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Experimental Physics IIa - Electric potential and capacitors 1
Experimental Physics EP2a
Electricity and Wave Optics
– Capacitors –– Dielectrics –
https://bloch.physgeo.uni-leipzig.de/amr/
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Experimental Physics IIa - Electric potential and capacitors 11
23r!
Electrostatic energy
2
12r!
13r!1
12
12 rkqV =
12
122 rkqqW =
23
2
13
13 r
kqrkqV += ÷÷
ø
öççè
æ+=
23
2
13
133 r
kqrkqqW
23
32
13
31
12
21
rqkq
rqkq
rqkqUW ++==
Electrostatic potential energy of a system of charges is the work needed to bring all
particles together from an infinite separation.
332211 21
21
21 VqVqVqU ++= å=
iiiVq21
3
R
qdqdq
RkqVdqdU ==
2121
0
1 QkRqdqkRUQ
-- == òQVU 2
1=
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Experimental Physics IIa - Electric potential and capacitors 12
Capacitance
RkQV =
VQCD
º úûù
êëé º FVC
d
spkE 4= EdV = dAQkp4=
kdACp4
=
kRC =
Capacitance does not depend on the charge or the electric potential.
Capacitance does depend on the configuration (size, shape, separation, geometrical arrangement) of the conductors.
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Experimental Physics IIa - Electric potential and capacitors 13
Capacitance of a coaxial cable
1R
2R
kQrLE pp 42 =×
L
ò-=D2
1
R
R
rdEV !!ò-=2
1
2R
R
drLrkQ
÷÷ø
öççè
æ=D
1
2ln2RR
LkQV
( )1
2ln2 RRkL
VQC =D
=
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Experimental Physics IIa - Electric potential and capacitors 14
Circuits
- +
- +
1C
2C
VCQ 11 =
VCQ 22 =
VQQ
VQC 21 +==
å= iCC||
- +1C 2C
+
--
+
-+
11 CQV =
22 C
QV =
QVV
QV
C211 +
== å -- = 11iser CC
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Experimental Physics IIa - Electric potential and capacitors 15
Energy stored in a capacitor
------------
++++++++++++
-
VdqdW D= ò D=Q
VdqW0
CQdq
CqW
Q 2
0 21
== ò ( )221 VC D=
EdV =DdA
kdAC 0
4e
p==
202
1 EAdW e= AduE=
202
1 EuE e= - energy density, is applicable for any field
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Experimental Physics IIa - Electric potential and capacitors 16
Dielectrics
k0EE = dielectric constant
k0VV = k0CC =
kee 0=permittivity
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Experimental Physics IIa - Electric potential and capacitors 17
Electric displacement field
! = #$%&'
polarization density (per unit volume)
permittivity
= #$() − +)'relative permittivity; dielectric constant
- = #$' + ! = #$)' = #'displacement electric field
/-01 = 2&34567&0,9:&&
! = 1;5⃗=
= = 1(5⃗ > ?3)
; = ! > ?3@
A
5⃗
?3?3 -
'
----
++
++
+; ;
1 =?3
01
2B30 = −/! > 01
DEuE!!
×=21
- energy densitywithin dielectric
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Experimental Physics IIa - Electric potential and capacitors 18
Forces on dielectrics
------------
++++++++++++
!" >!$
!$>!"
The normal components of the electric displacement vectors D are equal on both sides of the boundary surface between different dielectrics”
The tangential components of the E-field intensities are identical on both sides of the boundary surface between two dielectrics.
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Experimental Physics IIa - Electric potential and capacitors 19
Dielectric slab in a capacitor
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Experimental Physics IIa - Electric potential and capacitors 20
To remember!
Ø Electrostatic potential energy of a system of charges is the work needed to bring all particles together from an infinite separation.
Ø A capacitor is a device for storing charge and energy.
Ø It is defined as charge per potential.
Ø For parallel connection of capacitors, the potential difference is the same for all capacitors.
Ø For serial connection of capacitors, the potential differences for each capacitor are added.
Ø A non-conducting material is called dielectric.
Ø Dielectrics weaken the electric field and, therefore,increase capacitance by factor k, which is dielectricconstant.