91Ó°ÊÓ

If diffraction were the only limitation on resolution, what would be the smallest structure that could be resolved in an electron microscope using 10-keV electrons?

To resolve details of an object, you must use a wavelength that is about the same size, or smaller, than the details you want to observe. Suppose you want to study a molecule that is about \(1.000 \times 10^{-10} \mathrm{m}\) in length. (a) What minimum photon energy is required to study this molecule? (b) What is the minimum kinetic energy of electrons that could study this? (c) Through what potential difference should the electrons be accelerated to reach this energy?

A scanning electron microscope is used to look at cell structure with 10 -nm resolution. A beam of electrons from a hot filament is accelerated with a voltage of \(12 \mathrm{kV}\) and then focused to a small spot on the specimen. (a) What is the wavelength in nanometers of the beam of incoming electrons? (b) If the size of the focal spot were determined only by diffraction, and if the diameter of the electron lens is one fifth the distance from the lens to the specimen, what would be the minimum separation resolvable on the specimen? (In practice, the resolution is limited much more by aberrations in the magnetic lenses and other factors.)

An image of a biological sample is to have a resolution of \(5 \mathrm{nm} .\) (a) What is the kinetic energy of a beam of electrons with a de Broglie wavelength of \(5.0 \mathrm{nm} ?\) (b) Through what potential difference should the electrons be accelerated to have this wavelength? (c) Why not just use a light microscope with a wavelength of 5 nm to image the sample?

The phenomenon of Brownian motion is the random motion of microscopically small particles as they are buffeted by the still smaller molecules of a fluid in which they are suspended. For a particle of mass $1.0 \times 10^{-16} \mathrm{kg},\( the fluctuations in velocity are of the order of \)0.010 \mathrm{m} / \mathrm{s} .$ For comparison, how large is the change in this particle's velocity when the particle absorbs a photon of light with a wavelength of \(660 \mathrm{nm}\) such as might be used in observing its motion under a microscope?

At a baseball game, a radar gun measures the speed of a 144-g baseball to be \(137.32 \pm 0.10 \mathrm{km} / \mathrm{h} .\) (a) What is the minimum uncertainty of the position of the baseball? (b) If the speed of a proton is measured to the same precision, what is the minimum uncertainty in its position?

A hydrogen atom has a radius of about \(0.05 \mathrm{nm}\) (a) Estimate the uncertainty in any component of the momentum of an electron confined to a region of this size. (b) From your answer to (a), estimate the electron's kinetic energy. (c) Does the estimate have the correct order of magnitude? (The ground-state kinetic energy predicted by the Bohr model is \(13.6 \mathrm{eV} .\) )

A bullet with mass \(10.000 \mathrm{g}\) has a speed of $300.00 \mathrm{m} / \mathrm{s}\( the speed is accurate to within \)0.04 \% .$ (a) Estimate the minimum uncertainty in the position of the bullet, according to the uncertainty principle. (b) An electron has a speed of $300.00 \mathrm{m} / \mathrm{s}\(, accurate to \)0.04 \% .$ Estimate the minimum uncertainty in the position of the electron. (c) What can you conclude from these results?