Photons of energy 10 eV and be 5eV illuminated on a metal surface. The ratio of their wavelengths are ______

Explanation

To solve the problem of finding the ratio of the wavelengths of two photons with energies of 10 eV and 5 eV, we can follow these steps:

### Step 1: Understand the relationship between energy and wavelength
The energy (E) of a photon is related to its wavelength (λ) by the equation:
\[ E = \frac{hc}{\lambda} \]
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.

### Step 2: Rearranging the equation
We can rearrange the equation to express the wavelength in terms of energy:
\[ \lambda = \frac{hc}{E} \]

### Step 3: Calculate the wavelengths for both photons
Let’s denote:
- \( E_1 = 10 \, \text{eV} \) (for the first photon),
- \( E_2 = 5 \, \text{eV} \) (for the second photon).

Using the rearranged equation:
- For the first photon:
\[ \lambda_1 = \frac{hc}{E_1} = \frac{hc}{10 \, \text{eV}} \]
- For the second photon:
\[ \lambda_2 = \frac{hc}{E_2} = \frac{hc}{5 \, \text{eV}} \]

### Step 4: Find the ratio of the wavelengths
To find the ratio of the wavelengths \( \frac{\lambda_1}{\lambda_2} \):
\[ \frac{\lambda_1}{\lambda_2} = \frac{\frac{hc}{10 \, \text{eV}}}{\frac{hc}{5 \, \text{eV}}} \]

### Step 5: Simplify the ratio
The \( hc \) terms cancel out:
\[ \frac{\lambda_1}{\lambda_2} = \frac{5 \, \text{eV}}{10 \, \text{eV}} = \frac{5}{10} = \frac{1}{2} \]

### Final Answer
Thus, the ratio of their wavelengths is:
\[ \frac{\lambda_1}{\lambda_2} = \frac{1}{2} \]

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