Problem 51 - Master Resource Book in JEE Main Physics

Question

If the mass of neutral \(=1.7 \times 10^{-27} \mathrm{~kg}\), then the de- Broglie wavelength of neutral of energy \(3 \mathrm{eV}\) is \(\left(h=6.6 \times 10^{-34} \mathrm{~J}-\mathrm{s}\right)\) (a) \(1.6 \times 10^{-16} \mathrm{~m}\) (b) \(1.6 \times 10^{-11} \mathrm{~m}\) (c) \(1.4 \times 10^{-10} \mathrm{~m}\) (d) \(1.4 \times 10^{-11} \mathrm{~m}\)

Step-by-Step Solution

Answer

The de-Broglie wavelength is \(1.4 \times 10^{-10} \mathrm{~m}\), which corresponds to option (c).

Step 1: Energy Conversion

First, we need to convert the given energy from electronvolts to joules, since the de-Broglie formula requires the mass and velocity in SI units. The conversion factor is: \[1 \text{ eV} = 1.6 \times 10^{-19} \text{ J}\]. Thus, \(3 \text{ eV} = 3 \times 1.6 \times 10^{-19} \text{ J} = 4.8 \times 10^{-19} \text{ J}\).

Step 2: Determine Velocity of Particle

Using the relation \(E = \frac{1}{2}mv^2\), where \(E\) is the energy, \(m\) is the mass, and \(v\) is the velocity, solve for velocity \(v\). Rearrange the formula to \(v = \sqrt{\frac{2E}{m}}\). Substituting \(E = 4.8 \times 10^{-19} \text{ J}\) and \(m = 1.7 \times 10^{-27} \text{ kg}\), we get: \[v = \sqrt{\frac{2 \times 4.8 \times 10^{-19}}{1.7 \times 10^{-27}}} = \sqrt{5.647 \times 10^8} \approx 2.38 \times 10^4 \text{ m/s}\].

Step 3: Calculate de-Broglie Wavelength

The de-Broglie wavelength \(\lambda\) is given by \(\lambda = \frac{h}{mv}\). Using \(h = 6.6 \times 10^{-34} \text{ J} \cdot \text{s}\), \(m = 1.7 \times 10^{-27} \text{ kg}\), and \(v = 2.38 \times 10^4 \text{ m/s}\), we find \[\lambda = \frac{6.6 \times 10^{-34}}{1.7 \times 10^{-27} \times 2.38 \times 10^4} \approx 1.4 \times 10^{-10} \text{ m}\].

Step 4: Selection of Correct Option

After computing the de-Broglie wavelength, look at the options provided. The calculated wavelength \(1.4 \times 10^{-10} \text{ m}\) matches with option (c).

Key Concepts

Energy Conversion

Understanding energy conversion is fundamental in bridging various units of measurement. When dealing with particles and energy, it's common to see energy values given in electronvolts (eV), specifically in the context of quantum mechanics. However, for calculations involving other scientific quantities like velocity and mass in the International System of Units (SI), converting eV to joules (J) is necessary.
The relationship is straightforward:

• 1 eV = 1.6 × 10^-19 J

This conversion ensures consistency in units when using formulas that require SI units, such as the de Broglie wavelength formula. Thus, converting 3 eV to joules becomes a simple multiplication, yielding an energy of 4.8 × 10^-19 J. This conversion is crucial for further calculations, allowing physicists to work seamlessly within a coherent system of units that supports broader physical laws and formulas.

Particle Velocity

Velocity is a critical parameter in understanding the behavior of particles in motion, especially for quantum mechanics. To find the velocity of a particle, given its energy and mass, we use the classic kinetic energy formula:

• Energy (E) = 1/2  m v²

Rearranging this formula allows us to solve for velocity (v) as follows:

• v = \( \sqrt{\frac{2E}{m}} \)

With the converted energy in joules and the mass known, you can substitute these values into the equation. This step is vital in determining how fast the particle is moving, as velocity directly impacts the calculation of the de Broglie wavelength.
For instance, by substituting a particle's mass of 1.7 × 10^-27 kg and its energy of 4.8 × 10^-19 J, we find the velocity to be approximately 2.38 × 10⁴ m/s.

Quantum Mechanics

Quantum mechanics is the field of physics that explores the behavior of particles at the smallest scales. It reveals surprising truths about matter and energy, highlighting phenomena such as quantization, probabilities, and wave functions.
Within this realm:

• Particles can exhibit both wave-like and particle-like properties depending on how they are measured.
• Quantum mechanics challenges our classical perceptions and introduces concepts like superposition and entanglement.

This branch of physics is vital for understanding atomic and subatomic processes. In the example solution, quantum mechanics principles are applied to calculate a particle's de Broglie wavelength. This calculation allows you to see how waves and particles operate on small scales, which classical physics cannot accurately describe alone.

Wave-Particle Duality

Wave-particle duality is a fundamental concept in quantum mechanics. It proposes that particles, like electrons and photons, exhibit both wave-like and particle-like properties. This duality is not just a theoretical concept; it is evidenced through experiments like the double-slit experiment.

• The de Broglie wavelength formula, \( \lambda = \frac{h}{mv} \), supports this concept by linking a particle's wave nature to its momentum.
• This formula allows us to calculate a wavelength for any particle, implying that even massive particles have wave properties, although these are typically negligible on a human scale.

Using Planck's constant \( h \), the de Broglie wavelength becomes a practical tool in quantum physics to predict how quantum particles behave when observed as waves. In essence, wave-particle duality is central to many of today's technological advancements, from semiconductor physics to quantum computing.