With what voltages must we accelerate electrons in a electronmicroscope to resolve
(a) a virus of diameter 10nm
(b) an atom of diameter 0.1nm?

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3. a) Show that the phase velocity of the de Broglie waves of aparticle of mass m and de Broglie wavelength ? is given by .
b) Compare the phase and group velocities of an electron whose deBroglie wavelength is exactly 1x10-13 m.
4. To what voltages must we accelerate electrons (as in an electronmicroscope, for example) if we wish to resolve;
a) a virus of diameter 12 nm , b) an atom of 0.12 nm, c) a protonof diameter 1.2 fm?
5. a) The x-coordinate of an electron is measured with anuncertainty of 0.20 mm. What is the x- component of the electronâsvelocity, vx, if the minimum percentage uncertainty in asimultaneous measurement of vx is 1.0%.
b) Repeat part (a) for a proton. 6. Suppose Fuzzy, a quantummechanical duck, lives in a world in which h = 2? J.s. Fuzzy has amass
of 2.0 kg and is initially known to be within a region 1.0 mwide.
a) What is the minimum uncertainty in his speed? b) Assuming thisuncertainty in speed to prevail for 5.0 s, determine theuncertainty in position after
this time.
7. The seeing ability, or resolution, of radiation is determined byits wavelength. If size of an atom is of the order of 0.1 nm, howfast must an electron travel to have a wavelength small enough toâseeâ an atom?
8. An air rifle is used to shoot 1.0 g particles at 100 m/s througha hole of diameter 2.0 mm. How far from the rifle must an observerbe to see the beam spread by 1.0 cm because of the uncertaintyprinciple? Compare this answer with the diameter of the universe (? 1026 m).

Electron microscopes use the wave characteristic of fast electrons.The electrons are accelerated in an electric field which is createdby a potential difference of a few 10 kV for scanning electronmicroscopes and 100 â200 kV for transmission electron microscopes.Note: At these voltages the electrons are alreadyrelativistic.
Calculate the de Broglie wavelength of electrons in an electronmicroscope which was accelerated to a speed of 1000 km/s.

To resolve an object in an electron microscope, the wavelength of the electrons must be close to the diameter of the object. What kinetic energy must the electrons have in order to resolve a protein molecule that is 2.4nm in diameter?