Ratio of Energies of Two Radiations with Given Wavelengths

The energy of a radiation of wavelength $\lambda$ is given by Planck's equation: $E = \frac{hc}{\lambda}$

where:

• $h$ is Planck's constant,
• $c$ is the speed of light,
• $\lambda$ is the wavelength of the radiation.

For the first radiation with wavelength $\lambda_1$: $E_1 = \frac{hc}{\lambda_1}$

For the second radiation with wavelength $\lambda_2$: $E_2 = \frac{hc}{\lambda_2}$

Taking the ratio of the energies $\left(\frac{E_1}{E_2}\right)$: $\frac{E_1}{E_2} = \frac{\frac{hc}{\lambda_1}}{\frac{hc}{\lambda_2}} = \frac{\lambda_2}{\lambda_1}$

Given:

• $\lambda_1 = 2000 \text{ \AA}$
• $\lambda_2 = 6000 \text{ \AA}$

Substituting the given values into the ratio: $\frac{E_1}{E_2} = \frac{6000 \text{ \AA}}{2000 \text{ \AA}} = 3$

Thus, the ratio of the energies of these two radiations is $3$.