Question

In photoelectric effect, kinetic energy of photoelectrons varies from zero to K. If the probability of electrons having energy $\frac{K}{3}$ is maximum, then we may conclude that –

• A. Probability of photoelectrons with kinetic energy of $\frac{K}{2}$ is greater than those with kinetic energy of $\frac{2K}{3}$
• B. Probability of photoelectrons with kinetic energy $\frac{K}{2}$ is equal to those with kinetic energy of $\frac{K}{6}$
• C. Probability of electrons with lower kinetic energy is higher than those with higher kinetic energy
• D. Probability of electrons with lower kinetic energy is less than those with higher kinetic energy

Answer 1

Answer: (A)

Probability of photoelectrons with kinetic energy of $\frac{K}{2}$ is greater than those with kinetic energy of $\frac{2K}{3}$

Answer 2

The kinetic energy of photoelectrons, $E_k$, ranges from 0 to $K$. The probability of electrons having kinetic energy $\frac{K}{3}$ is maximum, meaning the probability density function $P(E_k)$ has a peak at $E_k = \frac{K}{3}$. For a unimodal distribution, values closer to the peak have a higher probability density.

Let's analyze the distances of the given kinetic energies from the peak energy $\frac{K}{3}$:

• For kinetic energy $\frac{K}{2}$: Distance from peak = $|\frac{K}{2} - \frac{K}{3}| = |\frac{3K - 2K}{6}| = \frac{K}{6}$.

• For kinetic energy $\frac{2K}{3}$: Distance from peak = $|\frac{2K}{3} - \frac{K}{3}| = |\frac{K}{3}| = \frac{K}{3}$.

Since $\frac{K}{6} < \frac{K}{3}$, the kinetic energy $\frac{K}{2}$ is closer to the peak energy $\frac{K}{3}$ than $\frac{2K}{3}$. Therefore, the probability density at $\frac{K}{2}$ is greater than the probability density at $\frac{2K}{3}$. This makes option A correct.

Option B compares $P(\frac{K}{2})$ and $P(\frac{K}{6})$. Distance of $\frac{K}{6}$ from the peak energy $\frac{K}{3}$ is $|\frac{K}{6} - \frac{K}{3}| = |-\frac{K}{6}| = \frac{K}{6}$. Both $\frac{K}{2}$ and $\frac{K}{6}$ are equidistant from the peak. However, without information about the symmetry of the probability distribution, we cannot conclude that their probabilities are equal. Thus, option B is not necessarily true.

Options C and D suggest that the probability is strictly decreasing or strictly increasing with kinetic energy, respectively. This contradicts the given information that the probability has a maximum at $\frac{K}{3}$. Therefore, options C and D are incorrect.