Maximum kinetic energy `(E_(k))` of a photoelectron varies with the frequency (v) of the incident radiation as

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Answer

To solve the problem of how the maximum kinetic energy \( E_k \) of a photoelectron varies with the frequency \( \nu \) of the incident radiation, we can use Einstein's photoelectric equation. Here’s a step-by-step solution:

### Step 1: Write down Einstein's photoelectric equation
The maximum kinetic energy \( E_k \) of a photoelectron is given by the equation:
\[ E_k = h\nu - \phi \]
where:
- \( E_k \) is the maximum kinetic energy of the photoelectron,
- \( h \) is Planck's constant,
- \( \nu \) is the frequency of the incident radiation,
- \( \phi \) is the work function of the material.

### Step 2: Analyze the equation
From the equation \( E_k = h\nu - \phi \), we can see that:
- The term \( h\nu \) represents the energy of the incoming photon.
- The term \( \phi \) represents the minimum energy required to remove an electron from the material (the work function).

### Step 3: Rearranging the equation
We can rearrange the equation to express it in a linear form:
\[ E_k = h\nu - \phi \]
This can be compared to the linear equation \( y = mx + c \), where:
- \( y \) corresponds to \( E_k \),
- \( x \) corresponds to \( \nu \),
- \( m \) (the slope) corresponds to \( h \),
- \( c \) (the y-intercept) corresponds to \(-\phi\).

### Step 4: Interpret the relationship
The relationship shows that the maximum kinetic energy \( E_k \) is linearly dependent on the frequency \( \nu \). As the frequency increases, the maximum kinetic energy of the emitted photoelectrons also increases linearly, provided that the frequency is above the threshold frequency (where \( E_k \) becomes positive).

### Step 5: Graphical representation
If we were to plot a graph with frequency \( \nu \) on the x-axis and maximum kinetic energy \( E_k \) on the y-axis, we would get a straight line:
- The slope of the line would be \( h \).
- The x-intercept would be at \( \nu = \frac{\phi}{h} \), which indicates the threshold frequency below which no photoelectrons are emitted.

### Final Relation
Thus, the maximum kinetic energy \( E_k \) of a photoelectron varies with the frequency \( \nu \) of the incident radiation as a linear function, represented by the equation:
\[ E_k = h\nu - \phi \]