Electric Field - ejwoo.com Hee University Korea ... Gauss’ law relates the net fluxϕof an...

Electric Field

Eung Je Woo

Department of Biomedical Engineering

Impedance Imaging Research Center (IIRC)

Kyung Hee University

Korea

[email protected]

mailto:[email protected]

2

References

• David Halliday, Robert Resnick, and Jearl Walker, Fundamentals of Physics, Wiley, 2014

• Matthew Sadiku, Elements of Electromagnetics, Oxford, 2014

• Jaakko Malmivuo and Robert Plonsey, Bioelectromagnetism, http://www.bem.fi/book/

http://www.bem.fi/book/

3

Fundamental Quantity

• Length (dimension or size) in meter (m)

• Time (sequence or duration or interval) in second (s)

• Mass in kilogram (kg)

• Charge in coulomb (C)

• Temperature in kelvin (K)

• Amount of substance in mole (mol)

• Luminous intensity in candela (cd)

• Mechanics

• Electromagnetics

• Optics

• Thermodynamics

• Chemistry

4

Charged Particle and Charge Density

• Free electron and hole are mobile

• Unbounded ion and molecule are mobile

• Bounded atom and molecule are immobile but may vibrate

• Polar molecule has no net charge but dipole moment and may rotate

• Mass (conserved)

• Charge (quantized and conserved)

• Size

• Position

, , ,m Q dr

5

Electric Materials

• Insulator• Negligible amount of mobile charged particles

• Rubber, plastic, glass, pure water

• A dielectric is an insulator that can be polarized by an external electric field

• Semiconductor• Intermediate between conductors and semiconductors

• Silicon, germanium

• Conductor• Significant amount of mobile charged particles

• Metals with free electrons, electrolyte with ions

• Superconductor• (Almost) Perfect conductors

6

Coulomb’s Law

(a) The two glass rods were each rubbed with a silk cloth and one was suspended by thread. When they are close to each other, they repel each other.

(b) The plastic rod was rubbed with fur. When brought close to the glass rod, the rods attract each other.

Ch21, HRW, Fundamentals of Physics, Wiley

7

Coulomb’s Law


(a) Two charged rods of the same sign repel each other.

(b) Two charged rods of opposite signs attract each other. Plus signs indicate a positive net charge, and minus signs indicate a negative net charge.

8

Coulomb’s Law


where ε0 = 8.85 × 10-12 C2/N.m2 is the permittivity constant. The ratio 1/4πε0 is often replaced with the electrostatic constant (or Coulomb constant) k=8.99×109 N.m2/C2. Thus k = 1/4πε0 .

Multiple Forces: If multiple electrostatic forces act on a particle, the net force is the vector sum (not scalar sum) of the individual forces.

9

Coulomb’s Law

)8.4(|rr|

)rr(Q

4

Q

|rr|4

)rr(QQ

|rr|4

)rr(QQ

|rr|4

)rr(QQF

N

1k3

k

kk

0

3

N0

NN

3

20

22

3

10

11

Q

Q1

Q2

QN

Qk

rr1F

rr 2F

rrkF

rr NF

Source Points

Field Point

10

Electric Field


• How does particle 1 “know” of the presence of particle 2? • That is, since the particles do not touch, how can particle 2 push on particle 1?• How can there be such an action at a distance?

• Particle 2 sets up an electric field at all points in the surrounding space, even if the space is a vacuum.

• If we place particle 1 at any point in that space, particle 1 knows of the presence of particle 2 because it is affected by the electric field particle 2 has already set up at that point.

• Thus, particle 2 pushes on particle 1 not by touching it as you would push on a coffee mug by making contact. Instead, particle 2 pushes by means of the electric field it has set up.

11

Electric Field

)12.4(|rr|

)rr(Q

4

1

|rr|4

)rr(Q

|rr|4

)rr(Q

|rr|4

)rr(QE

N

1k3

k

kk

0

3

N0

NN

3

20

22

3

10

11

Q=1C

Q1

Q2

QN

Qk

rr1E

rr2E

rrkE

rr NE

Source Points

Field Point

12

Electric Field

)c14.4(charge)olumev(aR4

dvE

)b14.4()chargeurfaces(aR4

dSE

)a14.4(charge)line(aR4

dlE

)c13.4()chargevolume(dvQdvdQ

)b13.4()chargesurface(dSQdSdQ

)a13.4()charge(linedLQdLdQ

R20

v

R20

S

R20

L

v vv

S SS

L LL

13

Point Charge in Electric Field


If a particle with charge q is placed in an external electric field E, an electrostatic forceF acts on the particle:

Ink-jet printer. Drops shot from generator Greceive a charge in charging unit C. An inputsignal from a computer controls the chargeand thus the effect of field E on where thedrop lands on the paper.

14

Electric Flux


Electric field vectors and field lines pierce an imaginary, spherical Gaussian surface that encloses a particle with charge +Q.

Now the enclosed particle has charge +2Q.

Can you tell what the enclosed charge is now?Answer: -0.5Q

15

Electric Flux


Now we can find the total flux by integrating the dot product over the full surface.The total flux through a surface is given by

The net flux through a closed surface (which is used in Gauss’ law) is given by

where the integration is carried out over the entire surface.

16

Flux through a Surface

A S

17

Flux through a Surface

적분감싸서에서

적분에서

S Surface :

)14.3(SdA

S Surface :

)13.3(SdAdSaAdScos|A|

S

SS nS

Closed Surface

18

Gauss’ Law


Gauss’ law relates the net flux ϕ of an electric field through a closed surface (a Gaussian surface) to the net charge qenc that is enclosed by that surface. It tells us that

We can also write Gauss’ law as

Two charges, equal in magnitude but opposite in sign, and the field lines that represent their net electric field. Four Gaussian surfaces are shown in cross section.

19

Electric Flux Density

Gauss’s Law states that the total electric flux (density) through any closed surfaceis equal to the total charge enclosed by that surface.

)38.4(aR4

dvD

)37.4()eargChofSheetInfinite(a2

D

)36.4(]C[SdD

)35.4(][C/m ED

R2v

ns

20

(4.43)D

)42.4(dvDdvSdD

)41.4(dvSdDQ

)40.4(dvQ enclosed charge Total

SdDd

)39.4(Q

v

Vv vS

v vS

v

S

enc

(Divergence Theorem)

20

Electric Field and Electric Flux

• Space with nothing

• Space with a single charged particle

• Space with two charged particles

• Space with multiple charged particles

• Space with a charge density distribution

Qx

y

z

0

Qrr

21

Electric Potential or Voltage

• Space with electric field E(r)

• Put a point charge at r1 from the infinity (a reference point)

• Move the point charge from r1 to r2

Qrr1

E(r)

Qrr2

22


)60.4(]voltC/Joule[LdEQ

WV

)59.4(]voltC/Joule[LdEQ

WV

LdEQW

(4.58))EQF(LdE-QLdFdW

B

AAB

B

AAB

B

A

The electric potential difference between points A and B is the total work done

per unit charge in moving the charge from A to B.

• VAB < 0 (>0) : Potential energy loss (gain) in moving Q from A to B.

• VAB is independent of the path taken.

23


(4.69)charge) (volume |'rr|

'dv)'r(

4

1)r(V

(4.68)charge) (surface |'rr|

'dS)'r(

4

1)r(V

(4.67)charge) (line |'rr|

'dl)'r(

4

1)r(V

(4.66)charge)(point |rr|

Q

4

1)r(V

|rr|4

Q

|rr|4

Q

|rr|4

Q)r(V

)65.4(|'rr|4

Q)r(V

)64.4(dLEV

r4

QV)63.4(

Vv

0

SS

0

LL

0

n

1k k

k

0

n0

n

2010

1

0

r

0

24


)76.4(VE

)75.4(

z

VE

y

VE

x

VE

dzz

Vdy

y

Vdx

x

VdV

dzEdyEdxELdEdV

z

y

x

zyx

25


t

BE**

(4.74))equations'Maxwell(0E

0Sd)E(LdE

theorems'Srokes Appling

)73.4()equations'Maxwell(0LdE

0LdEVV

VV

ABBA

BAAB

26

Gradient of Potential

)27.3(normaln:Gdn

dV|

dL

dV

cosGdL

dV

)26.3(0:dLcosGLdGdV

)25.3(V:az

Va

y

Va

x

VG

)24.3()adzadyadx(az

Va

y

Va

x

V

dzz

Vdy

y

Vdx

x

VdV

max

zyx

zyxzyx

방향등고선에은

최대값때일

기울기의

x

y

z

32F

r

1

4

1)z,y,x(F

2F1F

Slope of V (directional derivative of V)

Maximum when =0

Direction of n is normal to equipotential surface (contour)

27

Equipotential Surface or Line


Adjacent points that have the same electric potential form an equipotential surface, which can be either an imaginary surface or a real, physical surface.

28

Gradient of Scalar Field

)30.3()Spherical(aV

sinr

1a

V

r

1a

r

VV

)29.3()lCylindrica(az

Va

V1a

VV

)(Cartesianaz

Va

y

Va

x

VV

)28.3(az

Va

y

Va

x

VVVgrad

r

z

zyx

zyx

29

Slope in Three Dimension

30

How to find a minimum point?

31

Energy in Electrostatic Field

)86.4()VQVQVQ(2

1W

VQVQVQ

)VV(Q)VV(Q)VV(QW2

)85.4()VV(QVQ0

WWWW

)84.4()VV(QVQ0

WWWW

332211E

332211

323132321213121E

13121232

123E

32313212

321E

32

Energy in Electrostatic Field

)91.4(Vdv)D(2

1W

(4.90)charge) (volume xVd2

1W

)89.4(charge) (surfacexVd2

1W

(4.88)charge) (line VdL2

1W

)87.4(VQ2

1W

)86.4()VQVQVQ(2

1W

vE

3vE

2SE

LE

n

1kkkE

332211E

)98.4(dvwW

)97.4(]m/J[2

DE

2

1ED

2

1w

dv

dWw

)96.4(dvE2

1dvED

2

1W

dv)ED(2

1dv)VD(

2

1W)95.4(

EE

3

0

22

0E

EE

20E

vvE

33

Charge in Material Space

• Insulator• Negligible amount of mobile charged particles

• Rubber, plastic, glass, pure water

• A dielectric is an insulator that can be polarized by an external electric field

• Semiconductor• Intermediate between conductors and semiconductors

• Silicon, germanium

• Conductor• Significant amount of mobile charged particles

• Metals with free electrons, electrolyte with ions

• Superconductor• (Almost) Perfect conductors

34

Electric Current


As Fig. (a) reminds us, any isolated conducting loop—regardless of whether it has an excess charge — is all at the same potential. No electric field can exist within it or along its surface.

If we insert a battery in the loop, as in Fig. (b), the conducting loop is no longer at a single potential. Electric fields act inside the material making up the loop, exerting forces on internal charges, causing them to move and thus establishing a current. (The diagram assumes the motion of positive charges moving clockwise.)

Figure c shows a section of a conductor, part of a conducting loop in which current has been established. If charge dq passes through a hypothetical plane (such as aa’) in time dt, then the current i through that plane is defined as

𝑖 =𝑑𝑞

𝑑𝑡(definition of current)

35

Figure (a) shows a conductor with current i0 splitting at a junction into two branches. Because charge is conserved, the magnitudes of the currents in the branches must add to yield the magnitude of the current in the original conductor, so that

Figure (b) suggests, bending or reorienting the wires in space does not change the validity of the above equation. Current arrows show only a direction (or sense) of flow along a conductor, not a direction in space.

Electric Current


𝑖0 = 𝑖1 + 𝑖2 (Kirchhoff’s Current Law)

36

Charge in Motion, Electric Current

)4.5(SdJI

)3.5(SJI

)2.5(SJI

]msec//C[S

IJ

)1.5(sec]/C[dt

dQI

S

n

2n

na

J

SA

SaJJAI

ScosA

n

37

Current i (a scalar quantity) is related to current density J (a vector quantity) by

where dA is a vector perpendicular to a surface element of area dA and the integral is taken over any surface cutting across the conductor. The current density J has the same direction as the velocity of the moving charges if they are positive charges and the opposite direction if the moving charges are negative.

Streamlines representing currentdensity in the flow of chargethrough a constricted conductor.

Current Density


𝑖 = 𝐴

𝑱 ∙ 𝑑𝑨

38

Current Density


Current is said to be due to positive charges that are propelled by the electric field. In the figure, positive charge carriers drift at speed vd in the direction of the applied electric field E which here is applied to the left. By convention, the direction of the current density J and the sense of the current arrow are drawn in that same direction, as is the drift speed vd. The drift velocity vd is related to the current density by

Here the product ne, whose SI unit is the coulomb per cubic meter (C/m3), is the carrier charge density.

Conduction electrons are actually moving to the right but the conventional current iis said to move to the left.

𝑱 = 𝑛𝑒𝒗𝑑

39

Current in Filament

][A/sec/mDensityCurrent Convection:

(5.7)uJ

)6.5(uS

IJ

)5.5(uSt

yS

t

yS

t

QI

2

v

yvy

yvvv

40

Conduction Current by Electron

MassElectronm

DensityElectronn

FrequencyCollision:/1

TimeCollision

tyConductivi:m

ne

e

e2

m

newhere

)11.5(Laws'Ohm:EJ

Em

neJ

Ene)uen(m

Eenumn

umn)BuE(en

dt

udmnF

:EqMomentumElectron

e2

e

e2

e

e2e

ee

eee

41

Conductivity and Resistance

m:unit

sistivityRe:/1

)16.5(S

L

S

L

I

VR

)15.5(L

VE

)14.5(S

IJ

)13.5(L

VE

c

c

42

Conductivity, Resistivity and Resistance

22

2

0

0

, 0z L z

uu

z

u V u

V

u z zL

z

Vz u z

L

J a

zS

VI z dxdy S

L

J a

1= [ ]

V L LR

I S S

V

0 z = 0

z = LS

J

V

0

I

43

Conductivity and Resistance

V

I

I

L

S

MobileCharge

-+

e-

e-

V+_ V

I

I

-+

e-

e-

MobileCharge

V+

_

𝐄 = −𝛻𝑢, 𝐅 = 𝑞𝐄 = 𝑚𝐚, 𝐯𝑑 = 𝜇𝐄

𝐼 = 𝑅𝑉, 𝑅 =𝑉

𝐼=1

𝜎

𝐿

𝑆

𝐉 = 𝑐𝐯𝑑 = 𝑐𝜇𝐄 = 𝜎𝐄 = −𝜎𝛻𝑢

44

Ohm’s Law


I ~ V or I = V/R or V = RI

The power P, or rate of energy transfer, in an electrical device across which a potential difference V is maintained is

If the device is a resistor, the power can also be written as

𝑃 = 𝐼𝑉

𝑃 = 𝐼2𝑅 =𝑉2

𝑅

45

Emf Device of Circuit


To produce a steady flow of charge, you need a “charge pump,” a device that—by doing work on the charge carriers—maintains a potential difference between a pair of terminals. We call such a device an emf device, and the device is said to provide an emf, which means that it does work on charge carriers.

Figure shows an emf device (consider it to be a battery) that is part of a simple circuit containing a single resistance R. The emfdevice keeps one of its terminals (called the positive terminal and often labeled +) at a higher electric potential than the other terminal (called the negative terminal and labeled -). We can represent the emf of the device with an arrow that points from the negative terminal toward the positive terminal as in Figure. A small circle on the tail of the emf arrow distinguishes it from the arrows that indicate current direction.

46

Power

Power:Wattsec/J

sec/sec/mkg]P[

Work:Joulesec/mkg]W[

Force:Newtonsec/mkg]F[:unit

vFdt

LdF

dt

dWP

LdFdW

22

22

2

47

Joule’s Law: Ohmic Power Dissipation

)20.5(RI

VIdSJdLEP

2

SL

DensityPower:

)19.5(EJEdV

dPw

)18.5(dV)JE(P

dxdydzJE

dxdydz)v(E

vE)dxdydz(

v)EdQ(dP

2

P

v

v

dVdQ v

v

dV

48

Battery in Seawater

49

Current and Voltage

• White lines are current stream lines.• Black lines are equipotential lines.

+

-

+

-+

-

50

Ion Conduction and Conductivity

V

I

I

L

S

Ions-+

e-

e-

V+_ V

I

I

-+

e-

e-

Ions

V+

_

𝐄 = −𝛻𝑢, 𝐅 = 𝑞𝐄 = 𝑚𝐚, 𝐯𝑑 = 𝜇𝐄

𝐉 = 𝑐𝐯𝑑 = 𝑐𝜇𝐄 = 𝜎𝐄 = −𝜎𝛻𝑢

51

Nervous System

Malmivuo and Plonsey, 1995

52

Neuron (Excitable Cell) and Muscle


53

Neuron and Nerve Conduction


Nerve

Muscle

Action Potential

Neuronal Current

54

Intra-cardiac Nerve Conduction


55

Equivalent Dipole Current Source


Atrial Depolarization (80 ms)

Septal Depolarization (220 ms)

Apical Depolarization (230 ms)

Left Ventricular Depolarization (240 ms)

Left Ventricular Depolarization (250 ms)

Depolarized Ventricles (350 ms)

Ventricular Repolarization (450 ms)

Repolarized Ventricles (600 ms)

56

Body as an Active Volume Conductor


Conductivity Voltage Volume Source Density (Flow Source Density)

⋅ 𝐫, 𝑡 Φ 𝐫, 𝑡 = ⋅ 𝐉nc 𝐫, 𝑡 = − 𝐼𝐹 𝐫, 𝑡 in V

− 𝐫, 𝑡 Φ 𝐫, 𝑡 ⋅ 𝐧 = 0 on 𝜕V

𝐉 𝐫, 𝑡 = 𝐉nc 𝐫, 𝑡 − 𝐫, 𝑡 Φ 𝐫, 𝑡

Total Current Density Neuronal Current Density (Volume Dipole Moment Density)

Return Current Density

u

, r

-

+

57

Volume Conduction and Biosignal

• White lines are current stream lines.• Black lines are equipotential lines.

+

-

𝑢1+

𝑡1

+

-

+ 𝑢2

𝑡2

+

-

+ 𝑢3

𝑡3

𝑡𝑡1 𝑡2 𝑡3

𝑢

𝑢1

𝑢2

𝑢3

58

ECG (Electrocardiogram)

Medical Instrumentation: Application and Design, 3rd ed., by J. G. Webster

ECG

Amplifier

⋅ 𝐫, 𝑡 𝑢 𝐫, 𝑡 = −𝑓 𝐫, 𝑡 = ⋅ 𝐉nc 𝐫, 𝑡 in Ω

− 𝐫, 𝑡 𝑢 𝐫, 𝑡 ⋅ 𝐧 = 0 on 𝜕Ω

59

ECG (Electrocardiogram)

Grimnes and Martinsen, 2008

60

EEG (Electroencephalogram)

𝑓 𝐫, 𝑡

Amplifier

EEG

⋅ 𝐫, 𝑡 𝑢 𝐫, 𝑡 = −𝑓 𝐫, 𝑡 = ⋅ 𝐉nc 𝐫, 𝑡 in Ω

− 𝐫, 𝑡 𝑢 𝐫, 𝑡 ⋅ 𝐧 = 0 on 𝜕Ω

http://en.wikipedia.org/wiki/File:Spike-waves.png

http://en.wikipedia.org/wiki/File:Spike-waves.png

61



62



63

Dipole in Electric Field


The torque on an electric dipole of dipole moment p when placed in an external electric field E is given by a cross product:

(a) An electric dipole in a uniform external electric field E. Two centers of equal but opposite charge are separated by distance d. The line between them represents their rigid connection.

(b) Field E: causes a torque τ on the dipole. The direction of τ is into the page, as represented by the symbol (x-in a circle) .

64

Dielectrics

(a)

E=0d

-Q +Q

Center of negative charge

d

(b)

E

E

Polarization of Nonpolar Atom or Molecule

Polarization of Polar Molecule

65

Dipole and Potential

)81.4(|'rr|4

)'r-r(p)r(V

)80.4(r4

apV

(4.79)moment) (Dipole)Cm(dQp

)78.4(r

cosd

4

QV

)77.4(rr

rr

4

Q

r

1

r

1

4

QV

30

20

r

20

21

12

0210

(Natural matters contains both positive and negative charges (protons & electrons)

(HCl : H (hydrogen) – somewhat positive, Cl (chloride) – somewhat negative)

Q

Q

1r

2r

d

z

y

x cosd

P

P r

66

Point Charge and Dipole

E =qar4πε0r

2

V =q

4πε0r

E =p

4πε0r3(2cosθar + sinθaθ

V =Qd cosθ

4πε0r2

67

Polarization in Dielectrics

)23.5(]m/C[v

dQlim

P

)22.5(dQdQdQdQ

(5.21)]mC[(dipole)dQp

2

N

1kkk

0v

N

1kkknn2211

(Polarization 𝐏 is defined as the dipole moment per unit volume)

68

Polarization in Dielectrics

(5.32)EP

(5.31)PED

(5.30)D

)E(

)PE(

E

(5.29)E

0QQdvdS charge Total

P

aP)27.5(

)b28.5(surface) theinside (chargedvPdvQ

(5.28a)surface)on charge (bounddSSdPQ

0e

0

0

pv0v

0pvvt

bbv pvS ps

pv

nps

vv pvb

psb

)36.5(1

)35.5(

(5.34)ED

PE

(5.33)EE)1(D

0er

r0

0

r0e0

Dielectricconstant

Susceptibility

Free spacepermittivity

Permittivity

69

Capacitance

V+

-

+

-

+

-

+

-

+

-

+

-

+

-

+

-

+

-

+

-

+

-

+

-

+

-

+

-

+

-

+

-

+

-

+

-

V-

+

-

+

-

+

-

+

-

+

-

+

-

+

-

+

-

+

-

+

-

+

-

+

-

+

-

+

-

+

-

+

-

+

-

+

-+

-+

-

+

-+

-+

-+

-+

-

+

-+

-+

-

+

-

+

-+

-+

-

+

-+

-+

-+

- - - - - - - - -

+ + + + + + + + +

+ + + + + + + + +

- - - - - - - - -+Q

-Q

L

S

PolarizationPolar Molecules

e-

e-

e-

e- +Q

-Q

𝑞 𝑡 = 𝐶𝑣 𝑡 = 𝜀𝑆

𝐿𝑣 𝑡 , 𝑖 𝑡 =

𝑑𝑞(𝑡)

𝑑𝑡= 𝐶𝑑𝑣(𝑡)

𝑑𝑡

𝑣 𝑡 = 𝑉 cos𝜔𝑡 , 𝑖 𝑡 = −𝑉𝜔𝐶 sin𝜔𝑡

𝑣(𝑡) = 𝑅𝑒 𝑉𝑒𝑗𝜔𝑡 , 𝑖 𝑡 = 𝑅𝑒 𝑉𝜔𝐶𝑒𝑗 𝜔𝑡+90°

, 𝐙 =𝐕

𝐈=1

𝑗𝜔𝐶

70

Parallel-Plate Capacitor


Charging Capacitor

When a circuit with a battery, an open switch, and an uncharged capacitor is completed by closing the switch, conduction electrons shift, leaving the capacitor plates with opposite charges.

𝐶 =𝜀𝐴

𝑑

𝑄 = 𝐶𝑉

71

External Defibrillator

𝑈 =1

2𝐶𝑉2 (delivered energy)

72

Online Lectures by Walter Lewin at MIT

• Walter Lewin

• YouTube

• Videolectures.net

• Last lecture

• MIT News about removal of his online lectures

https://www.youtube.com/channel/UCiEHVhv0SBMpP75JbzJShqw

http://videolectures.net/walter_h_g_lewin/

https://www.youtube.com/watch?v=4a0FbQdH3dY

http://news.mit.edu/2014/lewin-courses-removed-1208

https://en.wikipedia.org/wiki/Walter_Lewin

http://web.archive.org/web/20140701083832/http://web.mit.edu/physics/people/faculty/lewin_walter.html

https://www.youtube.com/channel/UCiEHVhv0SBMpP75JbzJShqw

http://videolectures.net/walter_h_g_lewin/

https://www.youtube.com/watch?v=4a0FbQdH3dY

http://news.mit.edu/2014/lewin-courses-removed-1208

https://en.wikipedia.org/wiki/Walter_Lewin

http://web.archive.org/web/20140701083832/http:/web.mit.edu/physics/people/faculty/lewin_walter.html

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