De Broglie Hypothesis

The correct answer is (C): ɑ particle

The deBroglie wavelength is given by the equation: Since the proton, deuteron, and electron have the same energy, their deBroglie wavelengths depend on their masses. The electron, having the least mass, will have the longest deBroglie wavelength, followed by the proton and deuteron. The -particle, having the highest mass, will have the shortest wavelength. Thus, the correct order of wavelengths is .

Step 1: Understanding the Concept:
This problem requires the calculation of the de-Broglie wavelength of a macroscopic object. The de-Broglie hypothesis states that all matter has wave-like properties, and the wavelength ( ) is inversely proportional to the momentum ( ) of the object.

Step 2: Key Formula or Approach:
The de-Broglie wavelength ( ) is given by the formula: where is Planck's constant ( ), is the mass of the object, and is its velocity.

Step 3: Detailed Explanation:
Given data:
Mass, (It's crucial to convert to SI units).
Velocity, .
Planck's constant, .
Calculation:
First, calculate the momentum : Now, calculate the de-Broglie wavelength: Rounding the result, we get .

Step 4: Final Answer:
The de-Broglie wavelength of the ball is m.

Step 1: Understanding the Concept:
This problem connects the de-Broglie wavelength of a charged particle with the kinetic energy it gains when accelerated through a potential difference. We need to find the potential required for an alpha particle to have the same wavelength as a proton accelerated through potential V.

Step 2: Key Formula or Approach:
1. The kinetic energy (KE) gained by a particle with charge accelerated through a potential difference is . 2. The de-Broglie wavelength ( ) is related to momentum ( ) by . 3. Momentum is related to kinetic energy by . Combining these, we get an expression for the de-Broglie wavelength in terms of the accelerating potential:

Step 3: Detailed Explanation:
Let's denote the properties of the proton with subscript 'p' and the alpha particle with subscript ' '. - Proton: mass , charge . - Alpha particle ( ): mass , charge . For the proton: It is accelerated through potential . Its wavelength is: For the alpha particle: Let it be accelerated through a potential . Its wavelength is: Equating the wavelengths: We are given that we want the same wavelength, so . Squaring both sides (and canceling ): The terms and cancel out. Rearranging to solve for :

Step 4: Final Answer:
The required accelerating potential for the alpha particle is V/8.