Two separate monochromatic light beams `A` and `B` of the same intensity (energy per unit area per unit time) are falling normally on a unit area of a metallic surface. Their wavelength are `lambda_(A)` and `lambda_(B)` respectively. Assuming that all the the incident light is used in ejecting the photoelectrons, the ratio of the number of photoelectrons from beam `A` to that from `B` is

Options

A. `((lambda_(A))/(lambda_(B)))`
B. `((lambda_(B))/(lambda_(A)))`
C. `((lambda_(B))/(lambda_(A)))^(2)`
D. `((lambda_(A))/(lambda_(B)))^(2)`

Answer

To solve the problem, we need to determine the ratio of the number of photoelectrons emitted from two monochromatic light beams A and B, given that both beams have the same intensity but different wavelengths.

### Step-by-Step Solution:

1. **Understanding Intensity and Energy of Photons**:
The intensity \( I \) of a light beam is defined as the power per unit area. For monochromatic light, the energy of each photon is given by the equation:
\[ E = \frac{hc}{\lambda} \]
where \( h \) is Planck's constant, \( c \) is the speed of light, and \( \lambda \) is the wavelength of the light.

2. **Relating Intensity to Number of Photons**:
The intensity can also be expressed in terms of the number of photons \( n \) incident per second on a unit area:
\[ I = n \cdot E \]
Substituting the expression for energy, we get:
\[ I = n \cdot \frac{hc}{\lambda} \]
Rearranging this gives:
\[ n = \frac{I \lambda}{hc} \]
This equation shows that the number of photons \( n \) is directly proportional to the wavelength \( \lambda \) when intensity \( I \) is constant.

3. **Calculating the Ratio of Photoelectrons**:
Since the number of photoelectrons emitted is proportional to the number of incident photons (assuming all photons contribute to photoelectron emission), we can write:
\[ \frac{n_A}{n_B} = \frac{\frac{I \lambda_A}{hc}}{\frac{I \lambda_B}{hc}} = \frac{\lambda_A}{\lambda_B} \]
Here, \( n_A \) and \( n_B \) are the number of photoelectrons emitted from beams A and B, respectively.

4. **Final Result**:
Therefore, the ratio of the number of photoelectrons emitted from beam A to that from beam B is:
\[ \frac{n_A}{n_B} = \frac{\lambda_A}{\lambda_B} \]

### Conclusion:
The ratio of the number of photoelectrons emitted from beam A to that from beam B is given by the ratio of their wavelengths:
\[ \frac{n_A}{n_B} = \frac{\lambda_A}{\lambda_B} \]