Dual Nature Of Matter

1. Calculate the energy of one photon:
The energy of a photon is given by: where:
\begin{itemize} \item (Planck's constant),
\item (speed of light),
\item (wavelength).
\end{itemize} Substituting the values:
2. Calculate the total energy entering the pupil per second:
The intensity is given as .
The area of the pupil is .
The power entering the pupil is:
Since , the energy per second is:
3. Calculate the number of photons per second:
The number of photons per second is given by: Final Answer:

Step 1: Analyzing Assertion (A). The photoelectric effect demonstrates the particle nature of light because it involves the interaction of light with matter in such a way that light is treated as a collection of particles (photons). According to Einstein's explanation, each photon carries energy proportional to its frequency. This energy is transferred to electrons, allowing them to escape from the surface of the material, thus demonstrating the particle nature of light. Step 2: Analyzing Reason (R). The photoelectric current is not proportional to the frequency of the incident radiation. In fact, the current is proportional to the intensity of the incident radiation (i.e., the number of photons striking the surface per unit time). The frequency of the radiation determines the energy of the emitted photoelectrons, but it is the intensity (not the frequency) that controls the photoelectric current. The reason provided in Statement (R) is incorrect. Conclusion: Assertion (A) is true because the photoelectric effect demonstrates the particle nature of light. However, Reason (R) is false because the photoelectric current is proportional to the intensity (not the frequency) of the incident radiation.

The Einstein’s photoelectric equation is given by:
This equation indicates that the maximum kinetic energy ( ) of the emitted photoelectrons is zero when the frequency of the incident radiation is at or below the threshold frequency. As the frequency increases beyond the threshold frequency, increases linearly with .
The graph in option (C) correctly represents this relationship: it starts from zero at the threshold frequency and increases linearly thereafter.

Step 1: Einstein’s photoelectric equation is given by:
where:
- is the kinetic energy of the emitted photoelectron,
- is Planck's constant ( ),
- is the frequency of the incident radiation,
- is the work function of the material (the minimum energy required to release an electron from the surface).
This equation expresses the energy conservation in the photoelectric effect: the energy of the incoming photons is used to overcome the work function, and the excess energy is imparted as kinetic energy to the emitted electron. Step 2: Millikan's experiment validated this equation by measuring the stopping potential required to stop the emitted photoelectrons. He measured the maximum kinetic energy of the photoelectrons for various frequencies of incident light, which allowed him to confirm the linear relationship between the kinetic energy and frequency of the incident radiation, as predicted by Einstein’s equation.

The stopping potential is directly related to the kinetic energy of the photoelectrons. The stopping potential increases with the frequency of the incident radiation, and the maximum kinetic energy of the photoelectrons will occur for the metal with the largest stopping potential for a given frequency. In the graph, the stopping potential is highest for metal A at a given frequency, indicating that the kinetic energy of the photoelectrons is maximum for metal A.

(a) The de Broglie wavelength is given by: Comparing with the equation of a straight line ( ), the slope of the line is: Thus, the slope of the line represents , which is inversely proportional to . (b) From the equation: Since the slope of is greater than that of , it implies that is lighter, and is heavier. (c) No, this graph is not valid for a photon. The given equation for momentum: is not applicable to a photon, as a photon has zero rest mass and its momentum is given by:

When the intensity of incident radiation increases (for a constant frequency), the number of photons hitting the surface increases. This results in the emission of more photoelectrons per second, thereby increasing the photocurrent. This explanation aligns perfectly with the assertion.

The cut-off wavelength can be found using the photoelectric equation: where is Planck's constant, is the speed of light, and is the work function. Rearranging to solve for : Substitute , , and :

In the photoelectric effect, the kinetic energy of the emitted photoelectrons depends on the frequency (or wavelength) of the incident light, not its intensity. Hence, Assertion (A) is false. The photoelectric current depends on the intensity of the incident light and not its wavelength, so Reason (R) is also false. Thus, the correct answer is:

The de Broglie wavelength of a particle is given by: where: \begin{itemize} \item is Planck's constant, \item is the momentum of the particle. \end{itemize} If two particles have the same de Broglie wavelength, then their momenta must be the same, because is inversely proportional to . However, their masses may differ, and thus their speeds and energies can be different. For a moving electron and a moving proton, having the same wavelength implies: Thus, the correct answer is:

De Broglie Wavelength Relation:
The de Broglie wavelength is given by:
, where is Planck's constant and is the momentum. The kinetic energy (KE) is given by:
The kinetic energy gained by a charge accelerated through a potential is:
Since the de Broglie wavelengths are equal:
. Therefore, , which implies . Squaring both sides gives:
Substituting : We know that the mass of the alpha particle is approximately four times the mass of the proton, and the charge of the alpha particle is twice the charge of the proton:
and . Substituting these values:
Dividing both sides by : Therefore:

Explanation of the Photoelectric Effect:
- According to Einstein’s photoelectric equation: where is Planck’s constant, is the frequency of incident light, and is the work function of the material.
- The photoelectric effect demonstrates that:
- If the frequency of the incident light is less than a threshold frequency , no electrons are emitted.
- Increasing the frequency above increases the kinetic energy of emitted electrons.
- The intensity of light affects the number of electrons emitted, but not their kinetic energy.
Thus, the correct explanation for the quantum nature of light in the photoelectric effect is that there exists a minimum threshold frequency below which no electrons are emitted.

- Dual nature refers to the concept that particles exhibit both wave-like and particle-like properties.
- According to quantum theory, both photons (light quanta) and electrons can behave as particles and waves.
- Electron diffraction experiments clearly demonstrate the wave nature of electrons, while the photoelectric effect supports their particle nature.
- Similarly, photons show wave behavior (like interference and diffraction) and particle behavior (in photoelectric and Compton effects).
Let’s quickly refute the other options:
- (A) is incorrect because electrons do show diffraction.
- (B) is incorrect; both electrons and photons possess momentum: for photons.
- (D) is incorrect since all electromagnetic radiations consist of photons.
Final answer: Photons and electrons both exhibit dual nature.