4.6 \times 10^9}{100,000} \approx 4.6 \times 10^4 \text{ seconds}\]

Key Concepts

Electromagnetic Spectrum

The electromagnetic spectrum is like a rainbow of waves. It includes a wide range of wavelengths and frequencies.
From gamma rays, which have the smallest wavelengths, to radio waves, which have the longest.
Each type of wave varies in energy, with gamma rays being highly energetic and radio waves being the least.
These waves are made of electric and magnetic fields traveling together through space.
Understanding the electromagnetic spectrum helps us understand how different waves interact with matter.

• Gamma Rays: High energy, used in medical imaging.
• X-Rays: Used to look inside objects.
• Ultraviolet: Causes sunburn but also helpful in disinfection.
• Visible Light: This is what we see with our eyes.
• Infrared: Feels like heat, used in thermal imaging.
• Microwave: Used for cooking and communication.
• Radio Waves: Used for broadcasting radio and TV.

The type of wave is defined by its frequency and wavelength. The higher the frequency, the shorter the wavelength.

Radio Waves

Radio waves are a part of the electromagnetic spectrum with long wavelengths and low frequencies. These characteristics make them suitable for long-distance communication.
They are used in broadcasting, such as radio and television transmission, as well as in mobile phones and wireless networks.
In practical applications, radio waves carry signals by modulating either their amplitude (AM) or frequency (FM). Radio waves are harmless to humans at low frequencies.
When a hydrogen maser emits radio waves, they are precise, making them ideal for frequency standards.

• Long wavelengths make them ideal for traveling long distances.
• Radio waves are easy to generate.
• Can pass through the atmosphere.

These properties make radio waves a fundamental tool for various communications and technologies.

Frequency Measurement

Frequency measurement tells us how many times a wave repeats itself in one second.
It is measured in hertz (Hz), where 1 Hz equals one cycle per second. A cycle is one complete wave oscillation.
For instance, a radio wave with a frequency of 1,420,405,751.786 Hz means the wave oscillates that many times per second. With such precision, radio frequencies can be used in technologies like the hydrogen maser.
Here, high precision in measuring frequency is critical for accurate timekeeping. This accuracy reduces issues in scientific research, telecommunications, and navigation.

• Higher frequency means more cycles in a second.
• Used in determining the energy of electromagnetic waves.
• Radio wave frequencies can range from a few Hz to thousands of MHz.

Understanding frequency measurement is essential for developing precise technology.

Time Measurement

Time measurement has evolved using various devices and concepts.
In high precision environments, atomic clocks are used to measure time with incredible accuracy.
Time can be measured in various units like seconds, minutes, and hours. When using atomic clocks, the challenge lies in maintaining accuracy over extended periods.
They ensure minimal deviation, even with wear and environmental changes.

• Atomic clocks use atomic transitions for precision.
• Critical for navigation systems like GPS.
• Used in defining time standards worldwide.

Measuring time accurately aids scientific endeavors and technological innovations.

Microwave Bands

Microwave bands are specific frequency ranges in the electromagnetic spectrum.
They cover frequencies from 1 gigahertz (GHz) to 300 GHz, falling between radio and infrared waves.
Microwaves have various uses, from cooking food in ovens to transmitting data for satellite and cellular communications.
Also, masers operate within microwave frequencies to provide precise measurements.

• Used in radar technology and space communications.
• Microwave wavelengths allow for significant data transmission.
• Cheaper and more efficient compared to other electromagnetic waves.

Understanding microwave bands helps in comprehending how various technologies utilize these frequencies effectively.

Hydrogen Maser

A hydrogen maser is a device that generates exceptionally stable radio waves. It uses hydrogen atoms to maintain a highly accurate frequency standard.
This makes it one of the most stable oscillators available, often used in scientific research and communication systems.
In a hydrogen maser, hydrogen atoms are manipulated to excite and release specific radio frequencies.
This precision makes hydrogen masers ideal for atomic clocks and highly accurate frequency measurements.

• Offers unmatched frequency precision.
• Used in timekeeping and navigation.
• Helps maintain long-term stability in time measurement systems.

The hydrogen maser symbolizes the advances in technology that permit precise measurements over time.

Atomic Clock

Atomic clocks represent the pinnacle of time measurement technology.
They use the vibrations of atoms, usually cesium or hydrogen, to measure time with extraordinary precision.
Atomic clocks provide timekeeping that deviates by only a second over millions of years.
This accuracy is vital for various scientific and technological applications, including GPS, telecommunications, and synchronizing time across networks.
These clocks work by measuring the radiation emitted by electrons in an atom when they change energy levels.

• Atomic clocks ensure accurate GPS readings.
• Used worldwide as the standard time reference.
• Employ highly stable oscillators like hydrogen masers.

Understanding atomic clocks is essential for grasping how modern technologies keep us in sync globally.