Q52P - 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 $1 \mathrm{s}$ 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 $1\mathrm{hr}$ ? (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) The time taken by radio waves to complete one cycle is, $1.420\times {10}^9\mathrm{cycle}/\mathrm{s}$.

b) The number of cycles in $1\mathrm{hr}$ (frequency) is, $5.11\times {10}^{12}\mathrm{cycles}/\mathrm{hr}$.

c) The number of cycles that will occur during the age of the earth is, $2.06\times {10}^{26}$ cycle.

d) The number of seconds a hydrogen maser clock is off after a time interval equal to the age of the earth is, $4.6\times {10}^4 \mathrm{s}$.

Identification of given data

The given data can be listed below as,

The radio wave's frequency is,

$\begin{array}{rcl}{f}_R& =& 1420405751.786 \mathrm{Hz}\left(\frac{1\mathrm{cycle}/\mathrm{s}}{1 \mathrm{Hz}}\right).\\ & =& 1.420\times {10}^9\mathrm{cycle}/\mathrm{s}\end{array}$

The time for which clock turns off in 100000 years is, $t_{\mathrm{c}}=1 \mathrm{s}$.

The earth's age is, $T_E=4.6\times {10}^9\mathrm{yr}$

Concept of the Maser

The maser is a device that can produce a coherent, collimated and monochromatic beam of the microwave. It is done by using the principle of laser.

(a) Determination of is the time for one cycle of the hydrogen maser

The time period of a cycle of the radio wave in a hydrogen maser is given by,

$T=\frac{1}{f_R}$

Here, $f_R$ is the radio wave's frequency.

Substituting values,

$\begin{array}{rcl}T& =& \frac{1}{1.420\times {10}^9\mathrm{cycle}/\mathrm{s}}\\ & =& 7.04\times {10}^{-10} \mathrm{s}/1\mathrm{cycle}\\ & =& 7.04\times {10}^{-10} \mathrm{s}\end{array}$

Thus, the time taken by radio waves to complete one cycle is $7.04\times {10}^{-10} \mathrm{S}$.

(b) Determination of the number of cycles that occur in 1 hr

The number of cycles completed in a given time interval is given by,

$\mid f_c=\left(\frac{T_g}{T}\right)$

Here, $T_g$ is the given time to complete a cycle.

Substitute the values above,

$\begin{array}{rcl}{f}_c& =& \left(\frac{3600 \mathrm{s}/\mathrm{hr}}{7.04\times {10}^{-10} \mathrm{s}/\mathrm{cycle}}\right)\\ & =& 5.11\times {10}^{12}\mathrm{cycles}/\mathrm{hr}\end{array}$

Thus, the number of cycles that can be performed in a period of one hour is $5.11\times {10}^{12}$ cycles.

(c) Determination of the number of cycles that occurred during the age of the earth

The radio waves frequency is given by,

$f_{T_{\underset{¯}{E}}}=\frac{\text{ given time }}{\text{ time for one cycle }}$

Here, time interval is the age of the earth.

Substituting values,

$\begin{array}{rcl}{f}_{T_E}& =& \frac{\left(4.6\times {10}^9\mathrm{yr}\right)\left(3.156\times {10}^7 \mathrm{s}/\mathrm{yr}\right)}{7.04\times {10}^{-10} \mathrm{s}/\mathrm{cycle}}\\ & =& 2.06\times {10}^{26}\mathrm{cycle}\end{array}$

Thus, the number of cycles that will occur in the time interval equal to the age of the earth is $2.06\times {10}^{26}\mathrm{Cycle}$.

(d) Determination of the number of seconds a hydrogen maser clock be off after a time interval equal to the age of the earth

The number of second maser clock be off after time equal to the age of the earth can be expressed as,

$T_M=($ duration of clock be off $)\left(\frac{\text{ given time }}{\text{ No. of years after which clock turns off }}\right)$

Substitute all the values in the above,

$\begin{array}{rcl}{T}_M& =& (1 \mathrm{s})\left(\frac{4.6\times {10}^9\mathrm{yr}}{1\times {10}^5\mathrm{yr}}\right)\\ & =& 4.6\times {10}^4 \mathrm{s}\end{array}$

Thus, the number of seconds a hydrogen maser clock is off after a time interval equal to the age of the earth is $4.6\times {10}^4 \mathrm{s}$.

Final Solution

a) Radio waves can perform $1.420\times {10}^9$ cycle $/\mathrm{s}$.

b) In one hour, they can perform $5.11\times {10}^{12}$ cycles.

c) cycles that will occur during the age of the earth are $2.06\times {10}^{26}$.

d) A hydrogen maser clock will remain off, after a time interval equal to the age of the earth, is $4.6\times {10}^4 \mathrm{s}$.