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Unformatted text preview: PHYSICS 222
Introduction to Classical Physics II
Prof. Ruslan Prozorov
Iowa State University
Fall 2011 LECTURE 5
Capacitance and capacitors.
Dielectrics. Electric field energy. keep charges apart and you get
capacitance PHYS222  Prof. Ruslan Prozorov  Iowa State University 31 August 2011 2 capacitance remember from previous lecture: V d
0
Q
A
Capacitance is defined as: C  only geometry! 0
V
d
PHYS222  Prof. Ruslan Prozorov  Iowa State University 31 August 2011 3 capacitor
With the same
potential difference store more charge if
capacitance is larger A
Q 0 V C V
d PHYS222  Prof. Ruslan Prozorov  Iowa State University 31 August 2011 4 capacitors of different geometry: sphere
use the definition: Q
A
C 0
V
d look at the Gaussian surface between the shells.
From Gauss’s law, electric field between the
shells is:
1 Q E r 4 0 r 2 Electric potential between the shells is: Q
dr
Q 1 1
V Edr r 2 4 0 ra rb 4 0 ra ra
rb rb PHYS222  Prof. Ruslan Prozorov  Iowa State University so, the capacitance is: ra rb Q
C 4 0 V rb ra 31 August 2011 5 cylindrical capacitor 1
E
2 0 r rb dr
Q
V Edr r 2 0 L ln ra
2 0 r
r
rb a so, the capacitance is: rb a 2 0
Q
C V ln rb
ra PHYS222  Prof. Ruslan Prozorov  Iowa State University 31 August 2011 6 d the microphone
Current sensor
Current sensor
Battery
Battery (fixed
potential
difference) Moveable plate Fixed plate Sound waves incident pressure oscillations oscillating plate separation d
oscillating capacitance (C ~ 1/d)
oscillating charge on plate
(Q ~ C) oscillating current in wire (I =
dQ/dt) oscillating electrical signal in sensor
PHYS222  Prof. Ruslan Prozorov  Iowa State University 31 August 2011 7 real capacitors PHYS222  Prof. Ruslan Prozorov  Iowa State University 31 August 2011 8 very large capacitors capacitors are different from batteries: capacitors store charge, whereas
battery maintains voltage only up to a certain current (due to internal
resistance). The charge can be discharged very quickly reasulting in huge
currents. The array above right can store up to 50 MegaJouls (~ 10 kg TNT) PHYS222  Prof. Ruslan Prozorov  Iowa State University 31 August 2011 9 how to increase capacitance?
what if we put a dielectric in
between the plates? The applied
voltage will still be below the electric
breakthrough. Dielectric polarizes,
so the effective electric field
reduces.
Now, let’s charge a capacitor, so it
has charge Q on each plate. Then,
we insert a dielectric. The charge is
conserved, so:
dielectric constant Cempty Q
Q Cdielectric Vempty
Vdielectric PHYS222  Prof. Ruslan Prozorov  Iowa State University Cdielectric
K
1
Cempty
31 August 2011 10 molecular models PHYS222  Prof. Ruslan Prozorov  Iowa State University 31 August 2011 11 polarization and electric field lines PHYS222  Prof. Ruslan Prozorov  Iowa State University 31 August 2011 12 dielectric constants note that there are no perfect insulators. These is always some leakage electric
current, so that the capacitor slowly discharges internally! Only perfect vacuum
would work. PHYS222  Prof. Ruslan Prozorov  Iowa State University 31 August 2011 13 Gauss’s Law in dielectrics
electric
field is
reduced
due to
induced
surface
charge PHYS222  Prof. Ruslan Prozorov  Iowa State University 31 August 2011 14 in terms of electric field and surface charge
The polarization produces an
induced charge on the surfaces
of the dielectric that are in
contact with the capacitor
plates. E E 0 E polar i 0 E0 E K
K 0 i K PHYS222  Prof. Ruslan Prozorov  Iowa State University 1
i 1 K 31 August 2011 15 dielectric breakdown PHYS222  Prof. Ruslan Prozorov  Iowa State University 31 August 2011 16 permittivity
permittivity is defined as: K 0
for example, electric field of a point charge
inside a dielectric will be obtained by replacing 1 Q
E r 4 0 r 2
with 0 1 Q
E r 4 r 2
is called permittivity of vacuum PHYS222  Prof. Ruslan Prozorov  Iowa State University 31 August 2011 17 capacitors in series E0
d E1 , V1 E1 K1
2 Q Q
1 1
1 V V1 V2 E0
d
C1 C2
C C1 C2
E2 , V2 E2
K2
2 PHYS222  Prof. Ruslan Prozorov  Iowa State University 31 August 2011 18 capacitors in parallel Q1 Q2 Q
V V1 V2 C C1 C2
C1 C2 C PHYS222  Prof. Ruslan Prozorov  Iowa State University 31 August 2011 19 calculating capacitance
do it in parts 1 1 3 1 1 12 6 12 4 C
C 4 F C 3 11 4 18 F PHYS222  Prof. Ruslan Prozorov  Iowa State University 1 1 3 1 1 18 9 18 6 C
C 6 F
31 August 2011 20 energy stored in capacitors
The work required to bring small charge dQ from one plate to another is: Q
dW VdQ dQ
C
so, the total work to charge a capacitor from Q=0 to some Q is: W Q 0 Q
Q2
dQ C
2C this work is (ideally) converted into a potential energy (similar to
when we lift a weight to some heights). This energy is now stored
in a capacitor. Q 2 QV CV 2
U 2C
2
2
PHYS222  Prof. Ruslan Prozorov  Iowa State University 31 August 2011 21 size of a capacitor to store 50 MJ
Suppose the capacitor operates at 10 V 2U 2 50 106
C 2 106 F
V
102 CV 2
U
2 A
C d is it large? take our “best” dielectric from the earlier tables –
strontium titanate with K = 310 and spacing
between the plates of 1 mm. Then the area will
be: 106 103
A 3.2 m 2 310
Cd charge? Q CV 107 C without dielectric we would need 1000 m2 area
PHYS222  Prof. Ruslan Prozorov  Iowa State University 31 August 2011 22 energy of an electric field
The work done to charge a capacitor was done against an electric field
d dW EdQ
0 so, we may think of electric field energy. It is usually expressed as energy density: U
CV 2 1 u E2 Volume
2 Ad u 2
A C
, V Ed d PHYS222  Prof. Ruslan Prozorov  Iowa State University 31 August 2011 23 ...
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This note was uploaded on 11/14/2011 for the course PHYS 5863005 taught by Professor Meyer during the Fall '09 term at Iowa State.
 Fall '09
 MEYER

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