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 4.6×109 years? (d) By how many seconds would a hydrogen maser clock be off after a time interval equal to the age of the earth?By how many seconds would a hydrogen maser clock be off after a time interval equal to the age of the earth

Answer 1

Answers:

a) $T=7.04(10)^{-10} s$

b) $5.11(10)^{12} cycles$

c) $2.06(10)^{26} cycles$

d) 46000 s

Explanation:

a) Time for one cycle of the radio wave

We know the maser radiowave has a frequency $f$ of $1,420,405,751.786 cycles/s$

In addition we know there is an inverse relation between frequency and time $T$:

$f=\frac{1}{T}$ (1)

Isolating  $T$: $T=\frac{1}{f}$ (2)

$T=\frac{1}{1,420,405,751.786 cycles/s}$ (3)

$T=7.04(10)^{-10} s$ (4) This is the time for 1 cycle

b) Cycles that occur in 1 h

If $1h=3600s$ and we already know the amount of cycles per second $1,420,405,751.786 cycles/s$, then:

$1,420,405,751.786 \frac{cycles}{s}(3600s)=5.11(10)^{12} cycles$ This is the number of cycles in an hour

c) How many cycles would have occurred during the age of the earth, which is estimated to be $4.6(10)^{9} years$?

Firstly, we have to convert this from years to seconds:

$4.6(10)^{9} years \frac{365 days}{1 year} \frac{24 h}{1 day} \frac{3600 s}{1 h}=1.45(10)^{17} s$

Now we have to multiply this value for the frequency of the maser radiowave:

$1,420,405,751.786 cycles/s (1.45(10)^{17} s)=2.06(10)^{26} cycles$ This is the number of cycles in the age of the Earth

d) By how many seconds would a hydrogen maser clock be off after a time interval equal to the age of the earth?

If we have 1 second out for every 100,000 years, then:

$4.6(10)^{9} years \frac{1 s}{100,000 years}=46000 s$

This means the maser would be 46000 s off after a time interval equal to the age of the earth