Mastering Physics: ± Accelerating Electrons

Part A

Through what potential difference must electrons be accelerated (from rest) so that they will have the same wavelength as an x-ray of wavelength 0.160 ? Use 6.63×10−34  for Planck's constant, 9.11×10−31  for the mass of an electron, and 1.60×10−19  for the charge on an electron. ANSWER:    = 58.8     Part B Through what potential difference must electrons be accelerated so they will have the same energy as the x-ray in Part A? Use 6.63×10−34  for Planck's constant, 3.00×108  for the speed of light in a vacuum, and 1.60×10−19  for the charge on an electron. ANSWER:    = 7760

Mastering Physics: ± Uncertainty in the Atomic Nucleus

Rutherford's scattering experiments gave the first indications that an atom consists of a small, dense, positively charged nucleus surrounded by negatively charged electrons. His experiments also allowed for a rough determination of the size of the nucleus. In this problem, you will use the uncertainty principle to get a rough idea of the kinetic energy of a particle inside the nucleus.
Consider a nucleus with a diameter of roughly meters.

Part A
Consider a particle inside the nucleus. The uncertainty in its position is equal to the diameter of the nucleus. What is the uncertainty of its momentum? To find this, use .

Express your answer in kilogram-meters per second to two significant figures.

ANSWER:	= 2.10×10−20

Part B

The uncertainty sets a lower bound on the average momentum of a particle in the nucleus. If a particle's average momentum were to fall below that point, then the uncertainty principle would be violated. Since the uncertainty principle is a fundamental law of physics, this cannot happen. Using kilogram-meters per second as the minimum momentum of a particle in the nucleus, find the minimum kinetic energy of the particle. Use kilograms as the mass of the particle. Note that since our calculations are so rough, this serves as the mass of a neutron or a proton.

Express your answer in millions of electron volts to two significant figures.

ANSWER:	= 0.81

Compare this to the normal energy scale for electrons in an atom, which is on the order of single electron volts. The characteristic energy scale for the nucleus seems to be roughly one million times that for electrons in an atom. This difference can be seen in calculations of the energy output per unit mass from coal-burning power plants, which utilize chemical energy (energy associated with the electrons of an atom) compared to the energy output from nuclear reactors (power plants that harness the much higher energies of nuclei).

Mastering Physics: ± Simultaneous Measurements of Position and Velocity

Part A

The x coordinate of an electron is measured with an uncertainty of 0.200 . What is , the x component of the electron's velocity, if the minimum percentage uncertainty in a simultaneous measurement of is 1.00 ? Use the following expression for the uncertainty principle: , where is the uncertainty in the x coordinate of a particle, is the particle's uncertainty in the x component of momentum, and , where is Planck's constant.

Express your answer in meters per second to three significant figures.

ANSWER:	= 57.9

Part B
Repeat Part A for a proton.

Express your answer in meters per second to three significant figures.

ANSWER:	= 3.16×10−2

Mastering Physics: ± Resolution of an Electron Microscope

An electron microscope is using a -keV electron beam. An atom has a diameter of about meters.

Part A
What is the wavelength of electrons in this microscope?

Express your answer in nanometers to three significant figures.

ANSWER:	= 3.88×10−2

Compare this wavelength to the typical size of an atom.

 

Part B
Can an individual atom theoretically be resolved using this electron microscope?

ANSWER:	yes no

Note that in practice a resolution better than cannot be achieved with an electron microscope. This is due, in part, to the fact that the focal length of a magnetic lens depends on the electron speed, which is never the same for all electrons in the beam.

 

 

Part C
Suppose we replace the electron beam with a proton beam. What proton energy is needed to achieve the same resolution as the electron beam in Part A?

Express your answer in electron volts to three significant digits.

ANSWER:	0.545

In principle, we could use proton beams with relatively low energy to get great resolution. However, in practice, it is much easier to create an electron beam than it is to create a proton beam, which is why electron microscopes have become the norm.

 

Mastering Physics: Width of a Wave Function

A particle is described by a wave function , where and are real, positive constants.

Part A
If the value of is increased, what effect does this have on the particle's uncertainty in position?

ANSWER:

The particle's uncertainty in position will decrease. The particle's uncertainty in position will increase. There is no effect on the particle's uncertainty in position.

Part B
If the value of is increased, what effect does this have on the particle's uncertainty in momentum?

 ANSWER:

The particle's uncertainty in momentum will decrease. The particle's uncertainty in momentum will increase. There is no effect on the particle's uncertainty in momentum.