The relationship between the energy of radiation and its wavelength is given by:
[tex]\[ E = \frac{h \cdot c}{\lambda} \][/tex]
where:
- [tex]\( E \)[/tex] is the energy of the radiation,
- [tex]\( h \)[/tex] is Planck's constant,
- [tex]\( c \)[/tex] is the speed of light,
- [tex]\( \lambda \)[/tex] is the wavelength of the radiation.
Given two wavelengths:
- [tex]\(\lambda_1 = 8000 \, \text{Å}\)[/tex]
- [tex]\(\lambda_2 = 16000 \, \text{Å}\)[/tex]
The energies corresponding to these wavelengths are:
- [tex]\( E_1 \)[/tex] for [tex]\(\lambda_1\)[/tex],
- [tex]\( E_2 \)[/tex] for [tex]\(\lambda_2\)[/tex].
Since the energy of radiation is inversely proportional to its wavelength:
[tex]\[ E \propto \frac{1}{\lambda} \][/tex]
Therefore, we can express the ratio of [tex]\( E_1 \)[/tex] to [tex]\( E_2 \)[/tex] as:
[tex]\[ \frac{E_1}{E_2} = \frac{\lambda_2}{\lambda_1} \][/tex]
Now, substitute the given wavelengths:
[tex]\[ \frac{E_1}{E_2} = \frac{16000}{8000} = 2 \][/tex]
This tells us that:
[tex]\[ E_1 = 2 \times E_2 \][/tex]
Thus, the value of [tex]\( t \)[/tex] in [tex]\( E_1 = t \times E_2 \)[/tex] is:
[tex]\[ \boxed{2} \][/tex]
