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The energy of a radiation of wavelength [tex]$8000 \AA$[/tex] is [tex]$E_1$[/tex] and the energy of a radiation of wavelength [tex]$16000 \AA$[/tex] is [tex]$E_2$[/tex]. If [tex]$E_1 = t \times E_2$[/tex], then the value of [tex]$t$[/tex] is:

Sagot :

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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]
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