Electric fields and potentials - Physics and Astronomy at TAMU
Electric fields and potentials - Physics and Astronomy at TAMU
Electric fields and potentials - Physics and Astronomy at TAMU
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LABORATORY II<br />
ELECTRIC FIELDS AND POTENTIALS<br />
In this labor<strong>at</strong>ory, you will learn how to calcul<strong>at</strong>e <strong>and</strong> measure the electric <strong>potentials</strong><br />
<strong>and</strong> electric <strong>fields</strong> in a region of space. The <strong>potentials</strong> <strong>and</strong> <strong>fields</strong> are produced by charges<br />
<strong>and</strong> conductors in the region. Each charge is a source of electric field, <strong>and</strong> each conductor<br />
cre<strong>at</strong>es a boundary of constant potential – an equipotential. The electric field is a vector<br />
field E r<br />
(r r ) - a vector whose magnitude <strong>and</strong> direction are defined for every point of space r in the vicinity of the source. The p<strong>at</strong>tern of this vector field is visualized by drawing lines<br />
of force, a p<strong>at</strong>tern of lines th<strong>at</strong> are everywhere parallel to the electric field, <strong>and</strong> whose<br />
transverse density is proportional to the magnitude of the electric field.<br />
Figure 1. Lines of force <strong>and</strong> equi<strong>potentials</strong> near an electric dipole.<br />
As an example of these concepts, Figure 1 shows the equi<strong>potentials</strong> <strong>and</strong> lines of force<br />
in the vicinity of an electric dipole – a pair of equal <strong>and</strong> opposite charges separ<strong>at</strong>ed by a<br />
distance.<br />
While it is possible to measure electric field directly, it is frequently easier to measure<br />
the potential difference between two points. The device used to measure potential difference<br />
is a voltmeter. The voltmeter measures the potential difference between its two terminals.<br />
By <strong>at</strong>taching conducting leads to these terminals <strong>and</strong> then touching the other ends<br />
of the two leads to two points in a region, you can measure the potential difference, or<br />
voltage, between those points.<br />
In the first problem, you will first learn to calcul<strong>at</strong>e the distribution of potential in a region<br />
of space, using the relax<strong>at</strong>ion technique. While you will use this powerful technique<br />
to divide up a two-dimensional region into a finite-element grid, the same technique can<br />
also be used in three dimensions, <strong>and</strong> even in four dimensions to calcul<strong>at</strong>e time-changing<br />
electric <strong>fields</strong> <strong>and</strong> <strong>potentials</strong>.<br />
In the succeeding problems, you will learn to actually measure the potential distribution<br />
in a 2-dimensional array of charges <strong>and</strong> conductors.
Objectives:<br />
Successfully completing this labor<strong>at</strong>ory should enable you to:<br />
Prepar<strong>at</strong>ion:<br />
• Use Gauss’ Law <strong>and</strong> Coulomb’s Law to predict the distribution of potential in<br />
the space around two-dimensional distributions of electric charge.<br />
• Use the method of relax<strong>at</strong>ion to numerically calcul<strong>at</strong>e the potential distribution<br />
resulting from example 2-dimensional charge distributions.<br />
• Extract the electric field distribution from the potential distribution by numerical<br />
<strong>and</strong> graphical techniques.<br />
• Experimentally map the distribution of electric potential using conductive lines<br />
on resistive paper.<br />
• Measure the potential distribution inside a closed conductor boundary, <strong>and</strong> the<br />
shielding of electric field within a closed conductor arising from external<br />
charges.<br />
• Experimentally track the equi<strong>potentials</strong> in the space around charge distributions<br />
using a potential balance.<br />
Read Cummings, Laws, Redish & Cooney, v.3, chapter 22-25. Note th<strong>at</strong> you will be<br />
using the concept of electric potential before it is introduced in lecture. You may obtain<br />
background for this purpose by reading chapter 25.<br />
2
Problem 1. Calcul<strong>at</strong>ion of potential distributions using the<br />
Method of Relax<strong>at</strong>ion<br />
Your team is assigned to develop a piece of software th<strong>at</strong> will use the technique of relax<strong>at</strong>ion<br />
to calcul<strong>at</strong>e electric potential distributions in 2 dimensions of space near charges<br />
<strong>and</strong> conductors.<br />
<strong>Electric</strong> potential is the work per unit charge th<strong>at</strong> must be exerted to move a charge<br />
from one loc<strong>at</strong>ion to another. The electric field is the force per unit charge exerted upon a<br />
charge due<br />
You will cre<strong>at</strong>e an EXCEL spreadsheet, using each cell to represent a p<strong>at</strong>ch of area in<br />
the problem. The number contained in each cell will represent the potential <strong>at</strong> th<strong>at</strong> loc<strong>at</strong>ion.<br />
You will encode the relax<strong>at</strong>ion technique in the formulae of each cell th<strong>at</strong> calcul<strong>at</strong>es<br />
its electric potential.<br />
The method of relax<strong>at</strong>ion<br />
r<br />
The potential V (r ) in a region of empty space (a region containing no net electric<br />
charges) varies smoothly – the potential <strong>at</strong> one spot is the average of the <strong>potentials</strong> <strong>at</strong><br />
nearby spots surrounding it. This simple property makes it possible to calcul<strong>at</strong>e the potential<br />
distribution by the technique of relax<strong>at</strong>ion.<br />
To underst<strong>and</strong> this property, divide a region of empty space into a 2-D grid as shown in<br />
Error! Reference source not found.. The grid elements have side length a in both x <strong>and</strong><br />
y. Of course this grid is just a 2-D slice of a 3-D region of space. Each grid element is actually<br />
a rectangular prism, extending a distance L into the plane of the problem, of size (a ×<br />
a × L)<br />
Figure 2. Connection of electric field <strong>and</strong> potential to the adjacent elements in a finiteelement<br />
grid.
Problem 1. Calcul<strong>at</strong>ion of potential distributions using the Method of Relax<strong>at</strong>ion<br />
Consider the nine contiguous grid elements shown above, placed in an electric field E r<br />
as shown. We will develop a technique for calcul<strong>at</strong>ing the potential in the center grid element<br />
(V 22 ) in terms of the <strong>potentials</strong> in the neighboring grid elements. In order to connect<br />
potential to electric field, we will use the definition of potential <strong>and</strong> Gauss’ Law. The<br />
definition of potential difference is<br />
r<br />
r r r r<br />
V<br />
−V<br />
≡ −∫ 2<br />
(<br />
2<br />
) (<br />
1)<br />
E ⋅ dr<br />
r<br />
1<br />
Gauss’ Law st<strong>at</strong>es th<strong>at</strong><br />
r r Q<br />
∫ E ⋅ dS =<br />
ε 0<br />
r<br />
So how do we use these connections in a finite grid in space?<br />
Let’s apply the principle of superposition, <strong>and</strong> examine the effects on the potential distribution<br />
from the x- <strong>and</strong> y-components of E r separ<strong>at</strong>ely. The component E x (red in Error!<br />
Reference source not found.) can only affect the differences between <strong>potentials</strong> in cells<br />
th<strong>at</strong> differ in their x position. The component E y (blue in Error! Reference source not<br />
found.) can only affect the differences between <strong>potentials</strong> in cells th<strong>at</strong> differ in their y position.<br />
V<br />
V<br />
32<br />
22<br />
−V<br />
−V<br />
22<br />
12<br />
= −E<br />
= −E<br />
xr<br />
xl<br />
a<br />
a<br />
(1)<br />
(2)<br />
V23 −V22<br />
= −E<br />
yta<br />
(3)<br />
V −V<br />
= −E<br />
a<br />
22<br />
12<br />
yb<br />
E xr <strong>and</strong> E xl refer to the x-component of electric field evalu<strong>at</strong>ed <strong>at</strong> the right <strong>and</strong> left<br />
boundaries of the center cell, indic<strong>at</strong>ed in green in Error! Reference source not found..<br />
E yt <strong>and</strong> E yb refer to the y-component of electric field evalu<strong>at</strong>ed <strong>at</strong> the top <strong>and</strong> bottom<br />
boundaries of the center cell, as indic<strong>at</strong>ed in green in Error! Reference source not<br />
found..<br />
Now we can apply Gauss’ Law to rel<strong>at</strong>e these <strong>potentials</strong>:<br />
r r<br />
∫ E ⋅ dS = E<br />
xr<br />
( aL)<br />
+ E<br />
yt<br />
( aL)<br />
+ Exl<br />
( −aL)<br />
+ E<br />
yb<br />
( −aL)<br />
( V32<br />
−V22<br />
) ( V22<br />
−V12<br />
) ( V23<br />
−V22<br />
) ( V<br />
= ( aL)<br />
+ ( −aL)<br />
+ ( aL)<br />
+<br />
− a<br />
− a<br />
− a<br />
⎡ V32<br />
+ V12<br />
+ V23<br />
+ V21<br />
⎤<br />
= 4L⎢V<br />
22<br />
−<br />
⎥<br />
⎣<br />
⎦<br />
22<br />
−V<br />
− a<br />
12<br />
)<br />
( −aL)<br />
Now suppose these cells do not contain any net charge: Q = 0 in Eq. 2. Then potential<br />
V 22 is the average of the <strong>potentials</strong> in its neighbors:<br />
V<br />
22<br />
V32<br />
+ V12<br />
+ V23<br />
+ V21<br />
= (5)<br />
4<br />
(4)<br />
4
Problem 1. Calcul<strong>at</strong>ion of potential distributions using the Method of Relax<strong>at</strong>ion<br />
This is the essential underpinning of the method of relax<strong>at</strong>ion. To implement it in as a<br />
basis to calcul<strong>at</strong>e the <strong>potentials</strong> throughout a region, we follow an orderly prescription:<br />
‣ start with a guess for the <strong>potentials</strong> in all cells.<br />
‣ define the conditions <strong>at</strong> the boundaries of the region being calcul<strong>at</strong>ed.<br />
‣ implement the averaging calcul<strong>at</strong>ions of Eq. 5.<br />
‣ Iter<strong>at</strong>e the above procedure repe<strong>at</strong>edly. On successive iter<strong>at</strong>ions, the <strong>potentials</strong><br />
will approach ever closer to the physical <strong>potentials</strong> th<strong>at</strong> are solutions to Gauss’<br />
Law.<br />
Now you are ready to set up a spreadsheet th<strong>at</strong> will implement this procedure.<br />
0<br />
0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0<br />
0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0<br />
0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0<br />
0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0<br />
0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0<br />
0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0<br />
0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0<br />
0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0<br />
0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0<br />
0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0<br />
0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0<br />
0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0<br />
0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0<br />
0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0<br />
0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0<br />
0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0<br />
0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0<br />
0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0<br />
0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0<br />
0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0<br />
0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0<br />
0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0<br />
0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0<br />
0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0<br />
0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0<br />
0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0<br />
0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0<br />
0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0<br />
0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0<br />
0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0<br />
0<br />
Figure 3. EXCEL spreadsheet configured for relax<strong>at</strong>ion method calcul<strong>at</strong>ion of <strong>potentials</strong>.<br />
5
Problem 1. Calcul<strong>at</strong>ion of potential distributions using the Method of Relax<strong>at</strong>ion<br />
Setting up the spreadsheet<br />
The condition for empty space is th<strong>at</strong> its potential is an average of the <strong>potentials</strong> in the<br />
neighboring cells. You can achieve th<strong>at</strong> condition in cell C3, for example, by entering the<br />
expression<br />
=( B3 +C2+C4+D3)/4<br />
You should form<strong>at</strong> the cells of the spreadsheet so th<strong>at</strong> they are square as they appear on<br />
the screen (th<strong>at</strong> way the simul<strong>at</strong>ion of 2-D space will not be distorted). You can then enter<br />
the expression for neighbor averaging in any one cell, <strong>and</strong> copy the expression into all the<br />
other cells of the array. EXCEL will increment the cell indices correctly so th<strong>at</strong> the<br />
neighbor rel<strong>at</strong>ion will be preserved in all cells.<br />
Keep clearly in mind th<strong>at</strong> you are actually modeling a 3-D problem, which has planar<br />
symmetry: the distribution of charge <strong>and</strong> potential in any x-y slice of the problem region is<br />
exactly the same. So if you place into the problem region a straight line equipotential, it<br />
actually represents a planar equipotential surface extending back into the paper. And if<br />
you place a point charge, it actually represents a line of uniform charge density.<br />
Boundaries of the problem region<br />
The cells th<strong>at</strong> form the 4 boundaries of the problem region must be tre<strong>at</strong>ed in a way<br />
th<strong>at</strong> does not require knowledge of the <strong>potentials</strong> outside the problem region. There are 3<br />
ways you can tre<strong>at</strong> boundary regions:<br />
‣ Neumann boundaries. You can set a boundary region to a constant fixed potential.<br />
To do this, just enter the desired value into the cells along th<strong>at</strong> boundary<br />
region.<br />
You can require the potential to be constant along the boundary region, but<br />
let the potential <strong>at</strong>tain a value th<strong>at</strong> is determined by the <strong>potentials</strong> in the interior<br />
cells. To do this, set up a cell outside the boundary region, give it a name (for<br />
example right), <strong>and</strong> enter as its formula the average of the row or column of<br />
cells th<strong>at</strong> are inside the problem region, directly adjacent to the boundary cells<br />
of the equipotential. If you want the column Z4-Z50 to be an equipotential, you<br />
might set up AA25 to be your cell for averaging, name it right, <strong>and</strong> enter the<br />
formula<br />
=SUM(Y4:Y50)/47<br />
= right in the formulae of cells Z4-Z50.<br />
for the formula of cell AA25, <strong>and</strong> then enter<br />
‣ Dirichlet boundaries. You can require th<strong>at</strong> the electric field along a boundary<br />
be parallel to th<strong>at</strong> boundary. This is the same as requiring th<strong>at</strong> the potential on<br />
each boundary cell be the same as the potential on the cell one step in from it.<br />
For example, if you want the column Z4-Z50 to have electric field parallel to<br />
th<strong>at</strong> boundary, you would enter =Y4 as the formula of cell Z4, <strong>and</strong> copy th<strong>at</strong><br />
formula to the entire column. EXCEL again maintains the indexing so th<strong>at</strong> the<br />
condition requires parallel field along th<strong>at</strong> columnar boundary.<br />
6
Problem 1. Calcul<strong>at</strong>ion of potential distributions using the Method of Relax<strong>at</strong>ion<br />
Charges in the interior<br />
Now you are set up to enter objects into your problem region. Suppose you want to<br />
simply place a charge in the interior <strong>and</strong> find the potential distribution th<strong>at</strong> it cre<strong>at</strong>es in the<br />
space around it. You need to first choose wh<strong>at</strong> kind of boundaries you want to have. If<br />
there are no other charges in the problem, then you probably would want to impose Neumann<br />
boundaries (equi<strong>potentials</strong>). Note th<strong>at</strong> the result will be the potential distribution of<br />
a charge in a metal box. Unless you tie the 4 sides to a common equipotential, each<br />
boundary will flo<strong>at</strong> to a different potential according to how close the charge is to th<strong>at</strong><br />
boundary.<br />
Now select a spot where you want to place the charge. In the contents of its cell, you<br />
want to add a term th<strong>at</strong> gener<strong>at</strong>es potential appropri<strong>at</strong>e to th<strong>at</strong> charge.<br />
We must take care about units of measure <strong>at</strong> this point. You are assuming some units<br />
for the unit size h of the physical cells th<strong>at</strong> each EXCEL cell represents, <strong>and</strong> units for the<br />
<strong>potentials</strong> you put into the cells for equi<strong>potentials</strong>, <strong>and</strong> now you need units for charges th<strong>at</strong><br />
you want to place within the problem region. Suppose th<strong>at</strong> you want to insert a net charge<br />
Q in the cell (m,n). The correspondence of units for charge requires a more careful analysis<br />
of the m<strong>at</strong>hem<strong>at</strong>ics of the relax<strong>at</strong>ion method. This analysis is provided in Appendix A.<br />
The bottom line is th<strong>at</strong> you should add a term equal to<br />
Q<br />
∆ V = .<br />
2ε 0<br />
h<br />
With Q[C], h[m], <strong>and</strong> ε 0 = 8.9x10 -12 , the potential offset ∆V will have the units of<br />
volts. Note th<strong>at</strong> this offset is added together with the expression for the averaging of the<br />
neighbor <strong>potentials</strong> in the cell where the charge is placed.<br />
Now suppose th<strong>at</strong> you want to distribute charge uniformly within a region inside your<br />
problem. You can do this most easily by naming a cell outside the problem region currit,<br />
adding the above term using the named variable currit for the potential offset ∆V (note th<strong>at</strong><br />
the charge you use in calcul<strong>at</strong>ing the value to put in the variable currit is the charge th<strong>at</strong><br />
will be placed in each cell of the charge distribution). Then enter the added term to the<br />
formula in one element in the charged region, <strong>and</strong> copy it to all the elements in the region.<br />
You can keep track of which cells contain charge by coloring those cells.<br />
Now you need to set up the spreadsheet so th<strong>at</strong> it iter<strong>at</strong>ively applies all formulae one<br />
time each time you hit the F9 key. You can do th<strong>at</strong> by going to<br />
Tools→Options→Calcul<strong>at</strong>ion: check Iter<strong>at</strong>ions, <strong>and</strong> enter 1 in the box for maximum iter<strong>at</strong>ions.<br />
Each time you hit the F9 key, EXCEL will execute one iter<strong>at</strong>ion of all calcul<strong>at</strong>ions<br />
throughout the array, <strong>and</strong> then stop.<br />
Now hit F9 <strong>and</strong> observe how the contents of the cells in your array change. Each iter<strong>at</strong>ion<br />
brings the relax<strong>at</strong>ion closer to the physical potential distribution th<strong>at</strong> obeys the laws of<br />
electrost<strong>at</strong>ics. You will probably need to hit F9 many times before the cell contents all settle<br />
to equilibrium values.<br />
Now take a piece of tracing paper <strong>and</strong> trace the equi<strong>potentials</strong> th<strong>at</strong> are produced by<br />
your charge in a box.<br />
7
Problem 1. Calcul<strong>at</strong>ion of potential distributions using the Method of Relax<strong>at</strong>ion<br />
Try changing the loc<strong>at</strong>ion of the charge in the box, for example place it near one<br />
boundary, then out in the center. See how the potential distribution changes. Is the distribution<br />
sensible in each case?<br />
Now try putting a dipole in your box. Again relax the <strong>potentials</strong>, <strong>and</strong> trace the equi<strong>potentials</strong>.<br />
Explain the p<strong>at</strong>tern th<strong>at</strong> you obtain.<br />
Equi<strong>potentials</strong> in the interior<br />
Now try modeling a parallel pl<strong>at</strong>e capacitor. Select two parallel strips of cells, preferably<br />
ending <strong>at</strong> a boundary. Make the boundaries on the side where the strips end, <strong>and</strong> the<br />
opposite boundary, into Dirichlet boundaries. You probably want the other two boundaries<br />
to be Neumann boundaries. Now set the <strong>potentials</strong> on the two pl<strong>at</strong>es to equal <strong>and</strong> opposite<br />
values, <strong>and</strong> iter<strong>at</strong>e to obtain the potential distributions.<br />
Displaying the equi<strong>potentials</strong><br />
You can use the graphing functions of EXCEL to nicely display the equi<strong>potentials</strong> in<br />
your problem. After the problem has converged, cursor-select all the cells in your problem<br />
region in the spreadsheet, <strong>and</strong> then select Chart Wizard→Surface. Under the surface chart<br />
option, there are 4 styles available. Select the lower right option, which produces contours<br />
of constant cell value in a 2-D plot. Adjust scales, binning, <strong>and</strong> colors as you wish.<br />
Clearing the spreadsheet<br />
After you work each problem, it is a good idea to clear the values in all elements of the<br />
spreadsheet. You can make provision for this by choosing a cell somewhere outside the<br />
bounds of your problem, <strong>and</strong> naming it clearit. Then in the algorithm in each interior cell,<br />
just multiply the whole term for the averaging of cells by clearit. When you put a 0 in the<br />
cell named clearit <strong>and</strong> iter<strong>at</strong>e the spreadsheet, all cells will be set to 0. Then put 1 in the<br />
cell named clearit, <strong>and</strong> proceed with your next problem.<br />
Limit<strong>at</strong>ions of the relax<strong>at</strong>ion method<br />
All finite element techniques have the limit<strong>at</strong>ion th<strong>at</strong> you must bound the problem with<br />
one of the above two conditions. In any given problem, you should choose the boundary<br />
conditions to best approxim<strong>at</strong>e wh<strong>at</strong> you expect the potential distribution <strong>at</strong> the boundary<br />
to be. You can minimize the effect of the boundaries in a region of interest by simply<br />
making the entire domain of the problem (the array in EXCEL) larger, so the boundaries<br />
are further away from the region you want to study.<br />
8
Problem 1. Calcul<strong>at</strong>ion of potential distributions using the Method of Relax<strong>at</strong>ion<br />
Use the tool to explore!<br />
Use your spreadsheet engine to calcul<strong>at</strong>e the potential distribution in each of the following<br />
configur<strong>at</strong>ions:<br />
‣ Line charge of +10 -6 C/m.<br />
‣ Line dipole: two line charges ±10 -6 C/m.<br />
‣ Parallel pl<strong>at</strong>e capacitor: pl<strong>at</strong>es extending to boundary on one side, ending in<br />
center region.<br />
‣ Two charged tubes, one problem with same sign charges <strong>and</strong> one with opposite<br />
sign.<br />
‣ Line of charge between two opposite parallel conducting pl<strong>at</strong>es. Can you use<br />
the method of images discussed in class to connect the potential distribution in<br />
this case with th<strong>at</strong> of a dipole?<br />
‣ Potentials inside a closed conducting boundary. Calcul<strong>at</strong>e the <strong>potentials</strong> inside<br />
a closed hollow conductor when there is a charge loc<strong>at</strong>ed somewhere outside.<br />
This is a called a Faraday cage.<br />
‣ Follow your nose – study some distributions th<strong>at</strong> you are curious about.<br />
9
Problem 1. Calcul<strong>at</strong>ion of potential distributions using the<br />
Method of Relax<strong>at</strong>ion<br />
You have calcul<strong>at</strong>ed the potential distribution in a number of configur<strong>at</strong>ions of charges<br />
<strong>and</strong> conductors. Now you would like to plot the equi<strong>potentials</strong> – the contours along which<br />
the potential is constant. For an insol<strong>at</strong>ed point charge, for example, the equi<strong>potentials</strong> are<br />
concentric spherical surfaces centered on the charge. For a charge distributed uniformly<br />
along a straight line, the equi<strong>potentials</strong> are concentric cylinders centered on the line.<br />
Now look <strong>at</strong> your EXCEL spreadsheets of the charge configur<strong>at</strong>ions th<strong>at</strong> you calcul<strong>at</strong>ed<br />
in Problem 1. You can trace visually the contours along which the potential is constant.<br />
We have devised a tool with which you can do this with more visual effect. It is an<br />
applic<strong>at</strong>ion in LabView, which calcul<strong>at</strong>es the iso-value contours in a 2-D table of d<strong>at</strong>a <strong>and</strong><br />
plots it in a 3-D ‘mountain range’ display.<br />
Here is how to use the applic<strong>at</strong>ion.<br />
1. First you need to prepare an image of your spreadsheet th<strong>at</strong> does not contain the<br />
formulae th<strong>at</strong> gener<strong>at</strong>ed it. To do this, you cursor-select the entire spreadsheet<br />
<strong>and</strong> COPY it to the clipboard. Then you open a new spreadsheet, give it a<br />
name. Use a st<strong>and</strong>ard form<strong>at</strong>: your last name, Lab3, then a name describing the<br />
problem. For example: mcintyre-lab3-dipole.xls. Now select PASTE SPE-<br />
CIAL, select VALUES, <strong>and</strong> paste only the values from your spreadsheet into<br />
the new one.<br />
2. Now open the applic<strong>at</strong>ion ‘3dvi’ from your desktop. A query box will open,<br />
<strong>and</strong> you then select the spreadsheet th<strong>at</strong> you just cre<strong>at</strong>ed.<br />
The applic<strong>at</strong>ion will then display your d<strong>at</strong>a in either of two fashions.<br />
This mode displays the d<strong>at</strong>a in a ‘mountain range’ display, in which x <strong>and</strong> y are the coordin<strong>at</strong>es<br />
of your spreadsheet, <strong>and</strong> z is the value of the potential.
Problem 1. Calcul<strong>at</strong>ion of potential distributions using the Method of Relax<strong>at</strong>ion<br />
This mode displays the ‘mountain range <strong>and</strong> overlays equipotential contours. The contours<br />
are calcul<strong>at</strong>ed in intervals of potential difference. You can control the interval by rightclicking<br />
the mouse while in this display, <strong>and</strong> selecting a desired value for contour spacing,<br />
as shown below.<br />
11
Problem 1. Calcul<strong>at</strong>ion of potential distributions using the Method of Relax<strong>at</strong>ion<br />
Now use these tools to make equipotential plots of each of the charge/conductor configur<strong>at</strong>ions<br />
th<strong>at</strong> you calcul<strong>at</strong>ed in Problem 1.<br />
Discuss the equipotential distributions in terms of the limiting cases, close to a charge<br />
distribution or conductor, <strong>and</strong> far from it.<br />
Lines of force are continuous lines drawn parallel to the electric field vector in the<br />
space of a problem. Everywhere th<strong>at</strong> a line of force crosses an equipotential, they are always<br />
<strong>at</strong> right angles to one another. You can easily see th<strong>at</strong> this is the case from the rel<strong>at</strong>ionship<br />
E r<br />
= −∇V<br />
r<br />
On each of your 2-D spreadsheets, sketch in red pen the equi<strong>potentials</strong> <strong>and</strong> the lines of<br />
force. Note th<strong>at</strong> such a graphic construction does not work as well on the ‘mountain range’<br />
plots from the LabView applic<strong>at</strong>ion, because the z-axis is not a space dimension but r<strong>at</strong>her<br />
an enhancement for visualiz<strong>at</strong>ion.<br />
12
PROBLEM 2: Mapping potential distribution using resistive paper<br />
You have calcul<strong>at</strong>ed the potential distribution <strong>and</strong> the equi<strong>potentials</strong> produced by various<br />
distributions of charges. Now you will experimentally map the potential distribution<br />
near charge distributions <strong>and</strong> compare with your calcul<strong>at</strong>ions.<br />
You will use two equivalent but quite different techniques to map <strong>potentials</strong>. The first<br />
technique relies upon the f<strong>at</strong> th<strong>at</strong> currents in a sheet of uniformly resistive paper flow in<br />
precisely the same way th<strong>at</strong> electric flux flows.<br />
Lines of flow follow lines of force: When charged bodies of different <strong>potentials</strong> are<br />
loc<strong>at</strong>ed in a medium in which some flow of charge can occur, the electric <strong>fields</strong> in the medium<br />
will cause the charges to be transported from one body to the other. Flowing charge<br />
is called current. As current flows, the potential drops along the direction of current flow.<br />
By measuring the potential distribution in the plane of the resistive paper, the p<strong>at</strong>tern is<br />
therefore exactly th<strong>at</strong> of the equi<strong>potentials</strong> due to the charged conductors even if there were<br />
no resistive paper there <strong>and</strong> no current flowing! The resistive paper provides a convenient<br />
means to map the potential distribution in the vicinity of charged conductors. The method<br />
is however restricted to modeling two-dimensional problems, in which the conductors are<br />
extended into the third dimension (out of the paper) so th<strong>at</strong> any cross-section parallel to the<br />
paper would show the same cross-cut of conductors.<br />
To maintain the difference of potential while current flows in the resistive paper, the conducting<br />
regions must be connected to a source of electromotive force – a b<strong>at</strong>tery or d.c.<br />
power supply. The flow lines of the current follow the p<strong>at</strong>hs of the lines of force, th<strong>at</strong> is,<br />
they are also <strong>at</strong> all points perpendicular to the equipotential surfaces.<br />
If you draw a line or shape upon a sheet of resistive paper using a conductive ink, the<br />
region of ink forms an equipotential: the electric potential is the same <strong>at</strong> all points of the<br />
region covered by the ink.<br />
If you now draw a second such region on the paper, it will form a second equipotential.<br />
Now if you connect a source of potential difference V (a b<strong>at</strong>tery or d.c. power supply)<br />
with its + terminal to one conductive region <strong>and</strong> its – terminal to the other region, you will<br />
cre<strong>at</strong>e a potential difference V between the two equi<strong>potentials</strong>.<br />
EQUIPMENT<br />
You are provided with sheets of uniformly resistive paper, on which a 2-dimensional<br />
grid is marked on 1 cm spacing. This grid provides you with a convenient map with which<br />
to measure the distribution of electric potential everywhere around conductive regions.<br />
You are also provided with conductive ink (an arom<strong>at</strong>ic solvent containing a silver<br />
powder). You can apply this ink to the paper by clearing the opening in the end of the tube<br />
<strong>and</strong> then squeezing gently as you move the tip over the region you wish to define as a conductor.<br />
You are also provided with stencils with which you can make conductor p<strong>at</strong>terns.
PROBLEM 2: Mapping potential distribution using resistive paper<br />
You are equipped with a b<strong>at</strong>tery th<strong>at</strong> sustains a potential difference between two loc<strong>at</strong>ions,<br />
sourcing current to flow between its leads as necessary to sustain th<strong>at</strong> potential difference.<br />
Please do not leave the b<strong>at</strong>tery connected to the resistive load for long periods of<br />
time, as it will drain the b<strong>at</strong>tery’s capacity.<br />
You are also equipped with a direct current (d.c.) power supply, which oper<strong>at</strong>es exactly<br />
like a b<strong>at</strong>tery, but enables you to control the potential difference <strong>at</strong> any value from 0 to the<br />
maximum capacity of the supply (20 V).<br />
You are equipped with a digital multimeter (DMM), which is an electronic instrument<br />
th<strong>at</strong> can be used to measure the potential difference between two probe leads connected<br />
between its + <strong>and</strong> - input terminals. It can also be used to measure the resistance to current<br />
flow along a resistive p<strong>at</strong>h, <strong>and</strong> the current th<strong>at</strong> flows along a current p<strong>at</strong>h th<strong>at</strong> includes the<br />
meter as a series element. Your TA will instruct you in the oper<strong>at</strong>ion of the instrument.<br />
PREDICTIONS<br />
You will be studying the potential distributions produced by the charge distributions<br />
th<strong>at</strong> you studied in Problem 1. In each case you should reason wh<strong>at</strong> you expect from the<br />
physics of electrost<strong>at</strong>ics <strong>and</strong> from your calcul<strong>at</strong>ions.<br />
METHOD QUESTIONS<br />
It is useful to have an organized problem-solving str<strong>at</strong>egy. The following procedures<br />
should help with the analysis of your d<strong>at</strong>a:<br />
1. You should prepare yourself to make the p<strong>at</strong>terns of silver paint shown in<br />
Figure 4 on resistive paper. In each case, you should plan a) wh<strong>at</strong> connections<br />
of the b<strong>at</strong>tery or power supply you will make to each p<strong>at</strong>tern.<br />
2. If you want to study the potential distribution due to two identical charged cylinders,<br />
with the same sign of charge on both, how would you design a conductor<br />
geometry to simul<strong>at</strong>e it. Remember, your power supply enables you to apply<br />
a potential difference between two objects, but you want to cre<strong>at</strong>e the same<br />
potential on both cylinders, with respect to some third object (wh<strong>at</strong> should it<br />
be?)<br />
14
PROBLEM 2: Mapping potential distribution using resistive paper<br />
Figure 4. Conductor p<strong>at</strong>terns for study using resistive paper modeling.<br />
MEASUREMENT<br />
For each distribution to be studied, you should prepare the p<strong>at</strong>tern of conductors on a<br />
clean sheet of resistive paper, <strong>and</strong> then affix the paper onto the cork board. Insert a metal<br />
pin into the conducting region, <strong>and</strong> use the resistance function of the DMM to verify th<strong>at</strong><br />
the metal pin is in good contact with all regions of th<strong>at</strong> conducting region.<br />
Attach electrical leads to the metal pins between which you wish to cre<strong>at</strong>e a potential<br />
difference. Attach the other end of the leads to the + <strong>and</strong> – terminals of the b<strong>at</strong>tery or<br />
power supply. You are now gener<strong>at</strong>ing a potential difference between those points, <strong>and</strong><br />
current is now flowing through the resistive paper.<br />
Set up the DMM for its voltmeter function <strong>and</strong> <strong>at</strong>tach the – terminal of the DMM to the<br />
15
PROBLEM 2: Mapping potential distribution using resistive paper<br />
– terminal of the b<strong>at</strong>tery or power supply. If you now touch the + terminal of the DMM to<br />
any point on the resistive paper, it will read the potential difference between th<strong>at</strong> point <strong>and</strong><br />
the – terminal of the voltage source.<br />
Prepare an EXCEL spreadsheet in which you will use a region of x-y cells to record the<br />
<strong>potentials</strong> <strong>at</strong> every point of the 1 cm grid of cross-hairs on the resistive paper. Measure the<br />
potential <strong>at</strong> each point <strong>and</strong> record it in the appropri<strong>at</strong>e cell in the spreadsheet.<br />
ANALYSIS<br />
Use the same LabView tools as in Problem 1 to analyze the equi<strong>potentials</strong> <strong>and</strong> electric<br />
<strong>fields</strong> in the region of each of your conductor geometries. Compare with the calcul<strong>at</strong>ions<br />
from Problem 1 <strong>and</strong> with your predictions.<br />
For the last distribution shown in Figure 4, if you apply a potential difference between<br />
a hollow closed conductor <strong>and</strong> an external conductor, wh<strong>at</strong> is the potential distribution inside<br />
the hollow conductor? Explain it in terms of Gauss’ Law.<br />
16
PROBLEM 3: Mapping equi<strong>potentials</strong> using the Overbeck appar<strong>at</strong>us<br />
When a test charge is moved along an equipotential line or over an equipotential surface,<br />
no work is done. Since no work is done in moving a charge over an equipotential surface<br />
it follows th<strong>at</strong> there can be no component of the electric field along an equipotential surface.<br />
Thus the electric field or lines of force must be everywhere perpendicular to the equipotential<br />
surface. Equipotential lines or surfaces in an electric field are more readily loc<strong>at</strong>ed<br />
experimentally than lines of force, but if either is known the other may be constructed<br />
as shown in Figure 1. The two sets of lines must everywhere be normal to one another.<br />
EQUIPMENT<br />
The Overbeck appar<strong>at</strong>us is shown in Figure 5. It consists of a resistive film <strong>and</strong> a p<strong>at</strong>tern<br />
of conductors th<strong>at</strong> is trapped within an insul<strong>at</strong>ing plastic lamin<strong>at</strong>ion. By placing a conducting<br />
probe in contact <strong>at</strong> a loc<strong>at</strong>ion on the plastic surface, the probe <strong>at</strong>tains the potential <strong>at</strong><br />
th<strong>at</strong> point on the resistive film bene<strong>at</strong>h.<br />
Figure 5. Overbeck appar<strong>at</strong>us for mapping equi<strong>potentials</strong>.<br />
The Overbeck appar<strong>at</strong>us consists of a field-mapping board, a U-shaped probe, six field<br />
pl<strong>at</strong>es (pictured in Figure 6), <strong>and</strong> two plastic templ<strong>at</strong>es. The p<strong>at</strong>terns on the two templ<strong>at</strong>es<br />
are a composite of the p<strong>at</strong>terns on the six field pl<strong>at</strong>es. Any one of the six field pl<strong>at</strong>e p<strong>at</strong>terns<br />
can be reproduced with the templ<strong>at</strong>es. Eight similar resistors are connected in series<br />
between the two binding posts on the fieldmapping board to eight points separ<strong>at</strong>ed by the<br />
same difference of potential.<br />
You will use a galvanometer to follow equi<strong>potentials</strong> on the surface. The galvanometer<br />
is a sensitive current meter, which is instantaneously connected across two points A <strong>and</strong><br />
B in a circuit by pressing a contact button. Current flows in the galvanometer, from the<br />
contact with higher potential to the contact with lower potential. The deflection of the<br />
needle indic<strong>at</strong>es which way current flows <strong>and</strong> therefore which contact has higher potential.
Problem 3. Mapping equi<strong>potentials</strong> using the Overbeck appar<strong>at</strong>us<br />
Figure 6. Conductor p<strong>at</strong>terns to be studied using the Overbeck appar<strong>at</strong>us.<br />
METHOD QUESTIONS<br />
You will set up an electrical circuit as shown in Figure 5. You have four conductor<br />
p<strong>at</strong>terns, as shown in Figure 6. As an example, the conductor geometry in Figure 5 is a<br />
reproduction of an actual test made in designing a part for a high voltage gener<strong>at</strong>or. The<br />
solid lines are the equipotential lines <strong>and</strong> the dashed lines are the equi<strong>potentials</strong> <strong>and</strong> the<br />
lines of force for a field existing between a blade electrode <strong>and</strong> a plane.<br />
MEASUREMENT<br />
Turn the field mapping board over <strong>and</strong> notice the two metal bars. Each bar has two<br />
threaded holes. Two of these holes hold plastic-headed thumb screws with knurled lock<br />
nuts. Remove the thumb screws <strong>and</strong> center any one of the field pl<strong>at</strong>es so the holes in the<br />
pl<strong>at</strong>e coincide with holes in the<br />
metal bars. Insert a thumb screw into each hole <strong>and</strong> turn it until it touches the board below.<br />
Turn the knurled lock nut to hold the field pl<strong>at</strong>e securely in place.<br />
Binding posts marked “B<strong>at</strong>.” <strong>and</strong> “Osc.” are loc<strong>at</strong>ed on the upper side of the board. Connect<br />
the potential source to the appropri<strong>at</strong>e binding post. Fasten a sheet of 8.5 x 11-inch<br />
graph paper to the upper side of the board. Secure the paper by depressing the board on<br />
18
Problem 3. Mapping equi<strong>potentials</strong> using the Overbeck appar<strong>at</strong>us<br />
either side <strong>and</strong> slipping the paper under the four rubber bumpers. Select the design templ<strong>at</strong>e<br />
containing the field pl<strong>at</strong>e configur<strong>at</strong>ion you have chosen. Place the design templ<strong>at</strong>e<br />
on the two metal projections (templ<strong>at</strong>e guides) above the paper edge <strong>and</strong> let the two holes<br />
on top of the templ<strong>at</strong>e slide over the projections. Trace the design corresponding to the<br />
field pl<strong>at</strong>e p<strong>at</strong>tern in place on the underside of the mapping board <strong>and</strong> remove the templ<strong>at</strong>e.<br />
Place the Field Mapping Board <strong>and</strong> the U-shaped probe on a lecture table or labor<strong>at</strong>ory<br />
bench. Carefully slide the U-shaped probe onto the mapping board with the ball end facing<br />
the underside of the filed mapping board. Connect one lead of the null-point detector<br />
(galvanometer or headphones) to the U-shaped probe <strong>and</strong> one to one of the banana jacks,<br />
numbered E1 through E7. Notice the knurled knob on top of the probe (next to the spotting<br />
hole) <strong>and</strong> the screw below the probe th<strong>at</strong> acts as a support leg.<br />
To make tracings, guide the probe with one finger of one h<strong>and</strong> resting lightly on the<br />
knurled knob, <strong>and</strong> a finger on the other h<strong>and</strong> lightly touching the nut of the leg. The leg<br />
slides on the table top <strong>and</strong> stabilizes the probe. Do not apply pressure to the probe, <strong>and</strong><br />
avoid squeezing its jaws. This causes unnecessary wear on the pl<strong>at</strong>es. Although some wear<br />
is inevitable, the pl<strong>at</strong>es will last longer if proper care is taken.<br />
Using the selected banana jack, move the U-shaped probe over the paper until you obtain<br />
a null reading on the galvanometer when it is momentarily connected. The circular<br />
hole in the top arm of the probe is directly above the contact point th<strong>at</strong> touches the graphite-co<strong>at</strong>ed<br />
paper. Record the loc<strong>at</strong>ion of the equipotential point using a pencil through the<br />
hole directly on the paper. Move the probe to another null-point position <strong>and</strong> record it.<br />
Continue this procedure until you have gener<strong>at</strong>ed a series of these points across the paper.<br />
Connect the equipotential points with a smooth curve to show the equipotential line of the<br />
banana jack.<br />
Connect the detector to a new banana jack <strong>and</strong> plot its equipotential line. Repe<strong>at</strong> until<br />
equipotential lines are plotted for all banana jacks E1 through E7. Since the potential difference<br />
is the same across each similar resistor, the equipotential lines will be spaced to<br />
show an equal potential drop between successive lines.<br />
The lines of force are perpendicular to these equipotential lines <strong>at</strong> every point. Using<br />
dashed lines, draw in the lines of force of the electric field being studied. After completion,<br />
select a different field pl<strong>at</strong>e <strong>and</strong> repe<strong>at</strong> the above procedure until all electric <strong>fields</strong> from the<br />
six field pl<strong>at</strong>es (shown in Figure 6) are drawn.<br />
19
Problem 3. Mapping equi<strong>potentials</strong> using the Overbeck appar<strong>at</strong>us<br />
ANALYSIS<br />
Compare the equi<strong>potentials</strong> obtained in this manner to those for similar charge distributions<br />
with the resistive paper <strong>and</strong> the distributions calcul<strong>at</strong>ed using the Relax<strong>at</strong>ion Method.<br />
Discuss the limit<strong>at</strong>ions of each method <strong>and</strong> how they may explain differences.<br />
1. Why are the equipotential lines near conductor surfaces parallel to the surface <strong>and</strong> why<br />
perpendicular to the insul<strong>at</strong>or surface mapped?<br />
2. Is it possible for two different equipotential lines or two lines of force to cross? Explain.<br />
3. Explain, with the aid of a diagram, why lines of force must be <strong>at</strong> right angles to equipotential<br />
lines.<br />
4. Under wh<strong>at</strong> conditions will the field between the pl<strong>at</strong>es of a parallel pl<strong>at</strong>e capacitor be<br />
uniform?<br />
5. How does the electric field strength vary with distance from an isol<strong>at</strong>ed charged particle?<br />
6. Sketch the equipotential lines for an isol<strong>at</strong>ed neg<strong>at</strong>ively charged particle, spacing the<br />
lines to show equal difference of potential between lines.<br />
7. Compare the sketch in answer to Question 6 with the mapped field of the parallel-pl<strong>at</strong>e<br />
capacitor. Account for the difference.<br />
8. Show th<strong>at</strong> the electric field strength is equal to the potential gradient.<br />
9. Wh<strong>at</strong> conclusions can you draw about the field strength <strong>and</strong> the current density <strong>at</strong> various<br />
parts of sheet II, Figure 6.<br />
10. How much work is done in transferring an electrost<strong>at</strong>ic unit of charge from the one<br />
terminal to the other terminal in this experiment?<br />
11. Explain the lack of symmetry in the field of sheet I, Figure 6.<br />
12. Sketch the field p<strong>at</strong>tern of two positively charged small spheres placed a short distance<br />
from each other.<br />
13. Explain the p<strong>at</strong>tern of the field found inside a Faraday Ice Pail.<br />
20
Appendix A. The Method of Relax<strong>at</strong>ion<br />
We want to develop a method to numerically calcul<strong>at</strong>e the potential Φ(x,y) in a 2-D<br />
geometry. We will implement it in a 2-D spreadsheet of cells (m,n).<br />
Cartesian coordin<strong>at</strong>es. Let’s first develop the method for Cartesian coordin<strong>at</strong>es (x,y).<br />
We will make a Taylor expansion of the Φ near a point (x 0 , y 0 ):<br />
Φ(<br />
x<br />
+ dx,<br />
y<br />
+ dy)<br />
= Φ(<br />
x<br />
2<br />
∂Φ ∂Φ 1 ⎛ ∂ Φ<br />
, y + + + ⋅<br />
⎜<br />
0<br />
) dx dy ( dx)<br />
2<br />
∂x<br />
∂y<br />
2 ⎝ ∂x<br />
2<br />
∂ Φ<br />
+ ( dy)<br />
2<br />
∂y<br />
2<br />
2<br />
0 0<br />
0<br />
2<br />
+<br />
2<br />
∂ Φ ⎞<br />
⋅ dxdy<br />
⎟<br />
∂x∂y<br />
+K<br />
⎠<br />
We put dx = dy = ± h (h is the mesh size) <strong>and</strong> add all 8 combin<strong>at</strong>ions dx=<br />
( −h,0,<br />
h)<br />
<strong>and</strong><br />
dy = ( −h,0,<br />
h)<br />
of the neighboring cells to obtain<br />
dx=−h,0,<br />
2 2<br />
8 ⎛ ∂ Φ ∂ Φ ⎞<br />
∑Φ( x ± ± = ⋅Φ + ⋅<br />
⎜ +<br />
⎟ ⋅<br />
0<br />
dx,<br />
y0<br />
dy)<br />
8 ( x0<br />
, y0<br />
)<br />
h<br />
2 2<br />
h;<br />
dy=−h,0,<br />
h<br />
2 ⎝ ∂x<br />
∂y<br />
⎠<br />
2<br />
+K<br />
or dividing by 8<br />
(1)<br />
1<br />
8<br />
∑Φ(<br />
x<br />
0<br />
dx=−h,0,<br />
h;<br />
dy=−h,0,<br />
h<br />
± dx,<br />
y<br />
0<br />
± dy)<br />
= Φ(<br />
x , y<br />
0<br />
0<br />
) +<br />
1<br />
2<br />
2<br />
⎛ ∂ Φ I Φ ⎞<br />
⋅<br />
⎜ +<br />
⎟ ⋅ h<br />
2 2<br />
⎝ ∂x<br />
∂y<br />
⎠<br />
2<br />
+K<br />
Now we are developing a technique to calcul<strong>at</strong>e Φ in a 2-D region of space, in which Φ<br />
∂Φ<br />
is independent of the third coordin<strong>at</strong>e z: = 0<br />
∂z<br />
We can therefore equ<strong>at</strong>e the expression in brackets to the Laplacian oper<strong>at</strong>or<br />
2 2 2<br />
2 ∂ Φ ∂ Φ ∂ Φ<br />
∇ Φ = + +<br />
2 2 2<br />
∂x<br />
∂y<br />
∂z<br />
If the cell of interest (x 0 ,y 0 ) contains a net charge Q, then we must connect this oper<strong>at</strong>or<br />
to charge, using Gauss’ theorem, which rel<strong>at</strong>es the flux of a vector F r out of a closed surface<br />
to a volume integral of the divergence of the vector inside the surface:<br />
∫<br />
r r<br />
F⋅dS<br />
=<br />
∫<br />
r r<br />
∇⋅FdV<br />
We can then use Gauss’ Law to rel<strong>at</strong>e the bracketed expression in Eq. (1) to the enclosed<br />
charge:<br />
r<br />
r<br />
2<br />
∫∇<br />
ΦdV<br />
= ∫∇ ⋅ ( ∇Φ)<br />
dV = −∫<br />
∇ ⋅ EdV = −∫<br />
E ⋅ dS = −<br />
r<br />
r<br />
Rel<strong>at</strong>ing the first to the last expression, we obtain for a cubic cell of side h:<br />
r<br />
r<br />
Q<br />
ε<br />
0
∇<br />
2<br />
Φ = −<br />
ρ Q<br />
= −<br />
ε ε h<br />
0<br />
0<br />
3<br />
We can then solve Eq. (1) for the potential in the cell of interest:<br />
1<br />
8<br />
( Φ + Φ + Φ + Φ + Φ + Φ + Φ + Φ )<br />
Φ<br />
m,<br />
n<br />
=<br />
m,<br />
n−1<br />
m−1,<br />
n m−1,<br />
n−1<br />
m−1,<br />
n+<br />
1 m+<br />
1, n−1<br />
m+<br />
1, n m+<br />
1, n+<br />
1 m,<br />
n+<br />
1<br />
+<br />
1<br />
2<br />
Q<br />
h<br />
ε<br />
0<br />
Spreadsheet lapcart.xls embodies this algorithm. You will add boundary conditions,<br />
equi<strong>potentials</strong>, <strong>and</strong> charges appropri<strong>at</strong>e to your problem.<br />
Cylindrical coordin<strong>at</strong>es. In cylindrical coordin<strong>at</strong>es (r,ϑ,z), the Laplacian oper<strong>at</strong>or has<br />
the form<br />
2<br />
2 ∂ Φ 1<br />
∇ Φ =<br />
2<br />
∂r<br />
2 2<br />
∂Φ ∂ Φ ∂ Φ<br />
+ ⋅ + +<br />
2 2<br />
r ∂r<br />
∂z<br />
∂ϑ<br />
Now we will model this cylindrical space in a mesh (r, z) – one boundary of the<br />
meshed problem will be the axis of rot<strong>at</strong>ion of the cylindrical geometry. We will assume<br />
∂Φ<br />
th<strong>at</strong> Φ is independent of ϑ: = 0<br />
∂ϑ<br />
Let’s take the boundary m=0 to be the axis. Then r = mh <strong>and</strong> z = nh. We can make a<br />
Taylor expansion in the variables r,z <strong>and</strong> again obtain Eq. (1) with the substitutions<br />
x→r, y→z. The expression in brackets is however no longer the Laplacian oper<strong>at</strong>or:<br />
2 2<br />
∂ Φ ∂ Φ 2 1 ∂Φ ρ 1 ∂Φ<br />
+ = ∇ Φ − ⋅ = − − ⋅<br />
2 2<br />
∂r<br />
∂z<br />
r ∂r<br />
ε r ∂r<br />
The new term can be expressed in terms of the <strong>potentials</strong> in the neighboring cells <strong>at</strong><br />
larger <strong>and</strong> smaller radius from (r 0 ,z 0 ):<br />
1 m+ 1,<br />
n<br />
− Φ<br />
m−1,<br />
n<br />
∂Φ Φ<br />
⋅ =<br />
r ∂r<br />
( mh)( 2h)<br />
We will now solve for Φ(r 0 ,z 0 ) as before, but adding this extra term:<br />
0<br />
Φ<br />
m,<br />
n<br />
=<br />
1 Q Φ<br />
+ +<br />
2 ε h<br />
1 m+<br />
1, n m−1,<br />
n<br />
( Φ<br />
m−1,<br />
n<br />
+ Φ<br />
m,<br />
n−1<br />
+ Φ<br />
m−1,<br />
n−1<br />
+ Φ<br />
m−1,<br />
n+<br />
1<br />
+ Φ<br />
m−1,<br />
n+<br />
1<br />
+ Φ<br />
m+<br />
1, n+<br />
1<br />
+ Φ<br />
m+<br />
1, n<br />
+ Φ<br />
m,<br />
n+<br />
1<br />
)<br />
8<br />
0<br />
4m<br />
− Φ<br />
Spreadsheet lapcyl.xls embodies this algorithm. As before you will insert boundary<br />
conditions, equi<strong>potentials</strong>, <strong>and</strong> charges appropri<strong>at</strong>e to your problem.<br />
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