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Lecture 9 Vector Magnetic Potential Biot Savart Law

Lecture 9 Vector Magnetic Potential Biot Savart Law. Prof. Viviana Vladutescu. Figure 1: The magnetic ( H -field) streamlines inside and outside a single thick wire. . Figure 2: The H -field magnitude inside and outside the thick wire with uniform current density .

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Lecture 9 Vector Magnetic Potential Biot Savart Law

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  1. Lecture 9 Vector Magnetic PotentialBiot Savart Law Prof. Viviana Vladutescu

  2. Figure 1: The magnetic (H-field) streamlines inside and outside a single thick wire.

  3. Figure 2: The H-field magnitude inside and outside the thick wire with uniform current density

  4. Figure 3: The H-field magnitude inside and outside the thick conductors of a coaxial line.

  5. Vector Magnetic Potential A - vector magnetic potential (Wb/m)

  6. Figure 1: The vector potential in the cross-section of a wire with uniform current distribution.

  7. Figure 2: Comparison between the magnetic vector potential component  of a wire with uniformly distributed current and the electric potential V of the equivalent cylinder with uniformly distributed charge.

  8. Poisson’s Equation Laplacian Operator (Divergence of a gradient) Vector Poisson’s equation

  9. In electrostatics Poisson’s Equation in electrostatics

  10. Magnetic Flux The line integral of the vector magnetic potential A around any closed path equals the total magnetic flux passing through area enclosed by the path

  11. Biot Savart Law and Applications

  12. The Biot-Savart Law relates magnetic fields to the currents which are their sources. In a similar manner, Coulomb’s Law relates electric fields to the point charges which are their sources. Finding the magnetic field resulting from a current distribution involves the vector product, and is inherently a calculus problem when the distance from the current to the field point is continuously changing.

  13. Biot-Savart Law By using (see eq 6.31)

  14. In two steps

  15. Illustration of the law of Biot–Savart showing magnetic field arising from a differential segment of current.

  16. Example1 Component values for the equation to find the magnetic field intensity resulting from an infinite length line of current on the z-axis. (ex 6-4)

  17. Example 2 We want to find H at height h above a ring of current centered in the x – y plane.

  18. The component values shown for use in the Biot–Savart equation.

  19. The radial components of H cancel by symmetry.

  20. Solenoid Many turns of insulated wire coiled in the shape of a cylinder.

  21. For a set N number of loops around a ferrite core, the flux generated is the same even when the loops are bunched together.

  22. a b Example : A simple toroid wrapped with N turns modeled by a magnetic circuit. Determine B inside the closely wound toroidal coil.

  23. Ampere’s Law

  24. Electromagnets a) An iron bar attached to an electromagnet. b) The bar displaced by a differential length d.

  25. Applications Levitated trains: Maglev prototype Electromagnet supporting a bar of mass m.

  26. Wilhelm Weber (1804-1891). Electromagnetism.

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