Question

In his experiment on photoelectric effect, Robert A. Millikan found the slope of the cut-off voltage versus frequency of incident light plot to be \( 4.12 \times 10^{-15} \, \text{Vs} \). Calculate the value of Planck’s constant from it.

Hint:

The slope of the cut-off voltage versus frequency plot provides a direct way to calculate Planck's constant using the equation \( eV_{\text{cut}} = h f - \phi \).

Answer 1

In Robert A. Millikan's experiment on the photoelectric effect, the slope of the cut-off voltage versus frequency plot was found to be \( 4.12 \times 10^{-15} \, \text{Vs} \). We are tasked with calculating the value of Planck's constant (\( h \)) from this information.

1. Understanding the Photoelectric Effect Equation:

The photoelectric effect is described by the equation:

\[ eV_{\text{cut-off}} = h \nu - \phi \] where:

• \( e \) is the elementary charge ( \( e = 1.6 \times 10^{-19} \, \text{C} \) ),
• \( V_{\text{cut-off}} \) is the cut-off voltage,
• \( h \) is Planck’s constant,
• \( \nu \) is the frequency of the incident light, and
• \( \phi \) is the work function of the material.

 

2. Relating the Slope to Planck’s Constant:

In the experiment, the plot of \( V_{\text{cut-off}} \) versus \( \nu \) is a straight line. From the photoelectric effect equation, rearranged as:

\[ V_{\text{cut-off}} = \frac{h}{e} \nu - \frac{\phi}{e} \]

This equation is in the form \( y = mx + c \), where:

• \( y = V_{\text{cut-off}} \),
• \( x = \nu \),
• \( m = \frac{h}{e} \) (the slope), and
• \( c = -\frac{\phi}{e} \) (the y-intercept).

Thus, the slope of the plot \( m = \frac{h}{e} \), so we can calculate Planck's constant using the given slope.

 

3. Calculating Planck’s Constant:

The slope is given as \( m = 4.12 \times 10^{-15} \, \text{Vs} \), and we know the value of \( e = 1.6 \times 10^{-19} \, \text{C} \). Using the equation:

\[ \frac{h}{e} = 4.12 \times 10^{-15} \, \text{Vs} \]

We can solve for \( h \):

\[ h = 4.12 \times 10^{-15} \times 1.6 \times 10^{-19} \, \text{Js} \]

4. Performing the Calculation:

\[ h = 6.592 \times 10^{-34} \, \text{Js} \]

5. Final Answer:

The value of Planck's constant \( h \) is approximately \({6.59 \times 10^{-34}} \, \text{Js}\).