1. Lecture 9 Vector Magnetic Potential Biot 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.

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

10. In electrostatics
Poisson’s Equation
in electrostatics

12. 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

13. Biot Savart Law and Applications

14. 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.

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

17. In two steps

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

19. 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)

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

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

22. The radial components of H cancel by symmetry.

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

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

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

26. Ampere’s Law

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

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

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