In a photoelectric effect, electrons are emitted

Options

A. with a maximum velocity proportional to the frequency of the incident radiation
B. at a rate that is independent of the intensity of the incident radiation
C. only if the frequency of the incident radiation is above a certain threshold value
D. only if the temperature of the emitter is high

Explanation

To solve the question regarding the photoelectric effect and the emission of electrons, we can follow these steps:

### Step 1: Understand the Photoelectric Effect
The photoelectric effect refers to the phenomenon where electrons are emitted from a material (usually a metal) when it is exposed to light (or electromagnetic radiation) of sufficient frequency.

### Step 2: Use Einstein's Photoelectric Equation
Einstein's photoelectric equation is given by:
\[ K_{max} = h\nu - W \]
where:
- \( K_{max} \) is the maximum kinetic energy of the emitted electrons,
- \( h \) is Planck's constant,
- \( \nu \) is the frequency of the incident radiation,
- \( W \) is the work function of the material (the minimum energy needed to remove an electron from the surface).

### Step 3: Relate Maximum Kinetic Energy to Maximum Velocity
The maximum kinetic energy can also be expressed in terms of the maximum velocity (\( v_{max} \)) of the emitted electrons:
\[ K_{max} = \frac{1}{2} mv_{max}^2 \]
where \( m \) is the mass of the electron. Setting these two expressions equal gives:
\[ \frac{1}{2} mv_{max}^2 = h\nu - W \]

### Step 4: Solve for Maximum Velocity
Rearranging the equation to find \( v_{max} \):
\[ v_{max} = \sqrt{\frac{2(h\nu - W)}{m}} \]

### Step 5: Analyze the Options
1. **Maximum Velocity Proportional to Frequency**:
- From the equation \( v_{max} = \sqrt{\frac{2(h\nu - W)}{m}} \), we see that \( v_{max} \) depends on the square root of \( (h\nu - W) \). Therefore, it is not directly proportional to \( \nu \) alone, making this option incorrect.

2. **Rate of Emission Independent of Intensity**:
- The rate of emission of electrons is given by \( n = \frac{I_a}{h\nu} \), where \( I_a \) is the intensity of the incident light. This shows that the rate of emission is directly proportional to the intensity, thus this option is also incorrect.

3. **Frequency Above Threshold Value**:
- The condition for electron emission is that the frequency \( \nu \) must be greater than a certain threshold frequency \( \nu_0 \) (where \( W = h\nu_0 \)). If \( \nu < \nu_0 \), no electrons are emitted regardless of the intensity. This option is correct.

### Conclusion
The correct statement regarding the photoelectric effect is that electrons are emitted only if the frequency of the incident radiation is above a certain threshold value.

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