Light of wavelength 4000 Å 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. 4 eV
B. 1 eV
C. 2 eV
D. 6 eV

Explanation

To solve the problem, we need to find the work function (φ) of the metal surface using the photoelectric effect equation. The steps are as follows:

### Step 1: Understand the given data
- Wavelength of light (λ) = 4000 Å = 4000 × 10^(-10) m
- Maximum kinetic energy of emitted photoelectrons (K.E. max) = 2 eV

### Step 2: Convert the kinetic energy from eV to joules
1 eV = 1.6 × 10^(-19) J
Thus, K.E. max in joules:
\[ K.E. \text{ max} = 2 \, \text{eV} \times 1.6 \times 10^{-19} \, \text{J/eV} = 3.2 \times 10^{-19} \, \text{J} \]

### Step 3: Use the photoelectric effect equation
The photoelectric effect is described by Einstein's equation:
\[ K.E. \text{ max} = h\nu - \phi \]
Where:
- \( h \) = Planck's constant = \( 6.63 \times 10^{-34} \, \text{Js} \)
- \( \nu \) = frequency of the light
- \( \phi \) = work function of the metal

### Step 4: Calculate the frequency (ν) of the light
The frequency can be calculated using the formula:
\[ \nu = \frac{c}{\lambda} \]
Where:
- \( c \) = speed of light = \( 3 \times 10^8 \, \text{m/s} \)

Substituting the values:
\[ \nu = \frac{3 \times 10^8 \, \text{m/s}}{4000 \times 10^{-10} \, \text{m}} = \frac{3 \times 10^8}{4 \times 10^{-7}} = 7.5 \times 10^{14} \, \text{Hz} \]

### Step 5: Calculate the energy of the incident photons (E)
The energy of a photon is given by:
\[ E = h\nu \]
Substituting the values:
\[ E = 6.63 \times 10^{-34} \, \text{Js} \times 7.5 \times 10^{14} \, \text{Hz} \]
\[ E = 4.9725 \times 10^{-19} \, \text{J} \]

Now, convert this energy to eV:
\[ E = \frac{4.9725 \times 10^{-19} \, \text{J}}{1.6 \times 10^{-19} \, \text{J/eV}} \approx 3.11 \, \text{eV} \]

### Step 6: Calculate the work function (φ)
Using the photoelectric equation:
\[ \phi = h\nu - K.E. \text{ max} \]
Substituting the values:
\[ \phi = 3.11 \, \text{eV} - 2 \, \text{eV} = 1.11 \, \text{eV} \]

### Final Answer
The work function of the metal surface is approximately **1.11 eV**.

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