Question

Consider two radiations of wavelengths} \[ \lambda_1 = 2000 \, \text{Å}, \quad \lambda_2 = 6000 \, \text{Å} \] The ratio of the energies of these two radiations \(\left(\frac{E_1}{E_2}\right)\) is _______ (Nearest integer).

Answer 1

Correct Answer: 3

\textcolor{red}{Step 1: Recall the formula for energy of radiation.}
The energy of a radiation is inversely proportional to its wavelength, given by the formula: \[ E = \frac{hc}{\lambda} \] where \(h\) is Planck's constant, \(c\) is the speed of light, and \(\lambda\) is the wavelength of the radiation. \textcolor{red}{Step 2: Express the ratio of energies.}
The ratio of the energies \(E_1\) and \(E_2\) for the two radiations is: \[ \frac{E_1}{E_2} = \frac{\frac{hc}{\lambda_1}}{\frac{hc}{\lambda_2}} = \frac{\lambda_2}{\lambda_1} \] \textcolor{red}{Step 3: Substitute the given values.}
Substituting \(\lambda_1 = 2000 \, \text{Å}\) and \(\lambda_2 = 6000 \, \text{Å}\): \[ \frac{E_1}{E_2} = \frac{6000}{2000} = 3 \]