Capacitance, Dielectrics, Lightning
Transcript of Capacitance, Dielectrics, Lightning
PHYS 221 General Physics II
Spring 2015 Assigned Reading:
Lecture
Capacitance, Dielectrics, Lightning
18.4 – 18.6 5
Recap: PHYS 221
Phys 221 Spring 2015 Lecture 05 2
Last Lecture• Electric force is conservative• Electric potential energy• Potential
PEelect Welect qEx
V PEelect
q
E Vx
Distanceif=∆x∆x parallel to E
Many medical applications: EKG and EEG measures ∆V between electrodes
E
Capacitor
Phys 221 Spring 2015 Lecture 05 3
• Capacitor is a device that stores electrostatic potential energy.
• A capacitor consists of 2 spatially separated conductors which can be charged to +Q and -Q.
Ad
Equipotential surfaces
Capacitance
Phys 221 Spring 2015 Lecture 05 4
• The capacitance is defined as the ratio of the charge on one conductor of the capacitor to the potential difference between the conductors.
Capacitance is measured in Faradays represented by the letter F which is equal to (coulomb/volt)
QCV
Capacitance between Parallel Plates
Phys 221 Spring 2015 Lecture 05 5
Calculate the capacitance. We assume +, - charge densities on each plate with potential difference V:
- - - - -+ + + +
A
d
The electric field between the plates is uniform and is given by (see Eq. 17.17 in the textbook):
E 0
Q0A
The potential difference (or voltage) between the plates is
V Ed Qd0A
C QV
0Ad
Capacitance between Parallel Plates
Phys 221 Spring 2015 Lecture 05 6
• Calculate the capacitance. We assume +, - charge densities on each plate with potential difference V:
- - - - -+ + + +
A
d
• Need Q:
• Need V:
– from definition
– Use Gauss’ Law to find E
Q A
V E x Ed
EA Qo
E QoA
C QV
QQ0 Ad
0 Ad
C depends only on the geometry of the capacitor (A,d)
E
Demo
Phys 221 Spring 2015 Lecture 05 7
Demo: capacitance versus d
C QV
0Ad
Example
Phys 221 Spring 2015 Lecture 05 8
What is the capacitance of a parallel plate capacitor made out of two square plates 10 x10 cm2 separated by 1 mm wide gap?
d=1 mm
A=0.01 m2
If we connect 1.5V battery to its plates, how much charge could be stored in this capacitor?
C oAd
C 8.85x1012 ( C 2
N 2m2 ) 0.01m2
103 m 88.5x1012 F 88 pF
pC 132C 5.11088 12 VCQ
i>Clicker questionBB BB
Phys 221 Spring 2015 Lecture 05 9
Does the capacitance C of a capacitor increase, decrease or remain the same when the charge q on it is doubled?
(A) Decreases
(B) Remains the Same
(C) Increases
Lecture 05
Energy Stored in a CapacitorCharging capacitor requires work:
• must move charges from one plate to another through rising ΔV
ΔV
• charging a capacitor involves work: W=1/2 Q ΔV
• this work is equal to the energy stored in a capacitor
Energy stored in a capacitor: PEcap 12
QVPhys 221 Spring 2015 10
Energy stored in a capacitor
Energy stored in a capacitor: PEcap 12
QV
Alternative equations for energy stored in a capacitor
Using VCQ PEcap 12
C V 2
Using CQV PEcap
Q2
2C
Phys 221 Spring 2015 Lecture 05 11
Where is energy actually stored?
The energy per unit volume is:
u PEcap
Ad
120E2
The electric energy is actually stored in the field, proportional to E2.This equation is true in general, for any electric field.
For a parallel-plate capacitor, the capacitance isso the energy stored is
C 0Ad
PEcap 12
C V 2 120Ad
V 2 120 Ad V
d
2
Energy density
Phys 221 Spring 2015 Lecture 05 12
Energy Stored in a Capacitor (demo)
Phys 221 Spring 2015 Lecture 05 13
C P.S.
Power supply (P.S.) supplies voltage V.
can produce a big jolt.
Application: defibrillator A fibrillating heart is one in which the cardiac muscles go into uncontrolled twitching and quivering. A defibrillator is used to stop this.
PEcap 12
CV 2
i>Clicker questionBB BB
A parallel plate capacitor given a charge q. The plates are then pulled a small distance further apart. What happens to the charge q on each plate
of the capacitor?
1) Increases 2) Constant 3) Decreases
Remember charge is real/physical. There is no place for the charges to go.
+q-q+
+
+
-
-
- dpullpull
Phys 221 Spring 2015 Lecture 05 14
A parallel plate capacitor given a charge q. The plates are then pulled a small distance further apart. Which of the following apply to the situation after the plates have been moved?
1)The capacitance increases True False
2)The electric field increases True False
3)The voltage between the plates increases True False
Another Question +q-q+
+
+
+
-
-
-
-
dpullpull
C = ε/d C decreases!
E= Q/(ε0A) Constant
V= Ed
Phys 221 Spring 2015 Lecture 05 15
i>Clicker questionBB BB
Phys 221 Spring 2015 Lecture 05 16
A parallel plate capacitor given a charge q. The plates are then pulled a small distance further apart. Which of the following apply to the situation after the plates have been moved?
U= ½ QV Q constant, V increased
The energy stored in the capacitor
A) increases B) constant C) decreases
Plates are attracted to each other, you must pull them apart, so the potential energy of the plates increases.
+q-q+
+
+
+
-
-
-
-
dpullpull
Equivalent Capacitors
Phys 221 Spring 2015 Lecture 05 17
VC1 C2
a
b
Q2Q1
C3
Q3
-QC
a
b
+Q
VC
a
b
Q
a
b
+Q
-Q
+Q
+Q-Q
-Q
V
Capacitors in Parallel
Phys 221 Spring 2015 Lecture 05 18
Capacitors connected in parallel can be replaced with an equivalent capacitor that has the same total charge, Q and the same applied potential V.
Parallel Combination: 1 2
1 2
Q QVC C
2
2 11
CQ QC
VC
1C
2
a
b
Q2Q1
C V
a
b
Q
Equivalent Capacitor:1 2 1 1 2
1
( )Q Q Q C CQCV V C V
1 2C C C
Capacitors in Series
Phys 221 Spring 2015 Lecture 05 19
Capacitors are in series when a potential difference that is applied across their combination is the sum of the resulting potential differences across each capacitor.
-Q
Ca b +Q
C1 C2
a b+Q -Q-Q +Q
• The charge on C1 must be the same as the charge on C2since applying a potential difference across ab cannot produce a net charge on the inner plates of C1 and C2 .
a bQVC
RHS:
1 21 2
abQ QV V VC C
LHS:1 2
1 1 1C C C
Example 1
Phys 221 Spring 2015 Lecture 05 20
C1 C2
C3
a
bC
a b
• How do we start?? Realize to first group C1 and C2
• Recognize C3 is in series with the parallel combination on C1 and C2. i.e.
3 1 2
1 1 1C C C C
3 1 2
1 2 3
( )C C CCC C C
Question
Phys 221 Spring 2015 Lecture 05 21
(a) V= (2/3)V0 (b) V = V0 (c) V = (3/2)V0
What is the relationship between V0 and V ?
-Q
d
(Area A)
V0
+Q
-Q
conductor
(Area A)
V
+Qd/3
d/3
Dielectrics
• Empirical observation: Inserting a non-conducting material between the plates of a capacitor changes the VALUE of the capacitance.
• Definition:The dielectric constant of a material is the ratio of the capacitance when filled with the dielectric to that without it.
Phys 221 Spring 2015 Lecture 05 22
Dielectrics
Phys 221 Spring 2015 Lecture 05 23
CC0
k values are always > 1 (e.g., glass = 5.6; water = 78) (the water had better be very pure and non-conducting)They INCREASE the capacitance of a capacitor (good, since it is hard to make “big” capacitors)They permit more energy to be stored on a given capacitor than otherwise with vacuum (i.e., air)
• Empirical observation: Inserting a non-conducting material between the plates of a capacitor changes the VALUE of the capacitance.
• Definition: The dielectric constant of a material is the ratio of the capacitance when filled with the dielectric to that without it.
Effect of Dielectrics on the Electric Field
Phys 221 Spring 2015 Lecture 05 24
real charge real charge
E’E ' electric field due to the dipoles; it opposes
E 0
E E0
E 'resultant field
applied field
++++++++
---------
Demo: Dielectrics & Capacitance
Phys 221 Spring 2015 Lecture 05 25
+
+
++++
++++
––––––
–––––
E0
E E
V V
Dielectric Strength
Phys 221 Spring 2015 Lecture 05 26
*The maximum value of the electric field that a dielectric material can tolerate before breaking down.
* It limits the voltage that can be applied to a capacitor. The maximum voltage is called the breakdown potential.
Dielectric Properties
Phys 221 Spring 2015 Lecture 05 27
material κ dielectric strength(kV/mm)
air (1 atm) 1.00059 3paraffin 2.1–2.5 10glass (Pyrex) 5.6 14mica 5.4 10–100polystyrene 2.55 24H2O (200 C) 80 ?titania ceramic 130strontium titanate
310 8
Dielectric Combinations
Phys 221 Spring 2015 Lecture 05 28
d/2
d/2
1
1
2
2d d
A
A
A/2 A/2
Energy stored in electric field
Electric field energy density:
u 120E2
Electric field energy is proportional to E2
This equation is true in general, for any electric fieldThis formula can be easily verified for a plane capacitor
• Energy in a capacitor is stored in the electric field U=PE
• Since we know the energy of a capacitor, we can calculate energy stored in electric field and the energy density (U/Volume)
Phys 221 Spring 2015 Lecture 05 29
Energy stored in electric field
221 VCUcap
20
21 Ed
dAUcap
202
1 AdEUcap Volume insidea capacitor
Electric field energy density:2
021 E
AdU
u cap
Energy in a capacitor is stored in the electric field
Since we know the energy of a capacitor, we can calculate energy stored in electric field
Phys 221 Spring 2015 Lecture 05 30