Sodium and copper have work functions 2.3 eV and 4.6 eV respectively . Then the ratio of threshold wavelengths are respectively.

Options

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

Answer

To solve the problem, we need to find the ratio of the threshold wavelengths for sodium and copper given their work functions. The work functions are given as follows:

- Work function of sodium (Φ₁) = 2.3 eV
- Work function of copper (Φ₂) = 4.6 eV

### Step 1: Understand the relationship between work function and threshold wavelength

The threshold wavelength (λ₀) is related to the work function (Φ) by the equation:

\[ \Phi = \frac{hc}{\lambda_0} \]

Where:
- \( h \) is Planck's constant (approximately \( 4.1357 \times 10^{-15} \) eV·s)
- \( c \) is the speed of light (approximately \( 3 \times 10^8 \) m/s)

Rearranging this equation gives us the threshold wavelength:

\[ \lambda_0 = \frac{hc}{\Phi} \]

### Step 2: Calculate the threshold wavelengths for sodium and copper

Using the relationship derived above, we can find the threshold wavelengths for sodium and copper.

For sodium:

\[ \lambda_{0, \text{sodium}} = \frac{hc}{\Phi_{\text{sodium}}} = \frac{hc}{2.3 \, \text{eV}} \]

For copper:

\[ \lambda_{0, \text{copper}} = \frac{hc}{\Phi_{\text{copper}}} = \frac{hc}{4.6 \, \text{eV}} \]

### Step 3: Find the ratio of the threshold wavelengths

Now, we can find the ratio of the threshold wavelengths:

\[ \frac{\lambda_{0, \text{copper}}}{\lambda_{0, \text{sodium}}} = \frac{\frac{hc}{\Phi_{\text{copper}}}}{\frac{hc}{\Phi_{\text{sodium}}}} = \frac{\Phi_{\text{sodium}}}{\Phi_{\text{copper}}} \]

Substituting the values of the work functions:

\[ \frac{\lambda_{0, \text{copper}}}{\lambda_{0, \text{sodium}}} = \frac{2.3 \, \text{eV}}{4.6 \, \text{eV}} = \frac{1}{2} \]

Thus, the ratio of the threshold wavelengths is:

\[ \lambda_{0, \text{copper}} : \lambda_{0, \text{sodium}} = 1 : 2 \]

### Final Answer

The ratio of the threshold wavelengths for sodium and copper is \( 1 : 2 \).

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