A photon of light with wavelength 700 nm has energy E. A photon of light of what wavelength would correspond to energy 2E ?

Explanation

To solve the problem, we need to find the wavelength of a photon that corresponds to an energy of \(2E\), given that a photon with a wavelength of 700 nm has energy \(E\).

### Step-by-Step Solution:

1. **Understand the relationship between energy and wavelength**:
The energy \(E\) of a photon is given by the formula:
\[ E = \frac{hc}{\lambda} \]
where:
- \(E\) is the energy of the photon,
- \(h\) is Planck's constant (\(6.626 \times 10^{-34} \, \text{Js}\)),
- \(c\) is the speed of light (\(3 \times 10^8 \, \text{m/s}\)),
- \(\lambda\) is the wavelength of the photon.

2. **Set up the equation for the given photon**:
For the photon with wavelength \(700 \, \text{nm}\) (which is \(700 \times 10^{-9} \, \text{m}\)), the energy is:
\[ E = \frac{hc}{700 \times 10^{-9}} \]

3. **Determine the energy for the second photon**:
We want to find the wavelength \(\lambda_2\) for a photon that has energy \(2E\):
\[ 2E = \frac{hc}{\lambda_2} \]

4. **Relate the two energies**:
Since we know that:
\[ E = \frac{hc}{700 \times 10^{-9}} \]
We can express \(2E\) as:
\[ 2E = 2 \left(\frac{hc}{700 \times 10^{-9}}\right) \]

5. **Set up the ratio of energies and wavelengths**:
Using the relationship \(E \propto \frac{1}{\lambda}\), we can set up the following ratio:
\[ \frac{E}{2E} = \frac{\lambda_2}{700 \times 10^{-9}} \]
This simplifies to:
\[ \frac{1}{2} = \frac{\lambda_2}{700 \times 10^{-9}} \]

6. **Solve for \(\lambda_2\)**:
Rearranging the equation gives:
\[ \lambda_2 = 700 \times 10^{-9} \times \frac{1}{2} = \frac{700 \times 10^{-9}}{2} = 350 \times 10^{-9} \, \text{m} \]
Converting to nanometers:
\[ \lambda_2 = 350 \, \text{nm} \]

### Final Answer:
The wavelength of the photon that corresponds to energy \(2E\) is \(350 \, \text{nm}\).

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