In the photoelectric effect, the maximum speed of electrons is found to be `6xx10^(5)ms^(-1)`. The wavelength used is `4000Å`. The work function of the metal is

Options

A. 2.2 eV
B. 2.076 eV
C. 2.3 eV
D. 2.4 eV

Answer

To find the work function of the metal in the photoelectric effect, we can follow these steps:

### Step 1: Calculate the maximum kinetic energy of the electrons
The maximum kinetic energy (K.E.) of the emitted electrons can be calculated using the formula:
\[ K.E. = \frac{1}{2} mv^2 \]
where \( m \) is the mass of the electron and \( v \) is the maximum speed of the electrons.

Given:
- \( v = 6 \times 10^5 \, \text{m/s} \)
- Mass of the electron \( m = 9.1 \times 10^{-31} \, \text{kg} \)

Substituting the values:
\[ K.E. = \frac{1}{2} \times 9.1 \times 10^{-31} \times (6 \times 10^5)^2 \]

### Step 2: Calculate \( (6 \times 10^5)^2 \)
Calculating the square of the speed:
\[ (6 \times 10^5)^2 = 36 \times 10^{10} = 3.6 \times 10^{11} \]

### Step 3: Substitute back to find K.E.
Now substituting back into the K.E. formula:
\[ K.E. = \frac{1}{2} \times 9.1 \times 10^{-31} \times 3.6 \times 10^{11} \]
\[ K.E. = 0.5 \times 9.1 \times 3.6 \times 10^{-20} = 16.38 \times 10^{-20} \, \text{J} \]
\[ K.E. = 1.638 \times 10^{-19} \, \text{J} \]

### Step 4: Convert K.E. to electron volts
To convert joules to electron volts, we use the conversion factor \( 1 \, \text{eV} = 1.6 \times 10^{-19} \, \text{J} \):
\[ K.E. = \frac{1.638 \times 10^{-19}}{1.6 \times 10^{-19}} \approx 1.024 \, \text{eV} \]

### Step 5: Calculate the incident energy of the photons
The energy of the incident photons can be calculated using the formula:
\[ E = \frac{12400 \, \text{eV} \cdot \text{Å}}{\lambda} \]
Given that \( \lambda = 4000 \, \text{Å} \):
\[ E = \frac{12400}{4000} = 3.1 \, \text{eV} \]

### Step 6: Use the photoelectric equation to find the work function
According to the photoelectric effect equation:
\[ E = K.E. + \phi \]
where \( \phi \) is the work function. Rearranging gives:
\[ \phi = E - K.E. \]
Substituting the values:
\[ \phi = 3.1 \, \text{eV} - 1.024 \, \text{eV} = 2.076 \, \text{eV} \]

### Final Answer
The work function of the metal is approximately:
\[ \phi \approx 2.076 \, \text{eV} \]

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