To determine the ratio of the number of photoelectrons emitted from two monochromatic light beams A and B, we need to consider the relationship between the wavelength of the light and the energy of the photons. The energy of a photon is inversely proportional to its wavelength, which is described by the equation:
Photon Energy and Wavelength
The energy (E) of a photon can be expressed using the formula:
E = h * f
where:
- h is Planck's constant (approximately 6.626 x 10^-34 J·s),
- f is the frequency of the light.
Since the frequency (f) is related to the wavelength (λ) by the equation:
f = c / λ
where c is the speed of light (approximately 3 x 10^8 m/s), we can substitute this into the energy equation:
E = h * (c / λ)
Calculating the Number of Photoelectrons
Now, let's denote the wavelengths of beams A and B as λA and λB, respectively. The energies of the photons from beams A and B can be expressed as:
- E_A = h * (c / λ_A)
- E_B = h * (c / λ_B)
When these beams strike the metallic surface, they eject photoelectrons. The number of photoelectrons emitted is directly proportional to the energy of the incident photons and the intensity of the light. Since both beams have the same intensity, we can express the number of photoelectrons (n) emitted from each beam as:
- n_A ∝ (Intensity / E_A)
- n_B ∝ (Intensity / E_B)
Finding the Ratio
To find the ratio of the number of photoelectrons emitted from beam A to that from beam B, we can set up the following ratio:
n_A / n_B = (Intensity / E_A) / (Intensity / E_B) = E_B / E_A
Substituting the expressions for E_A and E_B, we get:
n_A / n_B = (h * (c / λ_B)) / (h * (c / λ_A))
Notice that the constants h and c cancel out:
n_A / n_B = λ_A / λ_B
Final Result
Thus, the ratio of the number of photoelectrons emitted from beam A to that from beam B is:
n_A : n_B = λ_A : λ_B
This means that if the wavelength of beam A is longer than that of beam B, then more photoelectrons will be emitted from beam A, and vice versa. This relationship highlights the fundamental principle that shorter wavelengths (higher energy photons) will eject more electrons when the intensity is held constant.
