Dual Nature Of Matter

Step 1: Using the de-Broglie wavelength formula.
The de-Broglie wavelength is related to the momentum of the electron by the equation: Where is Planck's constant. The energy of the electron is related to its momentum by: Step 2: Using the change in wavelength.
Since the wavelength reduces by half, the momentum must increase by a factor of . Therefore, the energy increases by a factor of 3. Thus, the correct answer is (C) .

Step 1: Resolving the force.
Given
Step 2: Calculating y-component of force.

Step 3: Using Newton’s second law.

Step 4: Conclusion.
The acceleration along y-axis is .

Step 1: Energy of incident photon.

Step 2: Kinetic energy of emitted electron.
Since work function is negligible:

Step 3: de-Broglie wavelength of electron.

Step 4: Substituting .

Step 5: Conclusion.
The de-Broglie wavelength is .

Step 1: de-Broglie wavelength formula.
For an electron accelerated through potential difference :

Step 2: Relation between wavelength and potential difference.

Step 3: Effect of doubling the potential difference.
If is doubled:

Step 4: Conclusion.
The de-Broglie wavelength decreases to times its original value.

Step 1: Calculate frequency of incident light.


Step 2: Compare with threshold frequency.
Threshold frequency Incident frequency

Step 3: Apply photoelectric condition.
Photoelectric emission occurs only if Here, .

Step 4: Final conclusion.
Since the frequency of incident light is less than the threshold frequency, photoelectrons will not be emitted.

Step 1: Energy of photons.
The energy of a photon emitted during a transition between two energy levels in a hydrogen atom is given by the formula: where and are the initial and final energy levels, respectively.
Step 2: Calculate energies for the two transitions.
For the first transition (second to first energy level): For the second transition (highest to second energy level, ):
Step 3: Conclusion.
The ratio of energies is: Thus, the correct answer is (D).

Step 1: Relation between kinetic energy and wavelength.
According to photoelectric equation: Thus, kinetic energy of photoelectrons depends on wavelength, not on intensity.
Step 2: Dependence of photoelectron number on intensity.
The number of emitted photoelectrons is directly proportional to the intensity of incident light.
Step 3: Final conclusion.

Step 1: Write expression for photon energy.
The energy of incident radiation is given by: Step 2: Use photoelectric effect condition.
Since the work function is negligibly small, the entire photon energy is converted into kinetic energy of the electron: Step 3: Express momentum of emitted electron.
Step 4: Use de-Broglie wavelength relation.
Step 5: Substitute momentum and simplify.