Continuous charges – Electric field, Gauss’s law · Application of the Gauss’s law:...
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Experimental Physics IIa - Continuous charges 1 Experimental Physics EP2 Electricity and Wave Optics – Continuous charges – Electric field, Gauss’s law https://bloch.physgeo.uni - leipzig.de/amr/
Transcript of Continuous charges – Electric field, Gauss’s law · Application of the Gauss’s law:...
Experimental Physics IIa - Continuous charges 1
Experimental Physics EP2
Electricity and Wave Optics
– Continuous charges –Electric field, Gauss’s law
https://bloch.physgeo.uni-leipzig.de/amr/
Experimental Physics IIa - Continuous charges 2
Electric field
2,122,1
212,1 r̂
rqkqF =
!
1 22,1r!
0q+
0qFE!
!º úû
ùêëéCN
+
+
-
+
rrkqE ˆ2=
!
At any point the total electric field is equal to vector sum of the electric fields due to the individual charges.
å=i
ii
i rrkqE ˆ2
!qD
rrdqkE ˆ2ò=
!
Experimental Physics IIa - Continuous charges 3
Selected examples
Q2xdqkdE =
2xdx
kdEl
=
÷øö
çèæ -+
-== ò+
aalk
xdx
kEal
a
112 ll
)( alakQE+
=
2akQE =la >> - the rod can be considered as a point-like charge
Experimental Physics - Mechanics - Gravity 4
Interaction with ring
dmsGMdF 2= acos2 dms
GMdFx -=
aa coscos2ring
2 sGMm
dms
GMFx =-= ò
integration over the ring
222 xas +=sx
=acos
( ) 2/322
ring2ring
xa
xGMmsx
sGMm
Fx+
==
R
r
dm
s
axa
M
Fd!
xdF
ydF
ydF
Experimental Physics IIa - Continuous charges 5
Selected examples
qcosdEdEx =2rdqkdE =
rx
=qcos
( )òò +== 2/3222 ax
xdqkrx
rdqkEx
Q
( ) 2/322 ax
kxQEx+
=
Experimental Physics IIa - Continuous charges 6
Selected examples
s
( ) drrxr
kxdEx sp22/322´
+=
( )ò +
+=
R
xxr
xrdkxE
02/322
22 )(sp
( ) R
x xrkxE0
2/1222 -+-= sp
÷÷ø
öççè
æ
+-=
222
Rx
xxxkEx sp xR >> spkEx 2=
Experimental Physics IIa - Continuous charges 7
Electric flux
EAE =F
Electric flux is equal to number of electric lines passing through a given surface area.
úû
ùêë
é ×CmN 2
acosEAE =F
n̂
a
ò=Fsurface
E AdE!!
Ad!
E!
Experimental Physics IIa - Continuous charges 8
Electric flux
dAEAdE nE ò ò==F!!
- net flux through a closed surface
21 El-=F
22 El=F
0=Fnet
Experimental Physics IIa - Continuous charges 9
Gauss’s law
kqrrkq
E pp 44 22 ==F
ò òò ===F dAEEdAAdEsurface
E
!!
Electric flux is equal to number of electric lines passing through a given surface area.
The net flux through any closed surface enclosing charge q is equal to 4pkq.
The net flux through any closed surface enclosing no charge zero.
inE kqAdE p4==F ò!!
Divergence theorem
Experimental Physics IIa - Continuous charges 10
Application of the Gauss’s law: Insulators
inE kqAdE p4==F ò!!
kQrEdAEEdA pp 44 2òò ===
2rkQE in=
2rkQE =
3
3
arQQin = r
akQ
E 3=
Experimental Physics - Mechanics - Gravity 11
Gravitational force: extended objects
ïî
ïí
ì
<=
>-=
Rr0
Rrr̂2
g
g
Fr
GmMF!
!
ïïî
ïïí
ì
<-=
>-=
Rrr̂
Rrr̂
3
2
RGmMrF
rGmMF
g
g
!
!
r! r!
r!j
Experimental Physics IIa - Continuous charges 12
Application of the Gauss’s law: Insulators
+
EAE 2=F
spkAE 4=F
spkE 2=
Experimental Physics IIa - Continuous charges 13
Earnshaw's theorem
Any equilibrium configuration of point-like charges interacting solely via electrostatic interactions is unstable.
The theorem is applicable for any inverse-square law forces, including gravitational and magnetic.
-
+
+!
!
"#!$%
$&
'&$( ≠ *
Experimental Physics IIa - Continuous charges 14
Earnshaw's theorem
-
-
-
-
-
-
-! = #
Electric Field Lines
shielding the electric field – the Faraday cage
Experimental Physics IIa - Continuous charges 15
Experimental Physics IIa - Continuous charges 16
Application of the Gauss’s law: Conductors
Electrostatic equilibrium:
Ø Electric field within a conducting body is zero everywhere.
Ø Electric field outside a conductor is perpendicular to its surface and is 4pks.
Ø Electric charges within a conductor are locates on its surface.
Ø If surface is irregular, then electric field is greatest where the curvature is highest.
Experimental Physics IIa - Continuous charges 17
Application of the Gauss’s law: Conductors
01 =E
222rkQE =
24)2(
rQQkE +-
=
03 =EQ2-
Q
Charge distribution?
Experimental Physics IIa - Continuous charges 18
To remember!
Ø At any point the total electric field is equal to the vector sum of electric fields due to individual charges.
Ø Electric flux is equal to number of electric field lines penetrating given surface - the scalar product of electric field and the area element (pointing along the normal to the surface).
Ø The net flux through any closed surface enclosing charge q is equal to 4pkq.
Ø The net flux through any closed surface enclosing no charge is zero.
Ø At electrostatic equilibrium there is no any macroscopic charge transport.
Ø Electric field within a conductor is zero, all chargeis located at the surface.
