The relationship between energy (E) of wavelength `2000 A^(0) and 8000 A^(0)` , respectively is

Options

A. `E_1 = 4 E_2`
B. `E_1=2E_2`
C. `E_1=(E_2)/2`
D. `E_1=(E_2)/4`

Answer

To solve the problem regarding the relationship between the energy of two wavelengths, 2000 Å and 8000 Å, we can use the formula derived from Planck's theory.

### Step-by-Step Solution:

1. **Understanding the Formula**:
According to Planck's theory, the energy (E) of a photon is given by the equation:
\[ E = \frac{hc}{\lambda} \]
where:
- \(E\) is the energy,
- \(h\) is Planck's constant,
- \(c\) is the speed of light,
- \(\lambda\) is the wavelength.

2. **Setting Up the Energies**:
Let:
- \(E_1\) be the energy corresponding to the wavelength \(2000 \, \text{Å}\),
- \(E_2\) be the energy corresponding to the wavelength \(8000 \, \text{Å}\).

3. **Using the Energy Formula**:
From the formula, we can express the energies as:
\[ E_1 = \frac{hc}{2000} \]
\[ E_2 = \frac{hc}{8000} \]

4. **Finding the Ratio of Energies**:
To find the relationship between \(E_1\) and \(E_2\), we can take the ratio:
\[ \frac{E_1}{E_2} = \frac{\frac{hc}{2000}}{\frac{hc}{8000}} \]

5. **Simplifying the Ratio**:
The \(hc\) terms cancel out:
\[ \frac{E_1}{E_2} = \frac{8000}{2000} \]

6. **Calculating the Value**:
Now, simplifying the fraction:
\[ \frac{8000}{2000} = 4 \]
Therefore, we have:
\[ E_1 = 4 \times E_2 \]

7. **Conclusion**:
The energy corresponding to the wavelength of \(2000 \, \text{Å}\) is 4 times greater than that corresponding to \(8000 \, \text{Å}\).

### Final Answer:
Thus, the relationship between the energies is:
\[ E_1 = 4 \times E_2 \]