Modern Physics

Step 1: Understanding the problem.
The particle is moving in a circular path, and we are asked to calculate the magnitude of the average acceleration after half a revolution. The formula for average acceleration in circular motion is: where is the change in velocity and is the time taken for half the revolution.
Step 2: Calculation.
The change in velocity after half a revolution is the difference between the initial and final velocity vectors. Since the particle has moved half a revolution, the angle between the two velocity vectors is , and the magnitude of the average acceleration is:
Step 3: Conclusion.
Thus, the correct answer is (A) .

Step 1: Formula for beats.
The number of beats per second is given by the difference in the frequencies of the two waves: where and are the frequencies of the waves.
Step 2: Calculation of frequencies.
For the wave , the frequency is: For the wave , the frequency is: Thus, the number of beats per second is:
Step 3: Conclusion.
Thus, the number of beats produced per second is (D) .

Step 1: Understanding the fundamental forces.
In nature, there are four fundamental forces: gravitational force, electromagnetic force, strong nuclear force, and weak nuclear force. The range of each force differs significantly. The gravitational force has the longest range, effectively acting over vast distances. In contrast, the weak nuclear force has the shortest range, acting only at subatomic distances.
Step 2: Conclusion.
Thus, the maximum range corresponds to gravitational force, and the minimum range corresponds to weak nuclear force. The correct answer is (A).

Step 1: Write Einstein’s photoelectric equation.
The maximum kinetic energy of emitted photoelectrons is
Step 2: Express kinetic energy in terms of velocity.

Step 3: Use magnetic force as centripetal force.

Step 4: Solve for radius .

Step 5: Conclusion.
The radius of circular path is .

Step 1: Write de-Broglie wavelength formula.
For a particle accelerated through potential difference :
Step 2: Write wavelengths for electron and proton.

Step 3: Take ratio.

Step 4: Conclusion.
The required ratio is .

Step 1: Applying the photoelectric equation.
The work function of the photosensitive surface is given by the equation: which is derived from the energy conservation in the photoelectric effect.
Step 2: Analyzing the options.
- (A) : Incorrect as it does not match the derived equation. - (B) : Incorrect, this does not correspond to the work function equation. - (C) : Correct, this matches the derived work function equation. - (D) : Incorrect, this does not match the work function equation.
Step 3: Conclusion.
The correct option is (C) because it represents the correct expression for the work function.

Step 1: Understanding the photoelectric effect.
The photoelectric current depends on both the frequency of the incident light and its intensity. The photoelectric effect only occurs if the frequency of the incident light is greater than the threshold frequency.
Step 2: Frequency and intensity effects.
- If the incident frequency is reduced to of its original value, the frequency is below the threshold frequency, which means no photoelectrons will be emitted, and the photoelectric current will be zero. - Doubling the intensity does not help because the frequency is the key factor in the emission of photoelectrons.
Step 3: Conclusion.
Since the frequency is below the threshold, the correct answer is (D), the photoelectric current will be zero.

Step 1: Understanding the photoelectric effect.
In the photoelectric effect, the kinetic energy of the emitted photoelectrons is related to the intensity of incident radiation. However, the maximum kinetic energy of the photoelectrons does not depend on the intensity of the light, but rather on the frequency of the incident radiation. As the intensity increases, the number of emitted electrons increases, but the maximum kinetic energy remains constant.
Step 2: Conclusion.
Thus, the graph that shows a constant kinetic energy as intensity increases is the correct one, which corresponds to Graph C.

Concept:
Use relations for different processes:

• Isothermal:
• Adiabatic:
• Isobaric:

Step 1: For X (isothermal).
Step 2: For Y (adiabatic).
Step 3: For Z (isobaric).
Step 4: Given condition: final pressures equal. Let common final pressure =
Step 5: Ratio. Divide by 2: Multiply by 2:
Step 6: Conclusion.

Concept:
For partial dielectric filling: Step 1: Initial capacitance.
Step 2: Final capacitance. Increase by 50%:
Step 3: Simplify.
Step 4: Solve.
Step 5: Conclusion.

Step 1: Constant Volume Law
In a closed vessel, is constant, so .
Step 2: Substitution
.
.
.
Step 3: Solve for
.
.
Final Answer: (C)

Step 1: Bohr's Postulate
.
Step 2: de-Broglie Relation
. Substituting this into the postulate: .
Step 3: Analysis
For Hydrogen atom, and .
From , we get .
Final Answer: (D)

Step 1: Formula for Velocity
. Maximum speed .
Step 2: Given Condition
.
Step 3: Solve for x
.
.
Final Answer: (D)