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

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<b>Bài giảng:</b>

<b>TRƯỜNG ĐIỆN TỪ (CT361)</b>

<b> (ELECTROMAGNETICS)</b>

<b>Chapter 6: </b>

<b>Maxwell’s Equations for Time-Varying Fields</b>

<b>Giảng viên: GVC.TS. Lương Vinh Quốc Danh</b>

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•

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

•

Displacement Current

•

Boundary Conditions for Electromagnetics

<b>Content</b>

2

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<b>Maxwell’s Equations </b>

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<b>Faraday’s Law</b>

Electromotive force (voltage) induced

by time-varying magnetic flux:

- Michael Faraday conducted the

experiments in his Lab in London.

- Joseph Henry discovered the

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<b>Stationary Loop in Time-Varying B Field</b>

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<b>Moving Conductor in a Static B Field</b>

<i><b>Magnetic force on charge q moving </b></i>

<b>with velocity u in a magnetic field </b>

<b>B:</b>

This magnetic force is equivalent to the

electrical force that would be exerted on
<b>the particle by the electric field E_m</b> 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
<i><b>called a motional emf </b></i>

(6.22)

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<i>Luong Vinh Quoc Danh</i> 12

<b>Moving Conductor in a Static B Field (cont.)</b>

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<b>Example 6-3: Sliding Bar</b>

The length of the loop is related
<i><b>to u by x</b><b>₀</b><b> = ut. Hence</b></i>

<i><b>Note that B = zB</b></i>

<i>₀</i>

<i><b>x, B increases </b></i>

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<b>EMF in an Airliner</b>

<b>These questions are about the potential difference induced across the wings of </b>

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
<i>horizontal, and has a strength of 1.7 x 10</i>–4 _(T).

<i>The charge on an electron = – 1.6 x 10</i>–19_{C. The aircraft’s speed is 270 m/s. The}

wing span of the aircraft is 60 m.

70°
North

Earth’s

<i>B-field</i>

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<b>EMF in an Airliner (cont.)</b>

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.

<i><b>Vertical component of B field = (1.7 x 10</b></i>–4<i>_{) cos 20° = 1.6 x 10}</i>–4_(T).

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Aircraft have wire static

<i><b>dischargers on their trailing </b></i>

<i><b>wing tips so that the electrical </b></i>

<i><b>charge that builds up in the </b></i>

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.

<i>Luong Vinh Quoc Danh</i> 16

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<i>Luong Vinh Quoc Danh</i> 18

<b>EM Generator EMF</b>

Also:

<b>Segment 3-4 moves with velocity –u. Hence:</b>

As the loop rotates with an angular

<i><b>velocity ω about its own axis, segment 1–2 </b></i>

<b>moves with velocity u given by</b>

(6.31)

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<b>Applications: EMF Sensors</b>

•

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.

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<b>Displacement Current</b>

This term is
conduction
<i><b>current I</b></i><b>_C</b>

This term must
represent a current

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Defne the dispeacement current

as: The dispeacement

current does not

invoeve reae charges;
it is an equivaeent
current that depends
<b>on D/t</b>

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<b>Capacitor Circuit</b>

<b>Given</b>: Wires are perfect

conductors and capacitor
insueator materiae is perfect
dieeectricd

<i>For Surface S</i>₁:

<i> I</i>₁<i> = I</i>_1c<i> + I</i>_1d

<b>(D = 0 in perfect conductor)</b>

<i>For Surface S</i>₂:

<i>I</i>

₂

<i> = I</i>

_2c

<i> + I</i>

_2d

<i>I</i>

_2c<i> = 0 (perfect dielectric)</i>

<b>Conclusion: I₁</b><i><b> = I</b></i><b>₂</b> 22

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<i><b>What role </b></i>

<i><b>What role </b></i>

<i><b>electromagnetcs</b></i>

<i><b>_{electromagnetcs}</b></i>

<i><b> plays in </b></i>

<i><b>_{plays in}</b></i>

<i><b>science, technology, and society?</b></i>

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