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Fall 2016 ECE 305 Electromagnetic Theory Chapter 8 Magnetic Forces, Materials, and Devices Qiliang Li Dept. of Electrical and Computer Engineering, George Mason University, Fairfax, VA 1 • 8.1 Introduction 2 • 8.2 Forces due to Magnetic Fields 3 4 • Read example 8.1, 8.2 • Example 8.3 5 • Example 8.4 6 §8.3 Magnetic Torque and Moment The torque T on the loop is the vector product of the moment arm r and the force F . 7 8 9 §8.4 A magnetic Dipole • A bar magnet or a small filamentary current loop is referred as a magnetic dipole – Magnetic vector potential 10 §8.4 A magnetic Dipole Use: 11 FIGURE 8.7 The B lines due to magnetic dipoles: (a) a small current loop with m = IS, (b) a bar magnet with m = Qmℓ. 12 §8.5 Magnetization in Materials • The magnetization M, in amperes per meter, is the magnetic dipole moment per unit volume 13 • In the materials FIGURE 8.12 Magnetic dipole moment in a volume ∆ν: (a) before B is applied, (b) after B is applied. 14 • M is defined as 15 §8.6 Classification of Materials 16 §8.7 Magnetic Boundary Conditions 17 §8.8 Inductors and Inductances • Magnetic flux: Ψ = 𝑩 ∙ 𝑑𝑺 • Flux linkage 𝜆 = 𝑁Ψ • If the medium is linear, 𝜆 ∝ 𝐼 or 𝜆 = 𝐿𝐼 𝜆 𝐼 • L is a constant – inductance 𝐿 = = • Magnetic energy 𝑊𝑚 = • Or 𝐿 = 𝑁Ψ 𝐼 1 𝐿 𝐼2 2 2𝑊𝑚 𝐼2 18 continue • Consider magnetic interaction between two circuits: mutual inductance M12 is defined: 𝜆12 𝑁Ψ12 𝑀12 = = 𝐼2 𝐼2 • The energy is 1 1 2 2 𝑊𝑚 = 𝐿1 𝐼1 + 𝐿2 𝐼2 ± 𝑀12 𝐼1 𝐼2 2 2 • Inductance L=Lin + Lext • It can be shown that: 𝐿𝑒𝑥𝑡 𝐶 = 𝜇𝜀 * + or – sign depends on I1 and I2 flow such that the B fields of the two circuits strengthen each other or not. 19 Procedure to find the self-inductance L: • Choose a suitable coordinate system • Let the inductor carry current I • Determine B from Biot-Savart’s law (or from Ampere’s law if symmetry exits), then calculate Ψ = 𝑩 ∙ 𝑑𝑺 𝜆 𝐼 • Finally, find L from 𝐿 = = 𝑁Ψ 𝐼 20 §8.9 Magnetic Energy • In electric field 𝑊𝐸 = 1 2 𝑫 ∙ 𝑬𝑑𝑉 = 1 2 𝜖𝐸 2 𝑑𝑉 • Similarly, magnetic energy in the field of an 1 inductor is 𝑊𝑚 = 𝐿 𝐼2 2 • Or 𝑊𝑚 = 𝑤𝑚 𝑑𝑉, Where 𝑤𝑚 = 1 𝑩 2 ∙𝑯= 1 𝜇𝐻2 2 21 22 • What is the magnetic energy in a circuit? or or magnetic energy 23 Example 8.10 calculate the self-inductance per unit length of an infinitely long solenoid Solve: 𝑁𝐼 𝐵 = 𝜇𝐻 = 𝜇𝐼𝑛 = 𝜇 X X X X X X X X 𝑙 (remember K=current/length) Ψ = 𝐵𝑆 = 𝜇𝐼𝑛S ′ 𝜆 𝑙 Linkage per unit length 𝜆 = = 𝑛Ψ = 𝜇𝑛2 𝐼𝑆 So 𝐿′ = 𝜆′ 𝐼 = 𝜇𝑛2 𝑆 (H/m) 24 Example 8.11 Determine the self-inductance of a coaxial cable of inner r=a and outer r=b. Solve: From previous example X Eq. 7.29 𝜇𝐼𝜌 2 𝑩𝟏 = 𝒂𝝓 1 2 2𝜋𝑎 𝜇𝐼 𝑩𝟐 = 𝒂𝝓 2𝜋𝜌 25 Method 1 26 27 Method 2 This method is more straight forward. 28 Example 8.12: determine the inductance per unit length of a two-wire transmission line with separation distance d. Each wire with r=a. Solve: Method 1 d 29 Method 2 L = 2(𝐿𝑖𝑛 + 𝐿𝑒𝑥𝑡 ) 30 Example 8.13 Two coaxial circular wires of radius a and b (b>a) are separated by distance h (h>>a, b)as shown in Fig. 8.13. Find the mutual inductance between the wires Solve: b P h a 31 32 • Section 8.10, 8.11 and 8.12: self-reading • Example of application is levitating trains 33 §8.12 Application Note – Hall Effect What is Hall Effect? Which way should the charges go? 34