Three metals have work functions in the ratio 2 3:4. Graphs are drawn for all connecting stopping potential and incident frequency. The graphs have slopes in the ratio

Options

A. `2:3:4`
B. `4:3:2`
C. `6:4:3`
D. `1:1:1`

Answer

To solve the problem, we need to analyze the relationship between the stopping potential (V₀) and the incident frequency (ν) for three metals with given work functions in the ratio of 2:3:4.

### Step-by-step Solution:

1. **Understanding the Photoelectric Effect**:
The photoelectric effect is described by the equation:
\[ E = \phi + KE_{max} \]
where \(E\) is the energy of the incident photon, \(\phi\) is the work function of the metal, and \(KE_{max}\) is the maximum kinetic energy of the emitted electrons.

2. **Relating Energy to Frequency**:
The energy of the incident photon can also be expressed in terms of frequency:
\[ E = h\nu \]
where \(h\) is Planck's constant and \(\nu\) is the frequency of the incident light.

3. **Setting Up the Stopping Potential Equation**:
The stopping potential \(V₀\) is related to the maximum kinetic energy by:
\[ KE_{max} = eV₀ \]
where \(e\) is the charge of the electron. Thus, we can rewrite the photoelectric equation as:
\[ h\nu = \phi + eV₀ \]

4. **Rearranging the Equation**:
Rearranging the equation gives us:
\[ eV₀ = h\nu - \phi \]
or
\[ V₀ = \frac{h}{e}\nu - \frac{\phi}{e} \]

5. **Identifying the Slope**:
This equation is in the form of \(y = mx + c\), where:
- \(y = V₀\)
- \(x = \nu\)
- \(m = \frac{h}{e}\) (the slope)
- \(c = -\frac{\phi}{e}\) (the y-intercept)

6. **Analyzing the Work Function**:
Given that the work functions of the three metals are in the ratio \(2:3:4\), we can denote them as:
\[ \phi_1 = 2k, \quad \phi_2 = 3k, \quad \phi_3 = 4k \]
where \(k\) is a constant.

7. **Slope Independence from Work Function**:
The slope \(\frac{h}{e}\) does not depend on the work function \(\phi\). Therefore, for all three metals, the slope remains the same:
\[ m_1 = m_2 = m_3 = \frac{h}{e} \]

8. **Conclusion**:
Since the slopes for all three metals are equal, the ratio of the slopes is:
\[ m_1 : m_2 : m_3 = 1 : 1 : 1 \]

### Final Answer:
The slopes of the graphs for the three metals are in the ratio \(1:1:1\).

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