Class 12 > Unit # 25: Quantum Physics > Photo Electric Effect

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Photoelectric Effect, Features, and Einstein’s Equation

Video Lecture: Photoelectric Effect & Einstein's Explanation

Watch the complete video lecture explaining the photoelectric effect, experimental features, and derivation of Einstein's photoelectric equation:

Photoelectric Effect

When electromagnetic radiation like light is shined on certain metallic materials, electrons are emitted due to the absorption of light by the electrons on the surface of the material.

Experimental Arrangement

Let us consider an evacuated tube containing two electrodes—a cathode and an anode—connected with a variable voltage source $V$. A monochromatic light source is shined onto the cathode through a quartz window, and the anode (collecting electrode) is connected to either a positive or negative potential with respect to the cathode using a battery. An ammeter is connected in the circuit to record the current due to photoelectrons.

Fig: Experimental setup for studying the photoelectric effect.

Features of Photoelectric Effect

1. Intensity and Emission: For a constant potential difference between the cathode and anode, the number of electrons emitted from the cathode increases with an increasing intensity of radiation.

2. Saturation Current: For a constant intensity and frequency of incident radiation, the photoelectric current varies with the potential difference $V$ between the cathode and anode and reaches a constant (saturation) value beyond which further increases in potential difference do not affect the photoelectric current.
3. Retarding and Stopping Potential: If the anode is made more and more negative with respect to the photocathode surface, the current decreases. This negative potential difference is called the retarding potential. For a particular value of retarding potential, the photoelectric current becomes zero. This potential is called the cut-off or stopping potential ($V_0$). The stopping potential and hence the maximum kinetic energy ($\text{K.E.}_{\max}$) of photoelectrons is independent of the intensity of incident radiation and depends only on the frequency $f$ of the radiation.

4. Threshold Frequency: For each substance, there exists a characteristic frequency $f_0$ such that for radiation with a frequency below $f_0$, photoelectrons are not ejected from the surface. This frequency is called the threshold frequency, and the corresponding wavelength is called the threshold wavelength ($\lambda_0$).

5. Time Lag: The time lag between the incidence of radiation and the emission of a photoelectron is extremely small—less than $10^{-9}\text{ seconds}$.

Einstein’s Explanation of Photoelectric Effect

Albert Einstein proposed that radiation energy is not continuously distributed over the wave-front, but light energy consists of discrete quanta of energy (photons). When a photon penetrates the surface of the cathode, all of its energy is transferred to a single electron.

When a photon is incident on the surface of a material, some of its energy is spent in making the electron free, and the rest appears as the kinetic energy of the electron. The electrons at the surface of the material are most loosely bound and require minimum energy for their liberation. This minimum energy is called the work function ($\Phi$ or $W$) of the material.

Mathematically, the energy balance equation is written as:

$\text{Quantum Energy} = \text{Work Required to Liberate Electron} + \text{Maximum K.E. of Photoelectron}$

$hf = \Phi + \text{K.E.}_{\max}$     --- (i)

Where:

• $hf$ = Energy of the incident photon ($f$ is frequency, $h$ is Planck's constant)
• $\Phi = hf_0$ = Work function of the metal ($f_0$ is the threshold frequency), measured in Joules ($\text{J}$) or electron-volts ($\text{eV}$). It is a characteristic property of the metal surface.
• $\text{K.E.}_{\max} = \frac{1}{2}mv_{\max}^2 = eV_0$ ($e$ is electronic charge, $V_0$ is stopping potential)

Substituting $\Phi = hf_0$ and $\text{K.E.}_{\max} = eV_0$ into equation (i):

$hf = hf_0 + \text{K.E.}_{\max}$

$\text{K.E.}_{\max} = hf - hf_0$

$eV_0 = h(f - f_0)$

This equation is known as Einstein’s Photoelectric Equation.

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