Question

What is the ratio between the energies of two radiations one with a wavelength of \[6000\mathop A\limits^0 \] and the other with \[2000\mathop A\limits^0 \]?

Answer 1

Hint:To proceed with the problem, it is important to know about the relationship between energy and the wavelength and the definition of a photon. That is, the energy and the wavelength are inversely proportional to each other. Photon is a packet of energies of electromagnetic radiation also known as light quantum. Now let us solve the problem step by step considering the definition.

Formula Used:
The formula to find the energy associated with a single photon is given by,
\[E = h\nu \]
\[E = \dfrac{{hc}}{\lambda }\]……. (1)
here, \[\nu = \dfrac{c}{\lambda }\]
Where, \[h\] is Planck’s constant, \[c\] is speed of light and the value is \[{\rm{3 \times 1}}{{\rm{0}}^{\rm{8}}}{\rm{m/s}}\] and \[\lambda \] is the wavelength.

Complete step by step solution:
From equation (1) we can write,
\[{E_1} = \dfrac{{hc}}{{{\lambda _1}}}\]…….. (2) and
\[\Rightarrow {E_2} = \dfrac{{hc}}{{{\lambda _2}}}\]……… (3)
Now, taking the ratio of equations (2) and (3) we obtain,
\[\dfrac{{{E_1}}}{{{E_2}}} = \dfrac{{\left( {\dfrac{{hc}}{{{\lambda _1}}}} \right)}}{{\left( {\dfrac{{hc}}{{{\lambda _2}}}} \right)}}\]
\[\Rightarrow \dfrac{{{E_1}}}{{{E_2}}} = \dfrac{{{\lambda _2}}}{{{\lambda _1}}}\]
\[\Rightarrow {\lambda _1} = 6000\mathop A\limits^0 \] and \[{\lambda _2} = 2000\mathop A\limits^0 \]
\[\Rightarrow \dfrac{{{E_1}}}{{{E_2}}} = \dfrac{{2000}}{{6000}}\]
\[\Rightarrow \dfrac{{{E_1}}}{{{E_2}}} = \dfrac{1}{3}\]
\[\therefore {E_2} = 3{E_1}\]

Therefore, the ratio between the energies of the two radiations is obtained as \[\dfrac{1}{3}\].

Note: Planck's constant is used to describe the behaviour of particles and waves at an atomic scale. Planck's constant is one of the main reasons for the development of quantum mechanics. The energy E which is contained in a photon represents the smallest possible packet of energy in an electromagnetic wave, where it is directly proportional to the frequency f. An energy of a photon is equal to its frequency multiplied by the Planck constant. Due to the mass-energy equivalence, the Planck constant also relates mass to frequency. The value of Planck’s constant is given as \[h = 6.625 \times {10^{ - 34}}Js\].