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

• 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

Content

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Maxwell’s Equations

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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

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Stationary Loop in Time-Varying B Field

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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 Em given by

This, in turn, induces a voltage difference between ends and 2, with end being at the higher potential The induced voltage is called a motional emf

(6.22)

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Moving Conductor in a Static B Field (cont.)

Example 6-3: Sliding Bar

The length of the loop is related to u by x0 = ut Hence

Note that B = zB0 x, B increases

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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

70° North

Earth’s

B-field

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).

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

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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)

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.

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Displacement Current

This term is conduction current IC

This term must represent a current

Defne the dispeacement current

as: The dispeacement

current does not

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

Capacitor Circuit

Given: Wires are perfect

conductors and capacitor insueator materiae is perfect dieeectricd

For Surface S1:

I1 = I1c + I1d

(D = in perfect conductor)

For Surface S2:

I2 = I2c + I2d

I2c = (perfect dielectric)

Conclusion: I1 = I2 22

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What role

What role electromagnetcselectromagnetcs plays in plays in

science, technology, and society?

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