ELEC 3105 Basic EM and Power Engineering

Conductivity / Resistivity

Current Flow

Resistance

Capacitance

Boundary conditions

Conductivity and resistivityThe relaxation time model for conductivity works for most metals and

semiconductors.

In a conductor at room temperature, electrons are in random thermal motion, with mean time between collisions.

Random motion of the electron in the metal. Electron undergoes collisions then moves off in different direction.

0E

electron

collision

Conductivity and resistivityThe relaxation time model for conductivity works for most metals and semiconductors.

In a conductor at room temperature, electrons are in random thermal motion, with mean time between collisions.

Electrons acquire a small systematic velocity v* component in response to applied electric field

E

electron

collision

Conductivity and resistivityFor a weak electric field v* can be obtained.

m = mass of electron = carrier mobility (ELEC 2507)

E

electron

collision

EEm

q

m

Eq

m

Fv

*

{} units of

Vs

m2

Conductivity and resistivity

FOR A STRONG ELECTRIC FIELD

E

for low fields v* proportional to E

For strong electric fields, electrons acquire so much energy between collision that mean time between collisions is reduced.

v*

E

Conductivity and resistivityAs long as we stay in the weak electric field regime, i.e. the linear region of the curve in the previous slide, then the current density can be defined as:

E

v*

E

This regionENqJ

EJ

Nq Conductivity

1

Resistivity

Conductivity of elements

Current flowThe total amount of charge moving through a given cross section per unit time is the current, usually denoted by I.

Conductor ???

vdt

dq v

dt

dqI CURRENT

Current flowIf we consider the current per unit cross-sectional area, we get a value which can be defined any point in space

as a vector, typically denoted

vdt

dq v

J

cross-sectional area A

N charged particles per unit volume moving at v meters per second

dq = N q vdt A Charge moving through cross-sectional area A in time dt

Current flowThe charge density is simply this quantity divided by the unit time and area. The current density is:

vdt

dq v

vNqJ

cross-sectional area A

N charged particles per unit volume moving at v meters per second

dq = N q vdt A Charge moving through cross-sectional area A in time dt

dq = N q vdt A

Current flowThe total current through the end face can be obtained from the current density as an integration over the cross-sectional area of the conducting medium.

vdt

dq v

davNqdaJIAA

cross-sectional area A

TOTAL CURRENT

Current flow

The total charge passing through the cross-sectional area A over a time interval from t1 to t2 can be obtained from:

vdt

Q v

2

1

2

1

t

t A

t

t

dtdaJIdtQ

cross-sectional area A

TOTAL CHARGE

MOSFET

Resistance of conductors: any shape

ab

abab I

VR

b

a

ab dEV

AA

ab dAEdAJI

RESISTANCE

Resistance of conductors: any shape

ab

abab I

VR

b

a

ab dEV

A uniform rectangular bar

ELVab

Electric field is uniform and in the direction of a bar length L.

Resistance of conductors: any shape

ab

abab I

VR

A uniform rectangular bar

Electric field is normal to the cross-sectional area A.

AA

ab dAEdAJI

EAIab

Resistance of conductors: any shape

ab

abab I

VR

A uniform rectangular bar

EAIab ELVab

A

LRab

A

LRab

SUPERCONDUCTORS

Capacitance• Capacitance is a property of a geometric configuration, usually

two conducting objects separated by an insulating medium.

• Capacitance is a measure of how much charge a particular configuration is able to retain when a battery of V volts is

connected and then removed.

• The amount of charge Q deposited on each conductor will be proportional to the voltage V of the battery and some constant C,

called the capacitance.

V

QC Capacitance {C/V}

Parallel plate capacitor

Free space between plates o

z

z dEV0

o

sE

+Q

-Q

V = 0 volts

V = V voltsz

zVo

sz

Plate area APlate separation D

Between plates

A

Qs

At z = D DA

QV

o

DA

QV

o Rearrange

D

AC o

Capacitance of parallel plate capacitor

V

QC

CAPACITORS IN SERIES/ PARALLEL/ DECOMPOSITION

PARALLEL

C1C2

Ceq

21 CCCeq

C1

C2

SERIES

Ceq

21

111

CCCeq

CAPACITORS IN SERIES/ PARALLEL/ DECOMPOSITION

DECOMPOSITION

3

21

111

C

CC

Ceq

Ceq

C1

C2

C3

CAPACITANCE OF A COAXIAL TRANSMISSION LINE

a

bVV L

ba ln2 Prove this result as part of

next assignment.

If we consider as the charge per unit length on each of the two coaxial surface, then:

V

QC

ba

L

VVC

a

bC

ln

2

ab

C

ln

2

m

pF

ab

C r

ln

6.55

L

(ELEC 3909)

CHARGE CONSERVATION AND THE CONTINUITY EQUATION

v

vin dvQ Charge in volume v

Current through surface A A

dAJI

Also recall dt

dQI

The main ingredients to the pie

31

CHARGE CONSERVATION AND THE CONTINUITY EQUATION

v

v

v

dvt

dvJ

Then:

Current out of volume ist

QI in

From divergence theorem

v

v

A

dvt

dAJ

Using previous expressions

Av

dAJdvJ

tJ v

CHARGE CONSERVATION AND THE CONTINUITY EQUATION

Interpretation of equation: The amount of current diverging from am infinitesimal volume element is equal to the time rate of change decrease of charge contained in the volume. I.e. conservation of charge.

tJ v

In circuits: 0inI If no accumulation of charge at node.

33

CHARGE CONSERVATION AND THE CONTINUITY EQUATION

EJ

A charge is deposited in a medium.

tJ v

tE v

AlsofreeD

freeE

t

0

t Tt

oet

CHARGE CONSERVATION AND THE CONTINUITY EQUATION

A charge is deposited in a medium.

Tt

oet

If you place a charge in a volume v, the charge will redistribute itself in the medium (repulsion???). The rearrangement of charge is governed by the constant

T = REARRANGEMENT TIME CONSTANT

T

TCu, Ag=10-19 s Tmica=10 h

Boundary conditionsTangential Component of E

0c

dE

Around closed path (a, b, c, d, a)

0V

ELECTROSTATICS

1

2

Boundary

t̂

a

b

cd

2E

1E

2tE

1tE

Potential around closed path

Boundary conditionsTangential Component of E

1

2

Boundary

t̂

a

b

cd

2E

1E

2tE

1tE

0lim 120,0,

a

d

c

bc

cdabdEdEdE

ELECTROSTATICS

Boundary conditionsTangential Component of E

t̂

a

b

cd

0lim 11220,0,

a

d

c

bc

cdabdEdEdE

012

dEE

012 dEE tt

012 tt EE

12 tt EE The tangential components of the electric field across a boundary separating two media are continuous.

ELECTROSTATICS

Boundary conditionsTangential Component of E

012 tt EE

12 tt EE

At the surface of a metal the electric field can have only a normal component since the tangential component is zero through the boundary condition.

ELECTROSTATICS

01 tEt̂

a

b

cd

metal

02 E

1 nEE ˆ11

Boundary conditions Normal Component ofE

enclosed

c

qdAD

Gauss’s law over pill box surface

ELECTROSTATICS

12

Boundary

n̂A

2E

1E

2nE

1nE

n̂

n

dAdADDdADs

cn

210

lim

ELECTROSTATICS

1

2

Boundary

n̂A

2E

1E

2nE

1nE

n̂

n

Boundary conditions Normal Component ofE

41

dAdADDdADs

cn

210

lim

1

2A

2nE

1nE

dAdADD snn 21

021 dADD snn

snn DD 21

snnEE

2211

The normal components of the electric flux density are discontinuous by the surface charge density.

Boundary conditions Normal Component ofE

ELECTROSTATICS

42

snn DD 21

snD 1

022 nn DE

1nEE ˆ11

11

snE

Boundary conditions Normal Component ofE

ELECTROSTATICS

metal

02 E

At the surface of a metal the electric field magnitude is given by En1 and is directly related to the surface charge density.

Boundary conditionsNormal Component of D

snn DD 21

ELECTROSTATICS

Gaussian Surface

Air Dielectric

Gaussian surface on metal interface encloses a real net charge s.

Gaussian surface on dielectric interface encloses a bound surface charge sp , but also encloses the other half of the dipole as well. As a result Gaussian surface encloses no net surface charge.

snD 1

021 nn DD

21 nn DD

ELEC 3105 Basic EM and Power Engineering

Extra extra read all about it!

44

45

Electric fields in metals

Electric fields in metals

(a) no current Einside = 0 (b) with current Einside 0

Inhomogeneous dielectrics

We can consider an inhomogeneous dielectric as being made up of small homogeneous pieces, at the interfaces of which bound charge will accumulate.

x

D

Suppose that we have a dielectric whose permittivity is a function of x, and a constant D field is directed along x as well.

dielectric

Inhomogeneous dielectrics

We can consider an inhomogeneous dielectric as being made up of a stack of thin sheets of thickness dx and permittivity (x).

x

D In each sheet, positive charges will

accumulate on the right and negative ones on the left, according to the permittivity of the sheet.

Inhomogeneous dielectrics

We can consider an inhomogeneous dielectric as being made up of a stack of thin sheets of thickness dx and permittivity (x).

x

D The charges will mostly cancel by

adjacent sheets, but any difference in permittivity between adjacent sheets d will leave some net charge density.

Inhomogeneous dielectrics

We can consider an inhomogeneous dielectric as being made up of a stack of thin sheets of thickness dx and permittivity (x).

x

D We can express this net bound charge

easily as the difference in polarizations, so that we have:

dx

dP

dx

xPdxxPbound

Inhomogeneous dielectricsIn the more general case when the permittivity is varies in all directions, i. e. (x,y,x).

x

D

We can express this net bound charge easily as the difference in polarizations, so that we have:

Pbound

y

z

dx

dP

dx

xPdxxPbound

PED o

Inhomogeneous dielectricsIn the more general case when the permittivity is varies in all directions, i. e. (x,y,x).

x

D

Take divergence on each side:

y

z

PED o

PED o

freeboundtotal