ELC 2050: Fields & Wave Propagation

Department of Electronics and Electrical Communications Engineering

Introduced By:Eng. Mohamed Ossama Ashour

E-mail: vert4231@gmail.comFirst term 20-21

Agenda

© Mohamed O. Ashour 2018 Slide 2

• Review on Electromotive Force

• Faraday's Law of Electromagnetic Induction➢ A Moving Conductor in a Static Magnetic Field

➢ A Stationary Circuit in a Time-Varying Magnetic Field

➢ A Moving Circuit in a Time-Varying Magnetic Field

• Inductances and Magnetic Energy

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© Mohamed O. Ashour 2018 Slide 3

Electromotive Force• When any electrical energy source (electric batteries, electric generators,

photovoltaic cells or other devices) is connected in an electric circuit, it provides a driving force for the charge carriers (electrons). This force manifests itself as an equivalent impressed electric field intensity Ei.

• In case of electric batteries, The line integral of Ei from the negative to the positive electrode inside the battery is called the electromotive force (Emf) of the battery.

• The electromotive force of the battery:

Also, remember that electric field can be represented as

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Faraday's Law of Electromagnetic Induction

𝐵 = 𝐵𝑜 𝑢𝑧

A Moving Conductor in a Static (Uniform) Magnetic Field

𝐹𝑚𝑎𝑔 = 𝑞 𝑢 × 𝐵

where 𝐹𝑚𝑎𝑔 is the applied force on the free

moving charges in the conductor.

So, the work done inside the conductor:

.𝐹𝑚𝑎𝑔ׯ 𝑑𝑙 = 𝑞 𝑢 × 𝐵. 𝑑𝑙

The electromotive force: ℰ𝑚𝑜𝑡𝑖𝑜𝑛𝑎𝑙 = ׯ𝐹𝑚𝑎𝑔

𝑞. 𝑑𝑙 = 𝑢 × 𝐵. 𝑑𝑙

Y

XZ

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© Mohamed O. Ashour 2018 Slide 5

Faraday's Law of Electromagnetic Induction

From Faraday’s law:

ℰ𝑖𝑛𝑑𝑢𝑐𝑒𝑑 = −𝑁𝑑𝜙

𝑑𝑡(1)

A Stationary Circuit in a Time-Varying Magnetic Field

From electromotive force definition, (1) & (2):

ℰ𝑖𝑛𝑑𝑢𝑐𝑒𝑑 = 𝐸𝑖𝑛𝑑ׯ . 𝑑𝑙 = −𝑁𝑑

𝑑𝑡.𝐵 𝑑𝑠 = −𝑁

𝑑𝐵

𝑑𝑡. 𝑑𝑠 = −𝑁 ሶ𝐵 . 𝑑𝑠

From Stokes’ theorem: 𝛻 × 𝐸𝑖𝑛𝑑 = − ሶ𝐵 “Rotational Field”

From the definition of the time varying flux:𝜙 = .𝐵 𝑑𝑠 (2)

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Faraday's Law of Electromagnetic Induction

This is the most general case where its solution is a combination of the pervious 2 cases.

A Moving Circuit in a Time-Varying Magnetic Field

ℰ𝑖𝑛𝑑𝑢𝑐𝑒𝑑 = −𝑁න𝑑𝐵

𝑑𝑡. 𝑑𝑠

ℰ𝑡𝑜𝑡𝑎𝑙 = ℰ𝑚𝑜𝑡𝑖𝑜𝑛𝑎𝑙 + ℰ𝑖𝑛𝑑𝑢𝑐𝑒𝑑

ℰ𝑚𝑜𝑡𝑖𝑜𝑛𝑎𝑙 = න𝑢 × 𝐵. 𝑑𝑙

So, The time varying magnetic field can induce an Electric field. Which means that we now have 2 sources for 𝐸 :1) Primary sources: free electric charges (produces flowing field).2) Secondary sources: time varying magnetic field (produces rotating field).

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Sheet 3

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Inductances and Inductors

• Consider two neighboring closed loops, 𝐶1 and 𝐶2 bounding surfaces 𝑆1and 𝑆2 respectively. If a current 𝐼1 flows in 𝐶1, a magnetic field 𝑩1 will be

created. Some of the magnetic flux due to 𝑩1 will link with 𝐶2.

• We can define the mutual flux between these 2 loops as:

Φ12 = න𝑆2

𝑩1. 𝑑𝑠2 = 𝐿12𝐼1

where the proportionality constant 𝐿12 is called the mutual inductance between loops 𝐶1 and 𝐶2.

• Generally, We can define the flux linkage as:Λ12 = 𝑁2Φ12ELC 2050

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© Mohamed O. Ashour 2018 Slide 16

Inductances and Inductors

• In case the self inductance of loop 𝐶1 is required, it’s given as:

• The procedure for determining the self-inductance of an inductor is as follows:

1. Choose an appropriate coordinate system for the given geometry.2. Find 𝑩 from 𝐼 (the current in the conducting wire) by Ampere's

circuital law or Biot-Savart law.3. Find the linking magnetic flux,Φ, from B by integration.4. Find the flux linkage Λ by multiplying Φ by the number of turns. 5. Find L by taking the ratio 𝐿 = Λ / 𝐼.

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Magnetic Energy

• The magnetic energy can be represented as

OR

• In case of linear medium, this expression can be represented as

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Sheet 3 Answers

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