Experimental Physics IIa - Electric potential and capacitors 1

Experimental Physics EP2a

Electricity and Wave Optics

– Capacitors –– Dielectrics –

https://bloch.physgeo.uni-leipzig.de/amr/

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=

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.

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

=

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

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

Experimental Physics IIa - Electric potential and capacitors 16

Dielectrics

k0EE = dielectric constant

k0VV = k0CC =

kee 0=permittivity

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

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.

Experimental Physics IIa - Electric potential and capacitors 19

Dielectric slab in a capacitor

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.