Light of wavelength 4000A∘ is incident on a metal surface. The maximum kinetic energy of emitted photoelectron is 2 eV. What is the work function of the metal surface?

Options

A. 1 eV
B. 2.1 V
C. 4000 V
D. 3.1 V

Answer

To find the work function of the metal surface when light of wavelength 4000 Å is incident on it, we can use the photoelectric equation derived from Einstein's photoelectric effect theory. The equation is given by:

\[ K_{max} = E - \phi \]

Where:
- \( K_{max} \) is the maximum kinetic energy of the emitted photoelectron.
- \( E \) is the energy of the incident light.
- \( \phi \) is the work function of the metal.

### Step-by-Step Solution:

**Step 1: Convert the wavelength from Ångströms to meters.**

Given:
\[ \text{Wavelength} (\lambda) = 4000 \, \text{Å} = 4000 \times 10^{-10} \, \text{m} \]

**Step 2: Calculate the energy of the incident light using the formula:**
\[ E = \frac{hc}{\lambda} \]

Where:
- \( h \) (Planck's constant) = \( 6.626 \times 10^{-34} \, \text{J s} \)
- \( c \) (speed of light) = \( 3 \times 10^{8} \, \text{m/s} \)

Substituting the values:
\[ E = \frac{(6.626 \times 10^{-34}) \times (3 \times 10^{8})}{4000 \times 10^{-10}} \]

Calculating:
\[ E = \frac{1.9878 \times 10^{-25}}{4 \times 10^{-7}} = 4.9695 \times 10^{-19} \, \text{J} \]

**Step 3: Convert the energy from joules to electron volts.**

Using the conversion factor \( 1 \, \text{eV} = 1.6 \times 10^{-19} \, \text{J} \):
\[ E = \frac{4.9695 \times 10^{-19}}{1.6 \times 10^{-19}} \approx 3.1059 \, \text{eV} \]

**Step 4: Use the photoelectric equation to find the work function.**

Given:
\[ K_{max} = 2 \, \text{eV} \]

Substituting into the equation:
\[ K_{max} = E - \phi \]
\[ 2 = 3.1059 - \phi \]

Rearranging gives:
\[ \phi = 3.1059 - 2 \]
\[ \phi \approx 1.1059 \, \text{eV} \]

**Step 5: Round the work function to a reasonable value.**

Since we are looking for a simple numerical answer, we can round:
\[ \phi \approx 1 \, \text{eV} \]

### Final Answer:
The work function of the metal surface is approximately **1 eV**.

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