To tackle this question, we need to delve into the photoelectric effect, which describes how light can eject electrons from a material. The kinetic energy of the emitted electrons is influenced by the wavelength of the incident light. Let's break this down step by step.
Understanding the Photoelectric Effect
According to the photoelectric effect, when light of a certain wavelength strikes a material, it can impart energy to electrons. The energy of the incoming photons is given by the equation:
E = h * f
where E is the energy of the photon, h is Planck's constant, and f is the frequency of the light. The frequency is related to the wavelength (λ) by the equation:
f = c / λ
where c is the speed of light. Thus, we can express the energy in terms of wavelength:
E = h * (c / λ)
Calculating Kinetic Energy
The maximum kinetic energy (K) of the emitted electrons can be expressed as:
K = E - φ
where φ is the work function of the material, the minimum energy required to remove an electron from the surface. For the initial wavelength λ, we have:
K = h * (c / λ) - φ
Changing the Wavelength
Now, when the wavelength is changed to 3λ/4, we can find the new energy of the photons:
E' = h * (c / (3λ/4)) = (4h * c) / (3λ)
Substituting this into the kinetic energy equation gives us:
K' = E' - φ = (4h * c) / (3λ) - φ
Relating the New Kinetic Energy to the Old One
Now, we can express the new kinetic energy K' in terms of the original kinetic energy K:
K' = (4h * c) / (3λ) - φ = (4/3) * (h * (c / λ)) - φ
Since we know that K = h * (c / λ) - φ, we can substitute this into our equation:
K' = (4/3) * (K + φ) - φ
Now, simplifying this expression:
K' = (4/3)K + (4/3)φ - φ = (4/3)K - (1/3)φ
Comparing K' to K
From this, we can see that the new kinetic energy K' is greater than K if the work function φ is relatively small compared to the energy of the incoming photons. However, we need to analyze the options given in the question:
- (A) 3K/4
- (B) 4K/3
- (C) less than 4K/3
- (D) greater than 4K/3
Since we derived that K' = (4/3)K - (1/3)φ, we can conclude that if φ is small, K' could be slightly less than (4/3)K, but it will not be less than (3/4)K. Therefore, the most reasonable conclusion is that:
The correct answer is (C) less than 4K/3.
