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Electric Field - ejwoo.com Hee University Korea ... Gauss’ law relates the net fluxϕof an...
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Electric Field
Eung Je Woo
Department of Biomedical Engineering
Impedance Imaging Research Center (IIRC)
Kyung Hee University
Korea
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/
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
Ch21, HRW, Fundamentals of Physics, Wiley
(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
Ch21, HRW, Fundamentals of Physics, Wiley
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
Ch22, HRW, Fundamentals of Physics, Wiley
• 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
Ch22, HRW, Fundamentals of Physics, Wiley
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
Ch23, HRW, Fundamentals of Physics, Wiley
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
Ch23, HRW, Fundamentals of Physics, Wiley
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.
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
Ch23, HRW, Fundamentals of Physics, Wiley
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
Electric Potential or Voltage
)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
Electric Potential or Voltage
(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
Electric Potential or Voltage
)76.4(VE
)75.4(
z
VE
y
VE
x
VE
dzz
Vdy
y
Vdx
x
VdV
dzEdyEdxELdEdV
z
y
x
zyx
25
Electric Potential or Voltage
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
Ch24, HRW, Fundamentals of Physics, Wiley
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
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
Ch26, HRW, Fundamentals of Physics, Wiley
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
Ch26, HRW, Fundamentals of Physics, Wiley
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
Ch26, HRW, Fundamentals of Physics, Wiley
𝑖 = 𝐴
𝑱 ∙ 𝑑𝑨
38
Current Density
Ch26, HRW, Fundamentals of Physics, Wiley
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
Ch26, HRW, Fundamentals of Physics, Wiley
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
Ch27, HRW, Fundamentals of Physics, Wiley
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
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
53
Neuron and Nerve Conduction
Malmivuo and Plonsey, 1995
Nerve
Muscle
Action Potential
Neuronal Current
55
Equivalent Dipole Current Source
Malmivuo and Plonsey, 1995
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
Malmivuo and Plonsey, 1995
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
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
63
Dipole in Electric Field
Ch22, HRW, Fundamentals of Physics, Wiley
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
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
Ch25, HRW, Fundamentals of Physics, Wiley
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
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