An early model of the atom, proposed by Rutherford after his discovery of the atomic nucleus, had a positive point charge +Ze (the nucleus) at the center of a sphere of radius R with uniformly distributed negative charge -Ze. Z is the atomic number, the number of protons in the nucleus and the number of electrons in the negative sphere. a. Show that the electric field strength inside this atom is
b. What is E at the surface of the atom? Is this the expected value? Explain.
c. A uranium atom has Z = 92 and R = 0.10 nm. What is the electric field strength at r = 1 2 R?
It has been verified that the electric field strength inside the atom is
the electrical field is 0 at any surface of any atom
The electric field strength of uranium at r=R / 2 is
We use Gauss' Law to find the electric field inside the sphere.
is the Electric Flux
E is the Electric Field
d a is the Area
Q is the Total Charge
0 is the Permittivity
The charge inside the sphere from the electron can be evaluated as:
The total charge enclosed in a sphere of radius r is
r is the radius inside the atom.
The electric field from the electrons can thus be given by:
The electric field from nucleus is from a point charge Z e It is thus being given by:
Combining the above equations, we get:
Since atoms are neutral therefore it can be said that the electric field strength at the surface of the atom is 0
The electric field strength at the uranium surface is calculated as
Evaluating electric field for the values of
, ,, r=0.5 R
An infinite cylinder of radius has a linear charge density . The volume charge density within the cylinder is , where is a constant to be determined.
a. Draw a graph of versus for an -axis that crosses the cylinder perpendicular to the cylinder axis. Let range from to .
b. The charge within a small volume is . The integral of over a cylinder of length is the total charge within the cylinder. Use this fact to show that .
Hint: Let be a cylindrical shell of length , radius , and thickness . What is the volume of such a shell?
c. Use Gauss's law to find an expression for the electric field strength inside the cylinder, , in terms of and .
d. Does your expression have the expected value at the surface, ? Explain.
FIGURE Q24.2 shows cross sections of three-dimensional closed surfaces. They have a flat top and bottom surface above and below the plane of the page. However, the electric field is everywhere parallel to the page, so there is no flux through the top or bottom surface. The electric field is uniform over each face of the surface. For each, does the surface enclose a net positive charge, a net negative charge, or no net charge? Explain.
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