Dual Nature Of Radiation And Matter

The de-Broglie wavelength ( ) is related to kinetic energy (K) and mass (m) by the equation: where h is Planck's constant. Since the kinetic energy K is the same for all particles, the wavelength is inversely proportional to the square root of the mass ( ).
To have the longest wavelength, the particle must have the smallest mass. Comparing the masses:
Mass of electron ( ) ) ) -particle ( ). The electron has the smallest mass, and therefore it will have the longest de-Broglie wavelength.

Einstein's Photoelectric Equation: The equation is: or Physical Significance of Each Term: : This term represents the energy of the incident photon. According to Planck's quantum theory, light consists of discrete packets of energy called photons. The energy of each photon is proportional to its frequency ( ), where is Planck's constant. This is the total energy supplied to a single electron on the metal surface.
: This is the work function of the metal. It represents the minimum amount of energy required to just liberate an electron from the surface of the metal, overcoming the attractive forces holding it. The work function is a characteristic property of the material.
: This is the maximum kinetic energy of the emitted electron (photoelectron). It represents the excess energy from the photon that is converted into the electron's kinetic energy after it has escaped the surface. The kinetic energy is a maximum because some electrons deeper within the metal may lose energy through collisions before they escape.
Calculation:
We need to find the energy ( ) of the incident photon. Given:
Wavelength of light,
Constants:
Planck's constant,
Speed of light,
The energy of a photon is given by the formula: The energy of the incident photon is Joules.

First, we need to find the velocity ( ) and radius ( ) of the electron in the second orbit ( ) of a hydrogen atom.
The velocity of an electron in the orbit is , and the radius is .
A simpler approach is to calculate the time period of one revolution. The time period of revolution in the orbit is given by .
It can be shown that the angular frequency is .
The frequency of revolution is .
For the second orbit, .

This is complex. Let's use a simpler known relation. The speed of the electron in the first orbit is .
The speed in the orbit is . So, .
The radius of the first orbit is the Bohr radius, .
The radius of the orbit is . So, .
The time period of one revolution in the second orbit is: The total time the electron stays in the orbit is .
The number of revolutions is the total time divided by the time for one revolution: The electron will make approximately revolutions.