Sodium and copper have work functions `2.3 eV and 4.5 eV` respectively . Then the ratio of the wavelength is nearest

Options

A. `1:2`
B. `4:1`
C. `2:1`
D. `1:4`

Answer

To solve the problem of finding the ratio of the wavelengths corresponding to the work functions of sodium and copper, we can follow these steps:

### Step-by-Step Solution:

1. **Understand the Work Function**:
The work function (φ) is the minimum energy required to remove an electron from a material. For sodium (Na), φ₁ = 2.3 eV, and for copper (Cu), φ₂ = 4.5 eV.

2. **Use the Work Function Formula**:
The relationship between the work function and the wavelength (λ) of the emitted electrons is given by the equation:
\[ φ = \frac{hc}{λ} \]
where:
- \( h \) is Planck's constant (\( 4.135667696 \times 10^{-15} \) eV·s),
- \( c \) is the speed of light (\( 3 \times 10^8 \) m/s),
- \( λ \) is the wavelength.

3. **Set Up the Equations for Both Materials**:
For sodium:
\[ φ₁ = \frac{hc}{λ₁} \implies λ₁ = \frac{hc}{φ₁} \]
For copper:
\[ φ₂ = \frac{hc}{λ₂} \implies λ₂ = \frac{hc}{φ₂} \]

4. **Find the Ratio of Wavelengths**:
To find the ratio \( \frac{λ₂}{λ₁} \):
\[ \frac{λ₂}{λ₁} = \frac{hc/φ₂}{hc/φ₁} = \frac{φ₁}{φ₂} \]

5. **Substitute the Work Functions**:
Now substitute the values of the work functions:
\[ \frac{λ₂}{λ₁} = \frac{2.3 \, \text{eV}}{4.5 \, \text{eV}} = \frac{2.3}{4.5} \]

6. **Calculate the Ratio**:
Simplifying \( \frac{2.3}{4.5} \):
\[ \frac{2.3}{4.5} \approx 0.5111 \]
This can be approximated to \( \frac{1}{2} \).

7. **Final Ratio**:
Therefore, the ratio \( λ₂ : λ₁ \) is approximately \( 2 : 1 \).

### Conclusion:
The ratio of the wavelengths is approximately \( 2 : 1 \).