Einstein's Equation of Photoelectric Effect

The photoelectric effect is a phenomenon in which electrons are emitted from a material when it is exposed to electromagnetic radiation of sufficient energy, typically in the form of light. This effect was explained by Albert Einstein in 1905, for which he was awarded the Nobel Prize in Physics in 1921. Einstein's explanation extended the quantum theory proposed by Max Planck.

Understanding the Photoelectric Effect

When light of a certain frequency shines on the surface of a metal, electrons are ejected if the frequency of the light is above a certain threshold. The classical wave theory of light could not explain why light below a certain threshold frequency, no matter how intense, would not cause any electrons to be emitted.

Einstein proposed that light is composed of packets of energy called photons. The energy of each photon is directly proportional to the frequency of the light:

[ E = h \nu ]

where ( E ) is the energy of the photon, ( h ) is Planck's constant (( 6.626 \times 10^{-34} ) Js), and ( \nu ) (nu) is the frequency of the light.

Einstein's Photoelectric Equation

Einstein's equation for the photoelectric effect relates the kinetic energy of the emitted electrons to the frequency of the incident light and the work function of the metal. The equation is given by:

[ K.E. = h \nu - \phi ]

where:

• ( K.E. ) is the maximum kinetic energy of the emitted electrons,
• ( h ) is Planck's constant,
• ( \nu ) is the frequency of the incident light, and
• ( \phi ) is the work function of the metal, which is the minimum energy required to remove an electron from the surface of the metal.

The work function is a property of the material and is usually given in electron volts (eV).

Table of Key Points

Key Point	Description
Photon	A quantum of electromagnetic radiation with energy proportional to its frequency.
Work Function (( \phi ))	The minimum energy needed to remove an electron from the surface of a material.
Threshold Frequency (( \nu_0 ))	The minimum frequency of light required to eject an electron from the material.
Kinetic Energy (K.E.)	The energy of the ejected electrons, which can be measured as an electrical current.
Planck's Constant (h)	A fundamental constant with a value of ( 6.626 \times 10^{-34} ) Js.

Formulas

The threshold frequency (( \nu_0 )) is related to the work function by:

[ \phi = h \nu_0 ]

The maximum kinetic energy of the emitted electrons can also be expressed as:

[ K.E. = \frac{1}{2} m v^2 ]

where ( m ) is the mass of the electron and ( v ) is the velocity of the electron.

Examples

Example 1: Calculating Kinetic Energy of Emitted Electrons

Suppose a metal with a work function of ( 2.0 ) eV is exposed to light with a frequency of ( 1.0 \times 10^{15} ) Hz. We can calculate the maximum kinetic energy of the emitted electrons using Einstein's photoelectric equation.

First, convert the work function to joules:

[ \phi = 2.0 \, \text{eV} \times 1.602 \times 10^{-19} \, \text{J/eV} = 3.204 \times 10^{-19} \, \text{J} ]

Now, apply Einstein's equation:

[ K.E. = h \nu - \phi ] [ K.E. = (6.626 \times 10^{-34} \, \text{Js}) \times (1.0 \times 10^{15} \, \text{Hz}) - 3.204 \times 10^{-19} \, \text{J} ] [ K.E. = 6.626 \times 10^{-19} \, \text{J} - 3.204 \times 10^{-19} \, \text{J} ] [ K.E. = 3.422 \times 10^{-19} \, \text{J} ]

The maximum kinetic energy of the emitted electrons is ( 3.422 \times 10^{-19} ) joules.

Example 2: Determining the Threshold Frequency

If the work function of a metal is ( 4.5 ) eV, we can find the threshold frequency (( \nu_0 )) using the work function formula:

[ \phi = h \nu_0 ] [ \nu_0 = \frac{\phi}{h} ] [ \nu_0 = \frac{4.5 \, \text{eV} \times 1.602 \times 10^{-19} \, \text{J/eV}}{6.626 \times 10^{-34} \, \text{Js}} ] [ \nu_0 = \frac{7.209 \times 10^{-19} \, \text{J}}{6.626 \times 10^{-34} \, \text{Js}} ] [ \nu_0 \approx 1.088 \times 10^{15} \, \text{Hz} ]

The threshold frequency for this metal is approximately ( 1.088 \times 10^{15} ) Hz.

Conclusion

Einstein's equation of the photoelectric effect provided a quantum explanation for the observed experimental results, which could not be explained by classical physics. It showed that light has both wave-like and particle-like properties, and it was a significant step in the development of quantum mechanics. The photoelectric effect is fundamental to our understanding of the interaction between light and matter and has practical applications in devices like photodiodes and solar cells.