Question

The electric field of certain radiation is given by the equation $E=200\{\sin(4\pi\times10^{10}t)+\sin(4\pi\times10^{15}t)\}$ falls on a metal surface having work function 2.0 eV. The maximum kinetic energy (in eV) of the photoelectrons is (use $h=6.63\times10^{-34}\text{Js}$ and $e=1.6\times10^{-19}\text{C}$)

• A. 3.3
• B. 4.3
• C. 5.3
• D. 6.3
• E. 7.3

Hint:

In photoelectric effect problems with mixed radiations, ignore the lower frequency components as they will only produce slower electrons (or none at all).

Answer 1

The Correct Option is D

Concept:
According to Einstein's photoelectric equation: $K_{max} = hf - \Phi$, where $\Phi$ is the work function. When multiple frequencies are present, $K_{max}$ is determined by the highest frequency components of the radiation.

Step 1: Identify the highest frequency.
The radiation contains two frequencies from the arguments $(4\pi \times 10^{10})t$ and $(4\pi \times 10^{15})t$. Since $\omega = 2\pi f$: \[ f_1 = \frac{4\pi \times 10^{10}}{2\pi} = 2 \times 10^{10} \text{ Hz} \] \[ f_2 = \frac{4\pi \times 10^{15}}{2\pi} = 2 \times 10^{15} \text{ Hz} \] We use $f_2$ as it has higher energy.

Step 2: Calculate photon energy in eV.
\[ E_{photon} = \frac{hf}{e} = \frac{(6.63 \times 10^{-34}) \times (2 \times 10^{15})}{1.6 \times 10^{-19}} \] \[ E_{photon} \approx \frac{13.26 \times 10^{-19}}{1.6 \times 10^{-19}} = 8.2875 \approx 8.3 \text{ eV} \]

Step 3: Calculate $K_{max}$.
\[ K_{max} = E_{photon} - \Phi = 8.3 \text{ eV} - 2.0 \text{ eV} = 6.3 \text{ eV} \]