E field disk

University Electromagnetism:
Electrif field of a charged thin disk

1. E-field of a thin disk 1
Electrical Field of a thin Disk
© Frits F.M. de Mul

2. E-field of a thin disk 2
E-field of a thin disk
Available :
A thin circular disk with radius R
and charge density s [C/m
2
]
Question :
Calculate E-field in arbitrary
points a both sides of the disk

3. E-field of a thin disk 3
E-field of a thin disk
•Analysis and symmetry
•Approach to solution
•Calculations
•Conclusions
•Appendix: angular integration

4. E-field of a thin disk 4
Analysis and Symmetry
1. Charge distribution:
s [C/m
2
]
2. Coordinate axes:
Z-axis = symm. axis,
perpend. to disk
Z
X
Y
3. Symmetry: cylinder
4. Cylinder coordinates:
r, z, j
e
r
r
e
z
z
e
j
j

5. E-field of a thin disk 5
Analysis, field build-up
R 1. XYZ-axes
Z
Y
X
2. Point P on Y-axis
P
E
i
Q
i
r
i
3. all Q
i
’s at r
i

and j
i contribute E
i
to E in P4. E
i,xy
, E
i,z
E
i,z
E
i,xy
5. expect: S E
i,xy
= 0,
to be checked !!
6. E = E
z
e
z
only !

6. E-field of a thin disk 6
Approach to solution
R
Z
2. Distributed charges
dQ
3.
redE
2
04r
dQ
pe
=
dE
e
r
r
P
1. Rings and segments
4. dQ = s.dA= s (da.)(a dj)
a
dj
j
da
5. z- component only !
dE
xy
dE
z
6. )(
4
2
0
zz
r
dQ
eedE
r·=
pe

7. E-field of a thin disk 7
Calculations (1)Calculations (1)
R
Z
dQ
dE
e
r
r
P
a
dj
j
da
dE
z
a
z
P
1. )(
4
2
0
zz
r
dQ
eedE
r·=
pe
2. dQ = s.dA= s (da.)(a dj)
22
cos
P
P
za
z
+
==· a
zree3.
4.
( )
22
0
22
0
2
0
4
...
P
P
R
P
z
za
z
za
dada
E
++
=òò
pe
js
p

8. E-field of a thin disk 8
Calculations (2)Calculations (2)
R
Z
dQ
dE
e
r
r
P
a
dj
j
da
dE
z
a
z
P
4.
( )
22
0
22
0
2
0
4
...
P
P
R
P
z
za
z
za
dada
E
++
=òò
pe
js
p
ú
ú
ú
û
ù
ê
ê
ê
ë
é
+
-=
22
0
1
2
.5
Ry
y
E
P
P
z
e
s
6. If R -> infinity :
0
2e
s
=
z
E

9. E-field of a thin disk 9
ConclusionsConclusions
Z
P
E
P
zeE
0
2e
s
=
P
for infinite disk:
field strength
independent of
distance to disk =>
homogeneous field

10. E-field of a thin disk 10
Appendix: angular integration (1)Appendix: angular integration (1)
R
Z
dQ
dE
e
r
r
P
a
dj
j
da
dE
z
a
z
P
1. )(
4
2
0
zz
r
dQ
eedE
r·=
pe
2. dQ = s.dA= s (da.)(a dj)
acos=·
zr
ee
a
a
cos
)(
Pz
fr ==
atan
P
za= a
a
d
z
da
P
2
cos
=
3.
)cos(4
cos)cos)(tan(
22
0
2
ape
ajaaas
-
-
=
P
PP
z
z
ddzz
dE

11. E-field of a thin disk 11
Appendix: angular integration (2)Appendix: angular integration (2)
R
Z
dQ
dE
e
r
r
P
a
dj
j
da
dE
z
a
z
P
3.
)cos(4
cos)cos)(tan(
22
0
2
ape
ajaaas
-
-
=
P
PP
z
z
ddzz
dE
ò
=
R
z
d
E
0
04
sin2
.4
pe
aaps
[ ]0coscos)1(
2
..
max
0
--= a
e
s
ú
ú
û
ù
ê
ê
ë
é
+
-=
22
0
1
2
..
Rz
z
P
P
e
s
0
2
..
e
s
¾¾®¾
¥®R
the endthe end