The photoelectrons emitted from a metal surface:

Options

A. Are all at rest
B. Have the same kinetic energy
C. Have the same momentum
D. Have speeds varying from zero up to certain maximum value

Explanation

To solve the question about the photoelectrons emitted from a metal surface, we will analyze the situation step by step.

### Step 1: Understand the Photoelectric Effect
The photoelectric effect occurs when light (or electromagnetic radiation) hits a metal surface and causes the emission of electrons. The energy of the incoming photons must be greater than the work function (Φ) of the metal for electrons to be emitted.

**Hint:** Remember that the work function is the minimum energy required to remove an electron from the surface of the metal.

### Step 2: Energy of the Incoming Photons
The energy (E) of the incoming photons can be calculated using the formula:
\[ E = h \nu \]
where:
- \( h \) is Planck's constant (approximately \( 6.626 \times 10^{-34} \, \text{Js} \))
- \( \nu \) is the frequency of the incoming light.

**Hint:** The frequency of the light is directly related to its energy; higher frequency means higher energy.

### Step 3: Kinetic Energy of Emitted Electrons
When a photon strikes the metal and an electron is emitted, the kinetic energy (KE) of the emitted electron can be expressed as:
\[ KE = E - \Phi \]
where:
- \( KE \) is the kinetic energy of the emitted electron,
- \( E \) is the energy of the incoming photon,
- \( \Phi \) is the work function of the metal.

**Hint:** The kinetic energy of the emitted electrons depends on the difference between the energy of the incoming photon and the work function of the metal.

### Step 4: Variation in Kinetic Energy
Since different photons can have different energies (depending on their frequency), the emitted photoelectrons can have a range of kinetic energies. If the frequency of the incoming light is constant, the kinetic energy of the emitted electrons will vary based on how much energy is above the work function.

**Hint:** The maximum kinetic energy of the emitted electrons occurs when the energy of the incoming photon is just above the work function.

### Step 5: Momentum of Emitted Electrons
The momentum (p) of the emitted electrons can be calculated using the relation:
\[ p = \sqrt{2m \cdot KE} \]
where \( m \) is the mass of the electron. Since the kinetic energy varies, the momentum will also vary.

**Hint:** Remember that momentum is related to kinetic energy, so if kinetic energy varies, momentum will also vary.

### Conclusion
In summary, the photoelectrons emitted from a metal surface will have different kinetic energies and momenta due to the variation in the energy of the incoming photons relative to the work function of the metal. This results in a range of velocities for the emitted electrons.

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