Question

The value of Planck's constant is \[6.63 \times {10^{ - 34}}Js\]. The velocity of light is \[3.0{\text{ }} \times {10^8}m{s^{ - 1}}\]. Which value is closest to the wavelength of a quantum of light with frequency of \[8{\text{ }} \times {\text{ }}{10^{15}}se{c^{ - 1}}\]?
A: \[5 \times {10^{ - 18}}\]
B: \[4 \times {10^1}\]
C: \[3 \times {10^7}\]
D: \[2 \times {10^{ - 25}}\]

Answer 1

Hint: A quantum of light is known as a photon. Planck's constant, h, refers to the ratio of the energy of a photon to its frequency.
$E = h \times v$
$\therefore h = \dfrac{E}{v}$
Where, E is energy of a photon, h is planck’s constant, v is the frequency.

Complete step by step answer:
In the question we are given, Planck's constant (h), velocity of light (c), frequency of quantum of light (v) as listed below:
\[h{\text{ = }}6.63 \times {10^{ - 34}}Js\]
\[c = 3.0{\text{ }} \times {10^8}m{s^{ - 1}}\]
\[v = 8{\text{ }} \times {\text{ }}{10^{15}}se{c^{ - 1}}\]
Now we are required to find the wavelength of a quantum of light i.e. $\lambda = ?$
We will apply the wave equation i.e. $c = v\lambda $
From wave equation:
$v = \dfrac{c}{\lambda }$
Thus, $\lambda = \dfrac{c}{v}$
$\lambda = \dfrac{{3 \times {{10}^8}}}{{8 \times {{10}^{15}}}} = 37.5 \times {10^{ - 9}}m = 37.5nm \simeq 4 \times {10^1}nm$
(the closest value to the wavelength (in nm) of a quantum of light with frequency $8 \times 10^{15} s^{−1}$).

Thus, option B i.e. $4\times 10^1$ is correct.

Additional information: Planck postulated that the energy of light is proportional to the frequency, and the constant that relates them is known as Planck’s constant (h). His work led to Albert Einstein determining that light exists in discrete quanta of energy, or photons.

Note: In case of light waves, wavelength is considered to be a significant parameter and refers to the distance between the corresponding points (i.e. two identical points in the same plane) of two successive waves. Wavelength can be calculated by measuring the distance from crest to crest or from a trough to trough in case of transverse waves while in longitudinal waves, wavelength can be calculated by measuring the distance from rarefaction to rarefaction or compression to compression.