Einstein’s Photoelectric Equation – Derivation
The photoelectric effect explains the emission of electrons from a metal surface when light of suitable frequency falls on it. This phenomenon was explained by Albert Einstein using quantum theory.
Basic Idea
According to Einstein, light consists of tiny packets of energy called photons. The energy of one photon is:
$$ E = h\nu $$Where:
- \( h \) = Planck’s constant
- \( \nu \) = frequency of incident light
When a photon strikes a metal surface:
- Part of its energy removes the electron
- Remaining energy becomes kinetic energy of the emitted electron
Einstein’s Photoelectric Equation
Let:
- \( \phi \) = work function of the metal
- \( K.E. \) = kinetic energy of emitted electron
Using conservation of energy:
$$ h\nu = \phi + K.E. $$This is called Einstein’s photoelectric equation.
Derivation
Step 1 – Energy of Incident Photon
Energy carried by one photon is:
$$ E = h\nu $$Step 2 – Work Function
Minimum energy required to remove an electron from the metal surface is called the work function.
$$ \phi $$Step 3 – Remaining Energy
After emission, the remaining energy appears as kinetic energy of the electron.
$$ K.E. = h\nu - \phi $$This is the mathematical form of Einstein’s equation.
Maximum Kinetic Energy
If the emitted electron has maximum velocity \( v_{max} \):
$$ K.E._{max} = \frac{1}{2}mv_{max}^2 $$Substituting into Einstein’s equation:
$$ h\nu = \phi + \frac{1}{2}mv_{max}^2 $$Threshold Frequency
The minimum frequency required for photoelectric emission is called threshold frequency.
At threshold frequency:
$$ K.E. = 0 $$Therefore,
$$ h\nu_0 = \phi $$or
$$ \nu_0 = \frac{\phi}{h} $$Important Conclusions
- Photoelectric emission is instantaneous
- Kinetic energy depends on frequency
- Number of electrons depends on intensity
- Threshold frequency exists for every metal
Applications
- Photocells
- Solar cells
- Automatic doors
- Burglar alarms
- Light sensors
Key Formula Summary
$$ E = h\nu $$ $$ h\nu = \phi + K.E. $$ $$ K.E._{max} = h\nu - \phi $$ $$ K.E._{max} = \frac{1}{2}mv_{max}^2 $$ $$ \phi = h\nu_0 $$Conclusion
Einstein successfully explained the photoelectric effect using photon theory. His work proved the particle nature of light and laid the foundation for quantum physics.