Question

By what factor does the energy of a 1-nm X-ray photon exceed that of a 10-MHz radio photon? How many times more energy has a 1-nm gamma ray than a 10-MHz radio photon?

Answer 1

To solve this problem we will apply the concepts related to the relationship between energy and frequency, from the latter we will obtain similar expressions that relate to the wavelength to find the two energy states between the given values. Finally we will make the comparative radius between the two. The relation between energy and frequency is given as,

$E = hf$

Here,

E = Energy

h = Planck's constant

The relation between the speed of the electromagnetic waves (c), frequency (f) and wavelength ($\lambda$ ) is,

$c = f\lambda$

Rearrange the above equation for frequency f as follows

$f = \frac{c}{\lambda}$

Substitute,

$E = h\frac{c}{\lambda}$

The wavelength x-ray or gamma ray photon is

$\lambda = 1.0nm (\frac{1nm}{10^{9}nm})$

$\lambda = 10^{-9} m$

Therefore the energy would be,

$E_1 = \frac{hc}{\lambda}$

$E_1 = \frac{(6.63*!0^{-34}J\cdo s)(3*10^{8}m/s)}{10^{-9}m}$

$E_1 = 19.89*10^{-17} J$

The frequency is given as,

$f = 10MHz (\frac{10^6z}{1.0MHz})$

$f = 10^7Hz$

Now the second energy would be

$E_2 = hf$

$E_2 = (6.63*10^{-27}J\cdot s)(10^7Hz)$

$E_2 = 6.63*10^{-27}J$

Therefore the ratio between them is

$\frac{E_1}{E_2} = \frac{19.89*10^{-17}J}{6.63*10^{-27}J}$

$\frac{E_1}{E_2} = 3*10^{20}$

Therefore the energy of 1nm x ray or gamma ray photon is $3*10^{20}$ times more than energy of 10MHz radio photon