Laplace’s Equation and
Field/Potential Estimation
Alan Murray
Estimation?





Often we can “spot” some
equipotentials (electrodes)
Can we infer others from these?
Can we create field lines and
strengths from the equipotentials?
How accurate is this?
How useful is this?!
Alan Murray – University of Edinburgh
Estimation – the principle
Given some equipotentials
Can we infer the rest?
And then the field lines?
5V
1V
2V
3V
0V
E-Field lines
4V
Alan Murray – University of Edinburgh
General idea
Let’s say we know the
height of b, c, d and
e’s bums.
e
Can we estimate the
height of a’s bum?
b
a
c
The “estimate” is
100% accurate
d
Ha’s bum =
¼(Hb’s bum
+ Hc’s bum
+ Hd’s bum + He’s bum )
Alan Murray – University of Edinburgh
General idea
Let’s say we know the
height of b, c, d and
e’s bums.
Can we estimate the
height of a’s bum?
The “estimate” is
100% accurate
Ha’s bum =
¼(Hb’s bum
+ Hc’s bum
+ Hd’s bum + He’s bum )
Alan Murray – University of Edinburgh
General idea
Let’s say we know the
height of a, c, d and
e’s bums.
Can we estimate the
height of a’s bum?
Now the estimate is
not 100% accurate
How could it be made
more accurate?
Ha’s bum =
¼(Hb’s bum
+ Hc’s bum
+ Hd’s bum + He’s bum )
Alan Murray – University of Edinburgh
Now for some Maths …
E    V , D   E ,  .D  

 .V    V   ,  V   ,

  x, y , z  
 
2
2
2
 V 

2
 in full,  V ( x , y , z )  




This is Poisson’s Equation, which describes the variation of
V(x,y,z) in space when a charge density ρ(x,y,z) is present.
 V 0
2
in full,  V ( x , y , z)  0 
2
This is Laplace’s Equation, which describes the variation
of V(x,y,z) in space when no charge is present.
Alan Murray – University of Edinburgh
Method …
V4
V1
Divide the region of
interest into a grid …
finer grid means more
accuracy and more
calculation!
… then approximate 2V
V3
Alan Murray – University of Edinburgh
V2
Look at neighbouring grid points
2
d 2Vx d Vy
In 2D,  V 
 2 0
2
dx
dy
2
2
2
d Vx d Vy d  dV



dx 2 dy 2 dx  dx
Approximate
dV
dx
dV
dx
dV
dy
dV
dy

a

b

c

d
V
x
V
x
V
y
V
y
dV dV
,
dx
dy
V0  V1


V2  V0


V0  V3

b
c

d
 d  dV 
 

 dy  dy 

a
V4
(d)
V1
(a)
V0 ?
(c)
V3

V4  V0

Alan Murray – University of Edinburgh
(b)
V2
Look at neighbouring grid points
2
d 2Vx d Vy
In 2D,  V 
 2 0
2
dx
dy
2
2
2
d Vx d Vy d  dV



dx 2 dy 2 dx  dx
Approximate
d  dV 

 
dx  dx  0
d  dV

dx  dx
dV
dx
d  dV 

dy  dy  0

b
dV
dy


d

 d  dV 
 

 dy  dy 
d  dV 

,



 dy  dy 
dV
dx
(d)
V1
(a)
c
V4  V0 V0  V3


 

Alan Murray – University of Edinburgh
V0 ?
(c)
V2  V0 V0  V1





a
dV
dy
V4
V3

(b)
V2
Look at neighbouring grid points
d 2V
So, tidying things up,
dx 2
0
V2  V1  2V0 d 2V

,
2

dy 2

0
V4  V3  2V0
2
2
d 2Vx d Vy V2  V1  2V0 V4  V3  2V0 V1  V2  V3  V4  4V0
 V
 2 


2
2
2
dx
dy


2
2
Prof. Laplace says that  V  0, so:2
V1  V2  V3  V4  4V0
0
2
V1  V2  V3  V4
V0 
in 2D
4
V1  V2  V3  V4  V5  V6
V0 
in 3D
6
Alan Murray – University of Edinburgh
Look at neighbouring grid points
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V0 is the average of its 4 neighbours in 2D
V0 is the average of its 6 neighbours in 3D
Intuitively “correct” … and supported by theory.
Is it useful?
Often we have some equipotentials
• Fixed by conducting plates/electrodes/etc.


Use Laplace (approximately) to calculate the
other potentials roughly
Repeat until accuracy is achieved
• Or you go blue in the face!
Alan Murray – University of Edinburgh
Worked Example : Coarse Grid,
First Guesstimates
100V
40V
40V
40V
50V
50V
0V
-40V
0V
Alan Murray – University of Edinburgh
-100V
Worked Example : Coarse Grid,
Iterate …
100V
50V
=48V
=38V
40V
(100+50+40+0)/4
(100+48-40+40)/4
0V
-40V
40V
50V
0V
Alan Murray – University of Edinburgh
-100V
Worked Example : Coarse Grid,
Iterate …
100V
=38V
=48V
47V
50V
50V
-4V
-13V
Alan Murray – University of Edinburgh
-11V
-100V
Worked Example : Coarse Grid,
Iterate …
100V
=38V
=48V
47V
50V
50V
-4V
-13V
Alan Murray – University of Edinburgh
-11V
-100V
Iteration : Guidelines

Do not spent ages making initial
estimates
• All values will converge
• Sensible values will converge faster

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Always use the most recent values of
neighbouring potentials
If the accuracy required is “to the
nearest volt”, stop when the last
change to every value is <1V
Alan Murray – University of Edinburgh
Worked Example : Coarse Grid,
Final Values
=90°
100V
75V equipotential
48V
50V
45V
45V equipotential
-5V
-16V
E
48V
50V
0V equipotential
-5V
Alan Murray – University of Edinburgh
-100V
Can we calculate the electric field ..
approximate (-V/x, -V/y)?
100V
E
48V
50V
Ey = -[100-(-5)]/2
45V
48V
Ex = -[45-50]/2
-5V
-16V
 = 0.05
-5V
 E
Alan Murray – University of Edinburgh
50V
= (50, -1050)V/m
-100V
Finite Differences and Laplace


This is clearly tedious
Not with a computer it’s not
• Can make the grid arbitrarily accurate
• Can calculate for ages to achieve high
accuracy

Can include not-zero charge density
• and non-zero conductivity
• and AC signals

This is the technique used when a “closed
form” solution is not feasible
http://www.see.ed.ac.uk/~afm/teaching/em3/
Examples : MS Excel
Alan Murray – University of Edinburgh
Finite Differences and Laplace

Examples?
• Field lines in a MOSFET
• Shape and form of “plates” in a cathode ray
tube/Scanning Electron Microscope etc
• Fields, resistance and inductance of rails on
the London Underground
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This was done with help from Brian Flynn and me, by
one of our own M.Eng. Students!
Aiming to detect cracks in the line
Practically important or esoteric Electromagnetics?
The student was able to solve Maxwell#3 “by hand”
to calculate inductance.
Alan Murray – University of Edinburgh
Example : Train Tracks
Note adaptive mesh – finer where necessary
Alan Murray – University of Edinburgh
Example : Train Tracks
colour indicates electric field strength E
Alan Murray – University of Edinburgh
Example : Train Tracks
colour indicates current density J
Alan Murray – University of Edinburgh