Dual Nature Of Matter

The correct answer is (A) :


From (i) and (ii)

To solve the problem of finding the threshold wavelength for a given metal surface illuminated by light, we use the photoelectric effect equation. The photoelectric equation relates the kinetic energy of ejected electrons to the frequency of the incident light and the work function of the material:

where:

• is the kinetic energy of the ejected electron.
• is Planck's constant.
• is the frequency of incident light.
• is the threshold frequency.

The stopping potential is related to the kinetic energy by:

where is the elementary charge.

We also have the relationship between frequency and wavelength:

where is the speed of light.

Substitute this into the photoelectric equation:

For two different stopping potentials, we can write two equations:

1. For with stopping potential 8V:
2. For with stopping potential 2V:

Now, we solve these equations to find , the threshold wavelength.

Rewrite Equation 1:

Rewrite Equation 2:

Equating the two expressions for :

Isolate terms:

Simplify:

So, using Equation 1:

Thus, the threshold wavelength is 9 , and the correct answer is 9 .

To solve this problem, we need to understand how the photoelectric current produced is related to the intensity of light and the focal length of the lens used. When a convex lens is used to form an image on a photo-electric cell, the intensity of light at the image location depends on the focal length of the lens.

When the lens focuses light on the photoelectric cell, the energy incident on the cell is proportional to the intensity of the light (which in turn depends on the area over which the light is spread). The intensity (not photoelectric current, but intensity) can be given as:

where is the focal length of the lens. Therefore, the intensity of light at the image location is inversely proportional to the square of the focal length.

Initially, the focal length of the lens was 40 cm. When we replace the lens with another one having a focal length of 20 cm (while the diameter remains the same), the new focal length is half of the original one. Thus, the intensity increases by a factor of:

However, this increase in intensity leads to a proportional increase in the photoelectric current produced, according to the photoelectric effect principle. Therefore, if the initial current produced was , the new current will be:

Hence, the photoelectric current now is 2I.

Step 1: Understanding the Question:
We need to evaluate the correctness of four separate statements related to photons, electromagnetic waves, and de-Broglie wavelength.
Step 2: Detailed Explanation:
Statement (A):
Energy of a single photon is .
The number of photons in a beam of total energy is .
Given: J, m.
J·s, m/s.
This value is very close to . Statement (A) is correct.
Statement (B):
The energy of a photon is inversely proportional to its wavelength ( ).
Therefore, the ratio of energies is .
Given: nm, nm.
The statement claims . Statement (B) is incorrect.
Statement (C):
The relationship between frequency ( ), wavelength ( ), and the speed of light ( ) is .
Given: Hz.
This value is approximately m. Statement (C) is correct.
Statement (D):
The de-Broglie wavelength is given by , where is the momentum.
When a particle of charge is accelerated by a potential difference , its kinetic energy is .
Momentum is related to kinetic energy by .
So, .
Electrons and protons have the same magnitude of charge but different masses ( ). If they are accelerated by the same , their wavelengths will be different. Specifically, since mass is in the denominator, the more massive proton will have a shorter wavelength. Statement (D) is incorrect.
Step 3: Final Answer:
Statements (A) and (C) are correct, while (B) and (D) is incorrect. The best choice among the options is (C) which states A & C are correct.