ELEC 3105 BASIC EM AND POWER ENGINEERING

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Magnetostatics The Basics

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Magnetostatics• Stationary charge:

• vq = 0

• E 0 B = 0

•Moving charge:

• vq 0 and vq = constant

• E 0 B 0•Accelerating charge:

• vq 0 and aq 0

• E 0 B 0• Radiating field

A uniformly moving charge produces an electric and magnetic field.

A stationary charge produces an electric field only.

A accelerating charge produces an electric and magnetic field and a radiating electromagnetic field.

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Magnetostatics

Units and definitions

Magnetic field strength

Magnetic field vector

Magnetic induction

Magnetic flux densityGT 4101

Tesla Gauss

SI unit

211

m

WbT

Weber

B

H

HB

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Magnetostatics

Permeability

or Permeability of free space

Relative permeability for a medium

Permeability of the medium

m

Ho

7104

m

Wb

m

HExact constant

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ferromagnetic

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Ferromagnetic materials WHY?

What happens when we cycle the applied magnetic field.

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8

PERMITTIVITY AND PERMEABILITY

ELEC 3105 BASIC EM AND POWER ENGINEERING

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Magnetostatics 1st Postulate

Magnetostatics

POSTULATE 1 FOR THE MAGNETIC FIELD

A current element immersed in a magnetic field will experience a force given by:

dI

B

Fd

dBIFd

Units of Newtons {N}

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Magnetostatics

POSTULATE 1 FOR THE MAGNETIC FIELD

A current element experiences a force which is at right angles to the plane formed by the current element and magnetic field direction magnitude: sinIBddF

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BE

Magnetostatics vs Electrostatics

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Magnetostatics postulate 1 for the magnetic field

IBdFF

Consider a straight segment

Net force on the segment

Right hand rule for direction

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postulate 1 for the magnetic field

Magnetic force on a moving charge

Current density:

Current through cross section dA:

J

I

qv

d

dA

vJ

Volume charge density

dAJI

vqdAdvdAdJId

Where is an element of volume enclosing charge q dAd

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postulate 1 for the magnetic field

Magnetic force on a moving charge

Modify force equation:

Net force on charge q

J

I

q

v

d

dA

vqId

dBIFd

BvqFd

BvqF

X

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postulate 1 for the magnetic field

Magnetic force on a moving charge

Lorentz force

Often used to define the magnetic field. Force, charge, and charge velocity are measurable.

q

v

d

dA

BvqF

qv

FB

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Magnetic force on a moving charge BvqF

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Magnetic Electric

Force at right angle to v and B vectors

Force proportional to v

Can do no work on a charge

Force along electric field lines

Force independent of v

Can do work

rdFdW

dtv

Bvq

dtvBvqdW

0 always

0dW

Electric AND Magnetic

ELEC 3105 BASIC EM AND POWER ENGINEERING

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Hall effect

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Hall effect

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Hall Effect

J

I

q

v

magF

B3-D view of block

When a conductor that carries a current is placed in a uniform magnetic field, an electrostatic field appears whose direction is perpendicular both to the magnetic field and to the current. The electric field here is known as the Hall field and reaches equilibrium in the order of 10-14 s. The electric field is characterized also by the Hall voltage across the faces of the conductor.

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Hall Effect

J

I

q

v

x

y

z

zBB ˆ

magF

B

yvv ˆ

BvqFmag

xqvBFmag ˆ

3-D view of block

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Hall Effect

J

I

q

v

x

y

z

zBB ˆ

magF

B

yvv ˆ

Top view of blockLook onto this surfacefrom above

- - - - - - - - - - - -

+ + + + + + + + + +

Negative charge build up on this surface

Positive charge build up on this surface

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Hall Effect

I

q

v

magF

B

Top view of block

Accumulation of charge continues until induced electric force equals magnetic force.- - - - - - - - - - - -

+ + + + + + + + + +

Voltage difference across charge distribution

EF

magE FF

HallV

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Hall Effect

J

I

q

v

magF

B3-D view of block

w

t

wt

IJ

Current density

?????HallV

vqNI

Current with N density of carriers in the material.

get v

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Hall Effect

J

I

q

v

magF

B3-D view of block

w

t

BqN

JwtqqvBFmag

Velocity of the moving charge

?????HallV

qN

Jwtv

Magnetic force

Simplified magnetic force on the charge moving at velocity vB

N

JwtFmag

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Hall Effect

I

q

v

magF

B

Top view of blockAccumulation of charge continues until induced electric force equals magnetic force.

- - - - - - - - - - - -

+ + + + + + + + + +

Voltage difference across charge distribution

EF

w

qVqEF Hall

HE

wEV HHall

BN

JwtFmag

withw

magE FF

and

VE

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Hall Effect

I

q

v

magF

B

In the steady state

- - - - - - - - - - - -

+ + + + + + + + + +

EF

w

qVqEF Hall

HE

HallV

BN

JwtFmag

w

magE FF

then

qN

tJBwB

qN

JwtwVHall

2

qN

tJBwB

qN

JwtwVHall

2

BqN

IwVHall

wI

qNVB Hall

ELEC 3105 BASIC EM AND POWER ENGINEERING

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Magnetostatics 2nd Postulate

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Magnetostatics

POSTULATE 2 FOR THE MAGNETIC FIELD

A current element produces a magnetic field which at a distance R is given by:

d

R

RIBd o

2

ˆ

4

dI

B

Bd

Units of {T, G, Wb/m2}

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Magnetostatics POSTULATE 2 FOR THE MAGNETIC FIELD

Postulate 2 implies that the magnetic field is everywhere normal to the element of length

sin

4 2R

IddB o

d

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Magnetostatics Magnitude of the magnetic field

d

R

RIBd o

2

ˆ

4

2

1

RdB

Similar to:

sin

4 2R

IddB o

2

1

RdE

magQId Conceptually similar to a magnetic charge.

Magnetic charges have not yet been found.

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Magnetostatics For a closed path made up of current elements

C

o

R

RdIrB

2

ˆ

4

Id

rrR R

RR

ˆ

Biot-Savard Law

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Magnetostatics Magnetic field produced by extended conductor

v

o dAdR

RJrB

2

ˆ

4

voldvJdI

rrR

R

RR

ˆ

J

I

S

AdJI

v

volo dv

R

RJrB

2

ˆ

4

dA

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Magnetostatics Magnetic field produced by single moving charge conductor

2

ˆ

4 R

RvqrB o

vqdI

rrR

R

RR

ˆ

q

v

R

B

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Magnetostatics POSTULATE 1 and 2 FOR THE MAGNETIC FIELD

Magnetic field lines are continuous and close on themselves. There are no magnetic charges for the lines to start or end on. Magnetic forces and magnetic fields are at 90 degrees to their sources.

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Consequence of postulates 1 and 2

From postulate 2: A moving charge produces a magnetic field.

From postulate 1: A magnetic field produces a force on a moving charge.

Is it possible then that a moving charge generate a magnetic force on a second moving charge?

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Consequence of postulates 1 and 2

Is it possible then that a moving charge generate a magnetic force on a second moving charge?

Answer “YES”

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Consequence of postulates 1 and 2

2

12

12112212

ˆ

4 R

RdIdIF o

vqId

Recall for a moving charge that the following substitution is possible:

THEN

BECOMES

2

12

12112212

ˆ

4 R

RvqvqF o

12R 22vq

11vq

FF

2

12

12122112

ˆ

4 R

RvvqqF o

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Compare magnitude of magnetic and electric force

between two moving charges2

2124vqq

RF o

m

22vq

11vq

mFmF

eFeF

2124

1qq

RF

oe

2vF

Foo

e

m

coo

1

2

2

c

v

F

F

e

m

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Compare magnitude of magnetic and electric force

between two moving charges

22vq

11vq

mFmF

eFeF

em Fc

vF

2

2

The magnetic force on these two moving charges can be obtained from the elctric force and the velocity of the two charges. Concepts of relativity come into play.

maximum

Magnetic induction charging

Induction Stove