A light of `4 eV` incident on metal surface of work function `phi eV` . Another light of `2.5 eV` incident on another metal surface of work function `phi_2 eV` . Find the ratio of maximum velocity of photo electrons `(v_1/v_2)`.

Options

A. `(sqrt(4+phi_1/(2.5-phi_2))`
B. `(sqrt(4-phi_1/(2.5-phi_2))`
C. `(sqrt(3-phi_1/(2.5+phi_2))`
D. `(sqrt(6+phi_1/(2.5-phi_2))`

Answer

To solve the problem of finding the ratio of the maximum velocities of photoelectrons emitted from two different metal surfaces when illuminated by light of different energies, we will follow these steps:

### Step 1: Understand the Photoelectric Effect
The maximum kinetic energy (K.E.) of photoelectrons emitted from a metal surface due to incident light can be expressed using the equation:
\[ K.E. = E - \phi \]
where \( E \) is the energy of the incident light (in eV) and \( \phi \) is the work function of the metal (in eV).

### Step 2: Write the Kinetic Energy Equations
For the first metal surface with light energy of \( 4 \, \text{eV} \) and work function \( \phi_1 \):
\[ K.E._1 = 4 - \phi_1 \]

For the second metal surface with light energy of \( 2.5 \, \text{eV} \) and work function \( \phi_2 \):
\[ K.E._2 = 2.5 - \phi_2 \]

### Step 3: Relate Kinetic Energy to Velocity
The kinetic energy can also be expressed in terms of the maximum velocity \( v \) of the emitted photoelectrons:
\[ K.E. = \frac{1}{2} m v^2 \]
where \( m \) is the mass of the electron.

Thus, we can write:
\[ K.E._1 = \frac{1}{2} m v_1^2 \]
\[ K.E._2 = \frac{1}{2} m v_2^2 \]

### Step 4: Set Up the Ratio of Velocities
Now, we can set up the ratio of the maximum velocities \( v_1 \) and \( v_2 \):
\[ \frac{K.E._1}{K.E._2} = \frac{\frac{1}{2} m v_1^2}{\frac{1}{2} m v_2^2} \]
This simplifies to:
\[ \frac{v_1^2}{v_2^2} = \frac{K.E._1}{K.E._2} \]
Taking the square root gives us:
\[ \frac{v_1}{v_2} = \sqrt{\frac{K.E._1}{K.E._2}} \]

### Step 5: Substitute the Kinetic Energy Expressions
Substituting the expressions for kinetic energy:
\[ \frac{v_1}{v_2} = \sqrt{\frac{4 - \phi_1}{2.5 - \phi_2}} \]

### Final Answer
Thus, the ratio of the maximum velocities of the photoelectrons is:
\[ \frac{v_1}{v_2} = \sqrt{\frac{4 - \phi_1}{2.5 - \phi_2}} \]