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


• 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


• 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


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


• 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


Demo: capacitance versus d

C QV

0Ad

Example


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


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


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


Energy Stored in a Capacitor (demo)


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


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


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




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


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


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


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


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


(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.


Dielectrics


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


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


+

+

++++

++++

––––––

–––––

E0

E E

V V

Dielectric Strength


*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


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


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)


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


Documents

Capacitance, Dielectrics, Lightning