Question

Frequency of incident light on body is \(\mathrm{f}\). Threshold frequency of body is \(f_{0}\). Maximum velocity of electron \(=\ldots \ldots \ldots\).. where \(m\) is mass of electron. (A) $\left[\left\\{2 \mathrm{~h}\left(\mathrm{f}-\mathrm{f}_{0}\right)\right\\} / \mathrm{m}\right]^{(1 / 2)}$ (B) $\left[\left\\{2 \mathrm{~h}\left(\mathrm{f}-\mathrm{f}_{0}\right)\right\\} / \mathrm{m}\right]$ (C) \([2 \mathrm{hf} / \mathrm{m}]^{(1 / 2)}\) (D) \(\mathrm{h}\left(\mathrm{f}-\mathrm{f}_{0}\right)\)

Short answer

The maximum velocity of the electron is given by the formula: \(v = \sqrt{\frac{2h(f - f_{0})}{m}}\). Therefore, the correct answer is (A).

Step-by-step solution

Recall Einstein's Photoelectric Equation

According to Einstein's photoelectric equation, the maximum kinetic energy (K.E.) of the emitted electron is given by: $$K.E. = h(f - f_{0})$$ Here, h is Planck's constant, f is the frequency of the incident light, and \(f_{0}\) is the threshold frequency.

Express Kinetic Energy in terms of Mass and Velocity

We can express the kinetic energy in terms of mass (m) and velocity (v) as follows: $$K.E. = \frac{1}{2}mv^2$$

Equate the two expressions of Kinetic Energy

Now, we will equate the two expressions for kinetic energy and solve for the maximum velocity (v): $$\frac{1}{2}mv^2 = h(f - f_{0})$$

Solve for the Maximum Velocity

Isolate the variable v by multiplying both sides of the equation by the reciprocal of \(\frac{1}{2}m\): $$v^2 = \frac{2h(f - f_{0})}{m}$$ Now, take the square root of both sides to find the maximum velocity: $$v = \sqrt{\frac{2h(f - f_{0})}{m}}$$

Find the correct answer from the multiple-choice options

Comparing the derived equation to the given options: (A) $\left[\left\\{2 \mathrm{~h}\left(\mathrm{f}-\mathrm{f}_{0}\right)\right\\} / \mathrm{m}\right]^{(1 / 2)}$ (B) $\left[\left\\{2 \mathrm{~h}\left(\mathrm{f}-\mathrm{f}_{0}\right)\right\\} / \mathrm{m}\right]$ (C) \([2 \mathrm{hf} / \mathrm{m}]^{(1 / 2)}\) (D) \(\mathrm{h}\left(\mathrm{f}-\mathrm{f}_{0}\right)\) We can see that our derived equation is the same as option (A): $$v = \sqrt{\frac{2h(f - f_{0})}{m}}$$ So, the correct answer is (A).