Photoelectric Effect And Wave Theory Of Light

The de Broglie wavelength of a particle is given by the formula: Where: - is Planck's constant ( ), - is the momentum of the particle.
The momentum of a particle accelerated by a potential difference is given by: Where: - is the mass of the particle,
- is the charge of the particle ( ),
- is the potential difference.
Now, we calculate the momentum for both the proton and the -particle.
1. For the proton:
- Mass of proton ,
- Charge of proton .
The momentum of the proton is: 2. For the -particle:
- Mass of -particle ,
- Charge of -particle .
The momentum of the -particle is: 3. Ratio of the de Broglie wavelengths:}
Since the de Broglie wavelength is inversely proportional to the momentum, the ratio of the wavelengths is: Simplifying the ratio: Thus, the ratio of the de Broglie wavelengths is approximately 2.83.

The de-Broglie equation relates the wavelength of a particle to its momentum. It shows that matter exhibits wave-like properties, just like light. The equation is given by: Where:
- is the de-Broglie wavelength, which is the wavelength associated with the particle,
- is Planck's constant, with a value of ,
- is the momentum of the particle. Momentum is given by the product of mass and velocity ( ), so the equation becomes: This shows that the wavelength is inversely proportional to both the mass and the velocity of the particle. The de-Broglie wavelength is especially significant for subatomic particles, such as electrons, where the wavelength becomes comparable to atomic distances.
For macroscopic objects, the de-Broglie wavelength is incredibly small, so it does not have noticeable effects. However, for very small particles like electrons, neutrons, and protons, the wave-like nature is significant, and it explains phenomena like electron diffraction.
In conclusion, de-Broglie postulated that all matter has an associated wavelength, and the smaller the mass of the particle, the larger its wavelength at a given velocity. The wavelength of an object moving at high speeds (close to the speed of light) becomes comparable to the wavelength of light itself. This wave-particle duality is a fundamental concept in quantum mechanics.

Einstein's photoelectric equation relates the energy of the incident photons to the kinetic energy of the emitted photoelectrons. The equation is: Where: - is the maximum kinetic energy of the emitted electrons,
- is Planck's constant ( ),
- is the frequency of the incident light (photons),
- is the work function of the metal, which is the minimum energy required to remove an electron from the metal surface.
The equation shows that the maximum kinetic energy of the emitted electrons is the difference between the energy of the incident photons and the work function of the material.
Effect of Increasing Frequency by Times:
If the frequency of the incident light is increased by a factor of , the energy of the photons increases by the same factor, i.e., the energy of the photons becomes . The new maximum kinetic energy is given by: Thus, if the frequency is increased by times, the maximum kinetic energy of the emitted electrons also increases by a factor of .
Threshold Wavelength and Work Function:
The threshold wavelength is the wavelength of the incident light below which no photoelectron is emitted, even if the intensity of the light is increased. The threshold wavelength is related to the work function by the equation: Where: - is the speed of light,
- is the threshold wavelength.
This equation shows that a larger work function corresponds to a shorter threshold wavelength, i.e., the more energy required to release electrons from the metal, the shorter the wavelength of light needed to release those electrons.

In Young's double slit experiment, the formula for the distance of the nth bright fringe (denoted as ) from the central maximum is given by:
Where: - is the fringe number (for bright fringes, ),
- is the wavelength of light (given as or ),
- is the distance between the slits and the screen (given as ),
- is the distance between the slits (given as or ).
For the nth dark fringe, the distance from the central fringe is given by the formula:
To find the distance between the 8th bright and the 3rd dark fringes, we calculate the distance for the 8th bright fringe ( ) and the 3rd dark fringe ( ). 1. Distance for the 8th bright fringe: 2. Distance for the 3rd dark fringe: Now, to find the distance between the 8th bright and the 3rd dark fringes:
Thus, the distance between the 8th bright and the 3rd dark fringes is .

The de Broglie wavelength ( ) of a particle is related to its momentum ( ) by the equation:
Where:
- is the de Broglie wavelength,
- is Planck's constant ( ),
- is the momentum of the particle.
Since momentum is given by , where is the mass and is the velocity of the particle, we can write:
Now, consider a proton and an -particle (helium nucleus) accelerated by the same potential difference. The kinetic energy acquired by a particle when accelerated by a potential is given by:
Where is the charge of the particle and is the potential. For a proton, and for an -particle, . From this, we can find the velocity of each particle and subsequently the momentum.
For the proton, the velocity is:
For the -particle, the velocity is:
Now, the de Broglie wavelength for the proton ( ) and the -particle ( ) are:
Finally, the ratio of their de Broglie wavelengths is:
Since the mass of a proton is approximately 1836 times the mass of an electron, and the mass of an -particle , we get:
Thus, the ratio of the de Broglie wavelengths of the proton and -particle is approximately 2.83.

Microwaves are a form of electromagnetic radiation with wavelengths ranging from 1 mm to 1 m, which places them between radio waves and infrared light in the electromagnetic spectrum. Their unique properties make them ideal for various applications. Two important uses of microwaves are:
1. Communication:
Microwaves are widely used for communication, especially in satellite communication and mobile phone networks. In satellite communication, microwaves are transmitted through the atmosphere to relay signals between satellites and the Earth. These waves are capable of traveling long distances and can pass through clouds, rain, and fog, making them ideal for long-distance communication. Additionally, microwaves are used in radar systems to detect objects and measure distances, speeds, and velocities.
2. Cooking:
Microwaves are commonly used in microwave ovens to heat and cook food. The microwaves are absorbed by water molecules in the food, causing them to vibrate and produce heat. This results in the cooking or heating of food, with the process being much faster than traditional heating methods. Microwaves penetrate the food and excite water molecules, which helps in evenly heating the food from within.

The de Broglie wavelength of a particle is given by the formula:
Where:
- is Planck's constant,
- is the momentum of the particle.
The momentum of a particle is given by:
Where:
- is the mass of the particle,
- is the velocity of the particle.
Since both the electron and the proton are accelerated by the same potential difference , their kinetic energy is:
From this, we can find the velocity of the electron and proton. For the electron:
For the proton:
Now, the ratio of the de Broglie wavelengths of the electron and proton is:
Substituting the expressions for and :
Simplifying:
Thus, the correct answer is option (D) .

Matter Waves:
Matter waves, also called de Broglie waves, are the waves associated with particles that exhibit both wave-like and particle-like properties. According to de Broglie’s hypothesis, every moving particle, such as an electron or a proton, behaves like a wave and has an associated wavelength. This concept was proposed by Louis de Broglie in 1924 and is known as wave-particle duality. The de Broglie wavelength of a particle with momentum is given by the following formula: where: - is Planck’s constant ( ), - is the momentum of the particle ( , where is the mass and is the velocity of the particle). Dual Nature of Particles:
The concept of wave-particle duality was experimentally verified by the Davisson-Germer experiment, which demonstrated that electrons exhibit diffraction patterns, similar to light waves, when passed through a crystal lattice. Final Answer: The formula for de Broglie wavelength is: The experiment that shows the dual nature of particles is the Davisson-Germer experiment.