91Ó°ÊÓ

The given data can be listed below as follows,

• The radio waves’ frequency is,$f_R=1420405751.786\mathrm{Hz}\left(\frac{1\mathrm{cycle}/\mathrm{s}}{1\mathrm{Hz}}\right)=1.420×{10}^9\mathrm{cycle}/\mathrm{s}.$ .
• The time for which the clock turns off in 100000 years is,$t_c=1 s$.
• The earth’s age is, $T_E=4.6×{10}^9 yr$

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.

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 waves’ frequency.

Substituting values in the above expression, we get,

$$\begin{gathered} T=\frac{1}{1.420×{10}^9\mathrm{cycle}/\mathrm{s}} \\ =7.04×{10}^{−10}\mathrm{s}/1\mathrm{cycle} \\ =7.04×{10}^{−10}\mathrm{s} \end{gathered}$$

Thus, the time taken by radio waves to complete one cycle is $7.04×{10}^{-10} s$.

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

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

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

Substituting values in the above expression, we get,

$\begin{array}{r}{f}_c=\left(\frac{3600\mathrm{s}/\mathrm{hr}}{7.04×{10}^{−10}\mathrm{s}/cycle}\right)\\ =5.11×{10}^{12}\mathrm{cycles}/\mathrm{hr}\end{array}$

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

The radio waves’ frequency is given by,

${\mathrm{f}}_{{\mathrm{T}}_{\mathrm{E}}}=\frac{\mathrm{given}\mathrm{time}}{\mathrm{time}\mathrm{for}\mathrm{one}\mathrm{cycle}}$

Here, the time interval is the age of the earth.

Substituting values in the above expression, we get,

$$\begin{gathered} f_{T_E}=\frac{\left(4.6×{10}^9\mathrm{yr}\right)\left(3.156×{10}^7\mathrm{s}/\mathrm{yr}\right)}{7.04×{10}^{−10}\mathrm{s}/\mathrm{cycle}} \\ =2.06×{10}^{26}\mathrm{cycle} \end{gathered}$$

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

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

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

Substitute all the values in the above,

$$\begin{gathered} T_M=\left(1\mathrm{s}\right)\left(\frac{4.6×{10}^9\mathrm{yr}}{1×{10}^5\mathrm{yr}}\right) \\ =4.6×{10}^4\mathrm{s} \end{gathered}$$

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