Electric field calculations
We practice how to calculate the electric field created by charge
distributed over space
Basic idea: apply the superposition principle of electric field
We go from the fundamental principle E(r)= E1(r) + E2(r)
to fully exploit E ( r )  E 1 ( r )  E 2 ( r )  E 3 ( r )  ...
Electric field on the axis of a ring of charge
homogeneously
charged ring
Total charge Q
Radius a
Q
Line charge density  
r 
x a
2
2
 co n st .
P

dEx
2 a
dEy
Q
E ( x , y  0)  e x  dE x
with
dE x 
cos  dQ
4  0 r
2

x
dQ
4  0 r
3
E ( x, y  0)  e x 
0

x
dQ 
x
4  0 r
Q
4  r
3
ex
3
Brief discussion of limiting case x>>a
r 
x a
2
2
 co n st .
r 

Ring structure becomes less “visible” from distant point P
E-field of a point charge
E 
x
Q
4  0 r
3
ex 
x
4  0
Q
x
2
a
2

3/2
ex 
x
Q
4  0 x
x>>a
3
ex 
1
Q
4  0 x
2
ex
x a
2
2
 x
Electric field on the axis of uniformly charged plate
homogeneous charge per plate area  
Q
R
2
We consider the plate as a collection of rings
x
we take advantage of our ring solution E 
4  0
Q
x
2
a
2

3/2
ex
da
a
Every ring of radius 0<a<R contributes with dQ   2  ada
R
E  ex 
0
2   ada
x
4  0
x
2
a
2

 ex
3/2
x
R

2 0
0
2
E  ex
x
2 0
2
x R

x
ada
x
x R
2
dz
z
2
2
a
2

3/2
2
x  1
 ex
 
2 0  x
x   1 
 ex


2 0  z 
x
with
z  x  a  zdz  ada
2
2
2


 
1
  ex
2
2

2

x R 
0

1
1
R / x
2


1

Brief discussion of limiting case R

 
E  ex
1
2 0 

1
R / x
2


1 



 ex

2 0
result independent of x
R
E 

2 0
E 
field direction everywhere perpendicular to the sheet
homogeneous field

2 0
we use this limiting case to derive
the electric field of two oppositely charged infinite sheets
sheet 2
E=0
above sheet2
E2


E=/0
E1
between the sheets
E2
E1
sheet 1
E2
E1
E=0
below sheet1
Demonstration
For a nice intuitive approach to an understanding of the Wimshurst machine watch also
MIT Physics Demo -- The Wimshurst Machine
http://www.youtube.com/watch?v=Zilvl9tS0Og