Dual Nature Of Radiation And Matter

Step 1: Photon Momentum Formula.
The momentum of a photon is related to its energy and wavelength. The formula is: where: - is the momentum of the photon, - is the energy of the photon, - is the speed of light, - is Planck's constant, - is the wavelength of the photon.

Step 1: Stopping Potential and Intensity.
The stopping potential in the photoelectric effect depends on the energy of the incident photons, which in turn depends on the frequency of light. The stopping potential is given by: where is Planck's constant, is the frequency of the light, is the charge of an electron, and is the work function of the material.
Step 2: Effect of Distance on Intensity.
The intensity of light decreases with the square of the distance from the source. The intensity is inversely proportional to the square of the distance , i.e., Since the energy of individual photons is not affected by the distance, the stopping potential remains unchanged. The decrease in intensity (due to the increased distance) will result in fewer electrons being emitted, but the energy of the emitted electrons (and thus the stopping potential) remains the same.
Step 3: Conclusion.
The stopping potential remains the same even if the source is moved to because the energy of the photons is not affected by the distance.

Step 1: Understanding matter waves.
Matter waves are waves associated with particles of matter, as proposed by Louis de Broglie. He suggested that every moving particle, such as an electron, behaves as if it is a wave. These waves are called de Broglie waves. The wavelength of a matter wave is related to the momentum of the particle. This is described by the de Broglie equation.
Step 2: de Broglie equation.
The de Broglie equation is given by: where: - is the de Broglie wavelength, - is Planck's constant ( ), - is the momentum of the particle ( , where is the mass and is the velocity of the particl.
Step 3: Conclusion.
The de Broglie equation relates the wavelength of a matter wave to the momentum of the particle:

Step 1: Ideal p-n Junction Resistance.
In forward bias, the resistance of an ideal p-n junction is zero, as current flows easily. In reverse bias, the resistance is infinite because no current flows.

Step 2: Conclusion.
Hence, the resistance of an ideal p-n junction is zero in forward bias and infinite in reverse bias, making the correct answer .

Louis de Broglie proposed that every moving particle is associated with a wave. This wave is known as the matter wave or de Broglie wave. He suggested that particles such as electrons, protons, and even macroscopic objects exhibit both wave-like and particle-like behavior, which is a fundamental principle of quantum mechanics.
de Broglie Hypothesis:
The wavelength associated with a particle of momentum is given by:
where:
= Planck’s constant
= momentum of the particle =
Now, in terms of kinetic energy:
Kinetic energy
Solving for velocity:
Now,
Substitute into de Broglie equation:
Final Result:

Step 1: Difficulty 1 – Stability of the Atom.
In Rutherford’s model, the electrons orbit the nucleus in circular paths. According to classical electromagnetism, an electron in motion should radiate energy in the form of electromagnetic waves. This would cause the electron to lose energy, spiral inward, and eventually crash into the nucleus. Hence, the atom should not be stable, but it is. This is a major difficulty of Rutherford’s model.
Step 2: Difficulty 2 – Discrete Line Spectra.
Rutherford’s model could not explain the observed discrete line spectra of atoms. According to the model, electrons would emit a continuous spectrum as they lose energy, but experimentally, only specific wavelengths (discrete lines) were observed in the atomic spectra. This discrepancy led to the development of a new model by Niels Bohr.
Final Answer:
1. Stability of the atom (electrons should spiral into the nucleus).
Failure to explain the discrete line spectra of atoms.

Step 1: Planck's Constant Dimensions.
Planck's constant, , has the dimension of action, which is defined as energy × time. The energy has dimensions of force × distance, so Planck's constant has the dimension of force × time.

Step 2: Conclusion.
Thus, the correct answer is ( force × time.