The ratio of slopes of `K_(max)` vs. V and `V_(0)` vs. v curves in the photoelectric effect gives (v= freqency. `K_(max)`= maximum kinetic energy, `V_(0)`=stopping potential) :

Options

A. charge of electron
B. Planck's constant
C. work function
D. the ratio of Planck's constant and electronic charge

Answer

To solve the question regarding the ratio of slopes of the \( K_{max} \) vs. \( \nu \) and \( V_0 \) vs. \( \nu \) curves in the photoelectric effect, we will follow these steps:

### Step 1: Understand the Photoelectric Effect
The photoelectric effect is described by Einstein's photoelectric equation:
\[ h\nu = \Phi + K_{max} \]
where:
- \( h \) is Planck's constant,
- \( \nu \) is the frequency of the incident light,
- \( \Phi \) is the work function of the material,
- \( K_{max} \) is the maximum kinetic energy of the emitted electrons.

### Step 2: Derive the Equation for \( K_{max} \)
Rearranging the photoelectric equation gives:
\[ K_{max} = h\nu - \Phi \]
This equation shows that \( K_{max} \) is a linear function of \( \nu \) with a slope of \( h \).

### Step 3: Plot the Graph for \( K_{max} \) vs. \( \nu \)
When we plot \( K_{max} \) on the y-axis and \( \nu \) on the x-axis, the graph will be a straight line:
\[ K_{max} = h\nu - \Phi \]
The slope of this line (let's denote it as \( m_1 \)) is:
\[ m_1 = h \]

### Step 4: Understand the Stopping Potential
The stopping potential \( V_0 \) is related to the maximum kinetic energy by:
\[ K_{max} = eV_0 \]
where \( e \) is the charge of the electron.

### Step 5: Derive the Equation for \( V_0 \)
From the photoelectric equation, we can also express it as:
\[ h\nu = h\nu_0 + eV_0 \]
Rearranging gives:
\[ V_0 = \frac{h\nu}{e} - \frac{h\nu_0}{e} \]
This shows that \( V_0 \) is also a linear function of \( \nu \).

### Step 6: Plot the Graph for \( V_0 \) vs. \( \nu \)
When we plot \( V_0 \) on the y-axis and \( \nu \) on the x-axis, the graph will be:
\[ V_0 = \frac{h}{e}\nu - \frac{h\nu_0}{e} \]
The slope of this line (let's denote it as \( m_2 \)) is:
\[ m_2 = \frac{h}{e} \]

### Step 7: Calculate the Ratio of Slopes
Now, we find the ratio of the slopes:
\[ \text{Ratio} = \frac{m_1}{m_2} = \frac{h}{\frac{h}{e}} = e \]

### Conclusion
The ratio of the slopes of the \( K_{max} \) vs. \( \nu \) curve and the \( V_0 \) vs. \( \nu \) curve is equal to the charge of the electron \( e \).

### Final Answer
The ratio of slopes is \( e \) (the charge of the electron).

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