For each of the questions, four choices have been provided. Select the correct alternative. Velocity of the electron in electron microscope is \(1.6 \times 106 \mathrm{~m} / \mathrm{s}\). What would be the wavelength of the electron? (a) \(5.39 \AA\) (b) \(4.55 \AA\) (c) \(3.54 \AA\) (d) \(6.78 \AA\)

An electron microscope is a powerful tool used in scientific research to magnify objects up to 2 million times their original size. Unlike traditional light microscopes, which use visible light to illuminate the sample, electron microscopes use a beam of accelerated electrons as a source of illumination.

The fundamental principle behind the operation of an electron microscope is the wave nature of electrons. According to de Broglie's hypothesis, particles such as electrons exhibit both particle and wave properties. This dual nature allows for the high-resolution imaging in electron microscopes. The wavelength of these electron waves is much shorter relative to that of visible light, which enables the electron microscope to resolve much finer details.

Electron velocity within the microscope is crucial because it determines the wavelength of the electrons, and thus the resolving power of the microscope. High-velocity electrons have shorter wavelengths leading to higher resolution images. The microscope's electromagnetic lenses focus the electron beam onto the sample, allowing users to observe the miniscule structure of the specimen.

The velocity of an electron is a key factor in various applications such as electron microscopy and particle physics. In the context of an electron microscope, knowing the electron velocity is essential for calculating the de Broglie wavelength of the electron beam. The wavelength in turn is integral to understanding the resolving power of the microscope.

Electron velocity can be influenced by several factors including the electric potential through which they are accelerated. When electrons are accelerated through a potential difference, they gain kinetic energy which translates into velocity. The higher the accelerating voltage, the greater the kinetic energy and the higher the velocity of the electrons.

Understanding electron velocity also allows scientists to manipulate the electron beam for precise imaging or for experimentation in particle physics, where the speed of electrons might be nearly comparable to the speed of light, necessitating corrections based on the principles of relativity.

Planck's constant (\( h \: 6.626 \times 10^{-34} \: \text{Js} \: \text{m}^2 \: \text{kg/s} \)), named after Max Planck, is one of the fundamental constants in physics and plays a central role in quantum mechanics. It relates the energy of a photon to the frequency of its associated electromagnetic wave. The constant is an essential part of the equation used to calculate the energy of a photon: \( E = h u \), where \( E \) is the energy and \( u \) is the frequency.

In the de Broglie equation, Planck's constant is used to link the momentum of a particle to its wavelength: \( \text{wavelength} (\text{λ}) = \frac{h}{\text{momentum} (p)} \). This equation demonstrates the wave-particle duality of matter, and it is a key principle underlying the behavior of particles at the quantum level. Whether it's in calculating the behavior of electrons in an electron microscope or understanding the discrete energy levels of atoms, Planck's constant is an integral part of the fabric of quantum physics.