Problem 67 - Master Resource Book in JEE Main Physics

Question

The wavelength associated with a moving particle depends upon power \(p\) of its mass \(m, q\) th power of its velocity \(v\) and \(r\) th power of Planck's constant \(h\). Then the correct set of values of \(p, q\) and \(r\) is (a) \(p=1, q=-1, r=1\) (b) \(p=1, q=1, r=1\) (c) \(p=-1, p=-1, r=-1\) (d) \(p=-1, q=-\mathbf{L} r=1\)

Step-by-Step Solution

Answer

The correct set of values is (d) \(p=-1, q=-1, r=1\).

Step 1: Understand De Broglie's Hypothesis

De Broglie's hypothesis relates the wavelength \( \lambda \) of a moving particle to its momentum. The relation is given by: \( \lambda = \frac{h}{mv} \), where \( h \) is Planck's constant, \( m \) is mass, and \( v \) is velocity. This implies \( \lambda \) is proportional to \( \frac{h}{m^1v^1} \).

Step 2: Set Up the Proportional Relationship

We express the wavelength \( \lambda \) in terms of the given powers \( p, q, \) and \( r \) such that \( \lambda \propto m^p v^q h^r \).

Step 3: Equate De Broglie's Formula to the Given Powers

From De Broglie's formula, \( \lambda \propto \frac{h^1}{m^1v^1} = h^1 m^{-1} v^{-1} \). Thus, comparing both expressions, we get:- For mass: \( p = -1 \)- For velocity: \( q = -1 \)- For Planck's constant: \( r = 1 \).

Step 4: Match Values with Answer Choices

Based on derived values of \( p, q, r \) as \( p = -1, q = -1, r = 1 \), match this set to the given options. Choice (d) matches correctly: \( p = -1, q = -1, r = 1 \).

Key Concepts

Wavelength of Moving Particle

When we talk about the wavelength of a moving particle, we are referring to a concept that arises from the field of quantum mechanics. According to De Broglie's hypothesis, every moving particle has an associated wavelength, even if it's not visible to the naked eye.
This wavelength is commonly known as the De Broglie wavelength.This idea revolutionizes the way we understand matter, suggesting that particles like electrons exhibit wave-like properties, similar to light. The De Broglie wavelength is mathematically represented as \[ \lambda = \frac{h}{mv} \] where \( \lambda \) is the wavelength, \( h \) is Planck's constant, \( m \) is the mass of the particle, and \( v \) is its velocity. This fascinating concept allows us to comprehend that the wavelength is inversely related to the momentum of the particle. This means that as the product of mass and velocity (momentum) increases, the wavelength decreases.

Planck's Constant

Planck's constant \( h \) is a fundamental constant in physics that plays a critical role in quantum mechanics. It helps to bridge the old gap between the classical and quantum worlds, acting as a conversion factor between the energy carried by a photon and its electromagnetic wave frequency.Planck's constant has a precise value of approximately \( 6.626 \times 10^{-34} \) Joule seconds (Js). This value might seem very small, but it is paramount in calculations involving particles at the atomic and subatomic levels.
In the context of De Broglie's hypothesis, Planck's constant is vital as it appears directly in the equation for calculating the wavelength of moving particles. It helps establish the quantifiable relationship between momentum and wavelength, enabling scientists to make precise calculations about particle behavior.

• It depicts the granular nature of energy at the quantum level.
• Central to the Planck-Einstein relation and De Broglie's equation.

Momentum and Wavelength Relationship

The intriguing relationship between momentum and wavelength stems from De Broglie's hypothesis. As seen in the formula \( \lambda = \frac{h}{mv} \), momentum \( mv \) plays a crucial role in determining the wavelength of a particle.Momentum is a measure of an object's motion, and it is calculated as the product of its mass and velocity.

**Key Points:**

• The wavelength is inversely proportional to momentum, meaning as one increases, the other decreases.
• This relationship articulates the dual nature of particles, embodying both particle-like and wave-like characteristics.
• Allows the understanding of quantum scale phenomena where classical mechanics fails.

In essence, for a given Planck's constant, a heavier and faster particle (higher momentum) will have a smaller wavelength, exhibiting more of its particle nature, while a slower, lighter particle will have a longer wavelength, showcasing wave-like properties.