Question

Find the maximum kinetic energy of electrons ejected from a certain material if material's work function is 2.3 eV and the frequency of the incident radiation is 3.0 x 10¹⁵ Hz.
(Ans: 10.13 eV)

Answer 1

The Photoelectric Effect

When light of a frequency above a certain threshold frequency falls on a metal surface, electrons are emitted. This is called the Photoelectric Effect.

The Photoelectric Effect was explained by Albert Einstein, for which he won the Nobel Prize in Physics in 1921.

The explanation essentially assumes the Particle nature of light, and uses the concept of Quanta of Light, known as Photons.

The photons (or quanta) have a specific amount of energy which is solely dependent on their frequency. In mathematical formulation, Energy of a photon is:

$E = hf$

where $h$ is the Planck's Constant and $f$ is the frequency.

Metal surface has electrons. Electrons can absorb photons as a whole. If a photon has an energy sufficiently high, the electron may be emitted completely from the metal atoms.

The minimum energy needed for photoemission of electrons is known as Threshold Energy or Work Function. It is usually given in the form of electron-volt (eV).

The excess energy present in the photon is gained by the electron and it becomes its Kinetic Energy.

So, essentially, an electron absorbs a photon, gets emitted from the atom (because of the high energy absorbed) and the excess energy is converted to its kinetic energy.

So, the Mathematical Formulation of Photoelectric Effect is:

$\Large E = \Phi + K_{max}$

where

$E$ = Energy of photon

$\Phi$ = Work Function

$K_{max}$ = Maximum Kinetic Energy of Electron

$\rule{300}{1}$

Let's get to the Problem in Hand.

The Data:

$f = 3.0 \times 10^{15}\ Hz \\\\ \Phi = 2.3\ eV$

So, the Energy will be :

$\begin{aligned}E &= hf\\\\ &=6.626\times 10^{-34}\times 3.0\times 10^{15}\ J\\\\\therefore E&=19.878\times 10^{-19}\ J\end{aligned}\\\\\\\text{Now, 1 eV = 1.602\times 10^{-19} J} \\\\\\\implies E = \dfrac{19.878\times 10^{-19}}{1.602\times 10^{-19}}\ eV\\\\\\\implies E=12.408\ eV \\\\\\ \implies E \approx 12.4\ eV$

Now, we can use the Mathematical formulation of the Photoelectric Effect:

$E = \Phi + K_{max} \\\\\\ \implies 12.4 = 2.3 + K_{max} \\\\\\ \implies K_{max} = 12.4-2.3\ eV \\\\\\ \implies \Large \boxed{\sf K_{max}=10.1\ eV}$

Thus, The Maximum Kinetic Energy of the electrons is about 10.1 eV

Answer 2

Answer:

First find the value of E=hu then substitute it in the photoelectric equation