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CLFds
George Brown
This work is produced by The Connexions Project and licensed under the Creative Commons Attribution License †
Abstract Page 5 of 6 of a PLTL workshop on Coulomb law elds, for undergraduate physics.
Page 1^1 Page 2^2 Page 3^3 Page 4^4 Page 5 (Section ) Page 6^5 Copyright ©2005, G. Raymond Brown, Ph.D.
2 Problems
Work on these problems with your Peer Team members. Determine analytic solutions before substituting any numerical values to nd a numerical solution. Each problem is solved by use of either Equation 2
(
⇀ E
[⇀
r
]
= (^4) πε^10
∑N
n=
qn rn^2
^
rn) or Equation 3 (
⇀ E
[⇀
r
]
= (^4) πε^10
V ′
ρ^η η^2 dV^
′, or the lower-dimension analogues
involving σ := (^) dSdq′ or λ := (^) dLdq′ ). These basic relationships should form the starting point of your solutions, although other basic relationships that you have encountered before, or perhaps have to look up, may also be needed to complete the solutions.
2.1 Problem 1
Consider a disk in the xy plane with radius R = 3 cm and constant charge density σ = 3. (^4) cmC 2 , centered on the origin of coordinates. Find the electric eld everywhere on the z axis.
2.2 Problem 2
Repeat Problem 1 with the charge density changed to σ = σ 0 s
′ R , where^ s
′ (^) is the radial variable in the xy
plane: s′^ =
x′^2 + y′^2 , restricted to the charge distribution, and σ 0 is the charge density of Problem 1. ∗Version 1.3: May 9, 2006 7:39 pm GMT- †http://creativecommons.org/licenses/by/2.0/ (^1) "Coulomb Law Fields" (^2) "CLFds2" (^3) "CLFds3" (^4) "CLFds4" (^5) "CLFds6"
2.3 Problem 3
The gure below can be taken as a very crude model of a water molecule. The red disk represents the oxygen atom and the two green disks represent the hydrogen atoms in the water molecule. Given the coordinate system shown in the diagram, nd the electric eld everywhere on the x axis except the origin. In nature, and for an isolated water molecule, the bond distance d ≈ 1. 0 Å = 10 −^10 m, the angle between the hydrogen atoms is 2 α ≈ 712 π = 105 ◦^ and the charge q ≈ 109 e. These values can change signi cantly if other molecules are in the neighborhood of the water molecule. Note the coordinate system chosen: all charges lie in the xy plane, with the x axis bisecting the bond angle 2 α.
Isolated Water Molecule
Figure 1: A crude model of the water molecule, with hydrogen atoms represented by green disks and the oxygen atom represented by a red disk. The black dots represent eld points on the x axis.
2.4 Problem 4
Four equal point charges q lie at the corners of a regular tetrahedron centered on the origin, as diagrammed be-
low. The coordinates of the point charges are {a{ √^13 , 0 , 2 √^16 , }, a{− 2 √^13 , 12 , − 2 √^16 }, a{− 2 √^13 , 12 , − 2 √^16 }, a{ 0 , 0 , (^12)
3 2 }}, where a is the edge length of the tetrahedron, which is the distance separating any two of the charges.
Crudely Modeled Ammonium Ion
Figure 3: A very, very crude model of the ammonium ion, NH 4 +. The central blue sphere represents the nitrogen atom, the red spheres hydrogen atoms.
(b) For this model, consider a eld point located at the center of any one of the four triangular sides of the tetrahedron. The sum for the electric eld at this point reduces to contributions from only two of the ve point source charges. Why? Which two charges sum to make a nonzero contribution? Is the direction
of this electric eld pointing out of the tetrahedron or into it? What is the value of the electric eld at this eld point? Justify your answers.
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(^6) "Coulomb Law Fields" (^7) "CLFds2" (^8) "CLFds3" (^9) "CLFds4" (^10) "CLFds6"
