To tackle the question about the relationship between the velocities of photoelectrons emitted from identical photocathodes when exposed to light of different frequencies, we can draw on the principles of the photoelectric effect. The key here is to understand how the energy of the incoming photons relates to the kinetic energy of the emitted electrons.
Understanding the Photoelectric Effect
The photoelectric effect describes how light can eject electrons from a material. When a photon strikes a photocathode, it transfers its energy to an electron. The energy of a photon is given by the equation:
E = hν
where E is the energy, h is Planck's constant, and ν is the frequency of the light. The kinetic energy (KE) of the emitted electron can be expressed as:
KE = (1/2)mv²
where m is the mass of the electron and v is its velocity.
Relating Frequencies and Velocities
For two frequencies, ν₁ and ν₂, corresponding to velocities v₁ and v₂, we can write the energy equations for each case:
- For frequency ν₁: hν₁ = (1/2)m(v₁)²
- For frequency ν₂: hν₂ = (1/2)m(v₂)²
Deriving Relationships
From these equations, we can express the kinetic energies in terms of the frequencies:
- v₁² = (2hν₁)/m
- v₂² = (2hν₂)/m
Now, if we subtract these two equations, we get:
v₁² - v₂² = (2h/m)(ν₁ - ν₂)
This can be factored as:
(v₁ - v₂)(v₁ + v₂) = (2h/m)(ν₁ - ν₂)
Analyzing the Options
Now, let's analyze the provided options:
- (1) x - y = [2h/m(x - y)]^(1/2)
- (2) x² - y² = 2h/m(x - y)
- (3) x + y = [2h/m(x - y)]^(1/2)
- (4) x² + y² = 2h/m(x - y)
From our derived equation, we see that option (2) x² - y² = 2h/m(x - y) aligns with our findings, as it represents the difference of squares, which matches our derived relationship. Therefore, option (2) is the correct choice.
Conclusion
In summary, the relationship between the velocities of the photoelectrons and the frequencies of the light can be derived from the principles of the photoelectric effect. By analyzing the kinetic energy equations, we can determine that the correct relationship is given by option (2), confirming that the physics behind the photoelectric effect is both fascinating and fundamental to our understanding of light and matter interactions.
