Chapter 6: Maxwell’s Equations for Time-Varying Fields

The wings cut the flux lines of the Earth’s magnetic field, inducing an emf between the wing tips. As the aircraft flies through the field, the north-[r]

Bài giảng: TRƯỜNG ĐIỆN TỪ (CT361)

(ELECTROMAGNETICS)

Chapter 6: Maxwell’s Equations for Time-Varying Fields

Giảng viên: GVC.TS Lương Vinh Quốc Danh

Bộ môn Điện tử Viễn thông, Khoa Công Nghệ

• Dynamic Fields

• Faraday’s Law

• Stationary Loop in a Time-Varying Magnetic Field

• The Ideal Transformer

• Moving Conductor in a Static Magnetic Field

• Moving Conductor in a Time-Varying Magnetic Field

Maxwell’s Equations

Faraday’s Law: Time varying magnetic feed 

Eeectric feedd

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Faraday’s Law

Electromotive force (voltage) induced

by time-varying magnetic flux:

- Michael Faraday conducted the

experiments in his Lab in London.

- Joseph Henry discovered the

matter at about the same time

(1831).

Three types of EMF

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Stationary Loop in Time-Varying B Field

(6.8)

Example 6-1

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Example 6-1 (cont.)

Ideal Transformer

Moving Conductor in a Static B Field

Magnetic force on charge q moving

with velocity u in a magnetic field

B:

This magnetic force is equivalent to the

electrical force that would be exerted on

the particle by the electric field E m given

by

This, in turn, induces a voltage

difference between ends 1 and 2,

with end 2 being at the higher

potential The induced voltage is

called a motional emf

(6.22)

(6.23)

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Moving Conductor in a Static B Field (cont.)

A: area

Example 6-3: Sliding Bar

The length of the loop is related

to u by x 0 = ut Hence

Note that B = zB0 x, B increases

with x.

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EMF in an Airliner

These questions are about the potential difference induced across the wings of

an aeroplane flying through the Earth’s magnetic field An airliner is flying due East from North America to Europe The Earth’s magnetic field acts at 70° to the

horizontal, and has a strength of 1.7 x 10–4 (T)

The charge on an electron = – 1.6 x 10–19 C The aircraft’s speed is 270 m/s The wing span of the aircraft is 60 m

EMF in an Airliner (cont.)

The wings cut the flux lines of the Earth’s magnetic field, inducing an emf

between the wing tips As the aircraft flies through the field, the

north-pointing tip of the wing becomes positively charged and the south-north-pointing

tip becomes negatively charged

Vertical component of B field = (1.7 x 10–4 ) cos 20° = 1.6 x 10–4 (T)

V.

6 2

T) 10

6 1 ( m 60 s

m 270

emf

4 1

Aircraft have wire static

dischargers on their trailing

wing tips so that the electrical

charge that builds up in the

plane as it travels can be safely and continuously discharged into the air Without the dischargers, this static charge could damage sensitive computer instruments onboard

Static Charges on Airplane’s Wings

Example 6-5

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EM Generator EMF

Also:

Segment 3-4 moves with velocity –u Hence:

As the loop rotates with an angular

velocity ω about its own axis, segment 1–2

moves with velocity u given by

(6.31)

(6.32)

Applications: EMF Sensors

• Piezoelectric crystals generate a voltage across them proportional to

the compression or tensile (stretching) force applied across them

• Piezoelectric transducers are used in medical ultrasound,

microphones, loudspeakers, accelerometers, etc.

• Piezoelectric crystals are bidirectional: pressure generates emf, and

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Displacement Current

This term is conduction

Defne the dispeacement current

invoeve reae charges;

it is an equivaeent current that depends

on D/t

Displacement Current (cont.)

Capacitor Circuit

conductors and capacitor

insueator materiae is perfect

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Boundary Conditions

What role electromagnetcs plays in

science, technology, and society?

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EECS 332, Winter 1994