Question

For the photoelectric effect, the maximum kinetic energy \( E_k \) of the photoelectrons is plotted against the frequency \( \nu \) of the incident photons as shown in figure. The slope of the graph gives

• A. Planck’s constant
• B. Charge of electron
• C. Work function of the metal
• D. Ratio of Planck’s constant to electric charge

Answer 1

The Correct Option is A

The photoelectric effect is described by the equation:

\[ E_k = hf - \phi \]

where:
- \( E_k \) is the kinetic energy of the emitted electrons,
- \( h \) is Planck’s constant,
- \( f \) is the frequency of the incident photons,
- \( \phi \) is the work function of the metal.

When the equation is rearranged in the form of \( y = mx + b \):

\[ E_k = hf - \phi, \]

where:
- \( E_k \) corresponds to \( y \),
- \( f \) corresponds to \( x \),
- \( h \) (Planck’s constant) is the slope \( m \),
- \( -\phi \) is the y-intercept.

Thus, the slope of the graph, which represents the relationship between kinetic energy and frequency, gives Planck’s constant \( h \).

Answer 2

The problem asks to identify the physical quantity represented by the slope of the graph of maximum kinetic energy (\(E_k\)) of photoelectrons versus the frequency (\(\nu\)) of the incident photons in the photoelectric effect.

Concept Used:

The relationship between the maximum kinetic energy of photoelectrons and the frequency of incident radiation is described by Einstein's Photoelectric Equation. This equation is a direct application of the principle of conservation of energy to the photoelectric effect.

The equation is given by:

\[ E_k = h\nu - \phi_0 \]

Where:

• \( E_k \) is the maximum kinetic energy of the emitted photoelectrons.
• \( h \) is Planck's constant, a fundamental constant of nature.
• \( \nu \) is the frequency of the incident photons.
• \( \phi_0 \) is the work function of the metal, which is the minimum energy required to remove an electron from the surface of the material. The work function is a constant for a given metal.

Step-by-Step Solution:

Step 1: Analyze the form of Einstein's Photoelectric Equation.

The photoelectric equation is \( E_k = h\nu - \phi_0 \). We can compare this equation with the standard equation of a straight line, which is:

\[ y = mx + c \]

Here, \( y \) represents the quantity plotted on the vertical axis, \( x \) represents the quantity on the horizontal axis, \( m \) is the slope of the line, and \( c \) is the y-intercept.

Step 2: Correlate the variables in the photoelectric equation with the straight-line equation.

In the given graph, the maximum kinetic energy \( E_k \) is plotted on the y-axis, and the frequency \( \nu \) is plotted on the x-axis.

By comparing \( E_k = h\nu - \phi_0 \) with \( y = mx + c \), we can make the following correspondences:

• The dependent variable \( y \) corresponds to \( E_k \).
• The independent variable \( x \) corresponds to \( \nu \).
• The slope \( m \) corresponds to \( h \) (Planck's constant).
• The y-intercept \( c \) corresponds to \( -\phi_0 \) (the negative of the work function).

The equation can be rewritten to highlight this structure:

\[ \underbrace{E_k}_{y} = \underbrace{(h)}_{m} \underbrace{\nu}_{x} + \underbrace{(-\phi_0)}_{c} \]

Step 3: Interpret the slope of the graph.

Since the equation \( E_k = h\nu - \phi_0 \) is a linear equation in \( \nu \), the graph of \( E_k \) versus \( \nu \) is a straight line. The slope of this straight line is the coefficient of the independent variable \( \nu \), which is \( h \).

\[ \text{Slope} = \frac{\Delta E_k}{\Delta \nu} = h \]

This means that the slope of the graph is a universal constant, independent of the material used, and its value is equal to Planck's constant.

Final Computation & Result:

Based on the comparison of Einstein's photoelectric equation with the general equation of a straight line, the slope of the plot of maximum kinetic energy (\(E_k\)) versus frequency (\(\nu\)) is equivalent to Planck's constant.

Therefore, the slope of the graph gives Planck's constant (h).