1. Chung-Ang University Field & Wave Electromagnetics CH 6. Static Magnetic Fields

2. Chung-Ang University Field & Wave Electromagnetics 6-1 Introduction Static Electric Field Magnetic Force Electric Force q CD Pickup Control Fleming’s Left hand rule “Hunt for Red October” (magneto-hydro-dynamics) Particle accelerator (cyclotron) CRT monitor

3. Chung-Ang University Field & Wave Electromagnetics Nicola Tesla – extra-terrestrial? Have you ever heard of Nikola Tesla? You should get to know him. Maybe he was a brother - specialized in energy. He invented the Tesla Coil and was instrumental in discovering ways of propagating energy (electrical) wirelessly and over wires in the most efficient manner. A book on him has pictures and a lot of information on his many brilliant devices. There seems to be a lot of speculation that either he was a walk- in, arrived on our planet as a child (left on a doorstep), was in touch with higher beings from Mars, etc. There are several theories to explain his brilliance - all involving ETs in one way or another. Nikola Tesla was born in 1856 in Smiljan Lika, Croatia. He was the son of a Serbian Orthodox clergyman. Tesla studied engineering at the Austrian Polytechnic School. He worked as an electrical engineer in Budapest and later emigrated to the United States in 1884 to work at the Edison Machine Works. He died in New York City on January 7, 1943. During his lifetime, Tesla invented fluorescent lighting, the Tesla induction motor, the Tesla coil, and developed the alternating current (AC) electrical supply system that included a motor and transformer, and 3-phase electricity. – How come? Surprised!Tesla coil Tesla is now credited with inventing modern radio as well; since the Supreme Court overturned Guglielmo Marconi's patent in 1943 in favor of Nikola Tesla's earlier patents.Guglielmo Marconi's Believe or not?!– He invented a converter that converts cosmic energy to electrical energy and he mounted the converter on a car to drive electric car!

4. Chung-Ang University Field & Wave Electromagnetics Nicola Tesla – extra-terrestrial? high voltage discharge experiments, Colorado Springs Laboratory. Dec 31, 1899

5. Chung-Ang University Field & Wave Electromagnetics General Three-Beam Optical Pickup Organization

6. Chung-Ang University Field & Wave Electromagnetics 6-1 Introduction Elecromagnetic Force Lorentz’s Force Equation There is no constant such as !! ☞ are the fundamental quantities, not or !! With the help of the four Maxwell equations, the equation of continuity, and the Lorentz force equation, we can now explain all of the electromagnetic phenomena! Maxwell equations ( Ch. 7) the equation of continuity (Ch. 5) the Lorentz force equation (Ch.6)

7. Chung-Ang University Field & Wave Electromagnetics 6-2 Fundamental Postulates of Magnetostatics in Free Space “There are no magnetic flow sources, (no magnetic monopole) and the magnetic flux lines always close upon themselves” The Law of conservation of magnetic flux N S S N S N S N S N Magnetic poles cannot be isolated. S N S N S N S N

8. Chung-Ang University Field & Wave Electromagnetics 6-2 Fundamental Postulates of Magnetostatics in Free Space The magnetic flux lines follow closed paths from one end of a magnet to the other end outside the magnet, and then continue inside the magnet back to the first end. Path C is the contour bounding the surface S, I is the total current through S. Ampere’s Circuital Law the circulation of the magnetic flux density in free space around any closed path is equal to times the total current flowing through the surface bounded by the path. Ex.6-1, -2, -3, p228 (6-10)

9. Chung-Ang University Field & Wave Electromagnetics Ex.6-1 b I Function of position!!! x y

10. Chung-Ang University Field & Wave Electromagnetics Magnetic Flux Density Inside a Closely Wound Toroidal Coil (Ex. 6-2) 1. In order to calculate B, contour C should be taken such that B is constant on the contour! B=0 2. Determine coordinate system and the direction of B. 3. Integrate and calculate B. r BB b - ab+ a

11. Chung-Ang University Field & Wave Electromagnetics Infinitely long solenoid can be considered as a part of toroidal coil of infinite radius. Or as a special case of toroid -no magnetic field outside -B field inside must be parallel to the axis. Magnetic Flux Density Inside an Infinitely Long Solenoid (Ex. 6-3) 1. In order to calculate B, contour C should be taken such that B is constant on the contour! 2. Determine coordinate system and the direction of B. 3. Integrate and calculate B. n turns/m

12. Chung-Ang University Field & Wave Electromagnetics 6-3 Vector Magnetic Potential A vector field is determined to within an additive constant if both its divergence and its curl are specified every where. How to find a divergence of So, Laplacian of Vector Magnetic Potential We know that ☞ Helmholtz’s theorem Poisson’s EquationLaplace’s Equation Can we determine A with this equation?

13. Chung-Ang University Field & Wave Electromagnetics 6-3 Vector Magnetic Potential The Laplacian of a vector field is another vector field whose components are the Laplacian of the corresponding components of. Therefore, With the purpose of simplifying above equation to the greatest extent possible, Coulomb condition (gauge) vector Poisson’s equation In Cartesian coordinates, and becomes Now, we have

14. Chung-Ang University Field & Wave Electromagnetics 6-3 Vector Magnetic Potential Poisson’s equation in electrostatics So, we have the solution for This enables us to find the vector magnetic potential from the volume current density. Source point Field point

15. Chung-Ang University Field & Wave Electromagnetics 6-3 Vector Magnetic Potential Magnetic Flux Vector potential relates to the magnetic flux through a given area S that is bounded by contour C in a simple way; Thus, vector magnetic potential does have physical significance in that its line integral around any closed path equals the total magnetic flux passing through the area enclosed by the path.

16. Chung-Ang University Field & Wave Electromagnetics 6-4 The Biot-Savart Law We are interested in determining the magnetic field due to a current-carrying circuit. For a thin wire with cross-sectional area S, we have Magnetic flux density is then Unprimed curl operation implies differentiations with respect to the space coordinates of the field point, and the integral operation is with respect to the primed source coordinates.

17. Chung-Ang University Field & Wave Electromagnetics 6-4 The Biot-Savart Law we use the following identity So, Magnetic flux density is equal 0 Biot-Savart Law Sometimes it is convenient to write above equation in two steps: Ex. 4,5,6 p236

18. Chung-Ang University Field & Wave Electromagnetics B from a Current-Carrying Straight Wire (Ex. 6-4) (I)

19. Chung-Ang University Field & Wave Electromagnetics B from a Current-Carrying Straight Wire (Ex. 6-4) (II) (b) By applying Biot-Savart law Which method do you like, or using Biot-Savart law? It is easier to use to find E. Why?

20. Chung-Ang University Field & Wave Electromagnetics Magnetic flux density at the center of the square loop is equal to four times that caused by a single side of length L. Using the result of Ex. (6-4), B at the Center of a Square Loop (Ex. 6-5) Converting the direction  to z and multiplying 4, However, it takes considerable efforts to calculate B other than center!

21. Chung-Ang University Field & Wave Electromagnetics Cylindrical symmetry : only consider z-component B at a Point on the Axis of a Circular Loop (Ex. 6-6) Apply Biot-Savart law to the circular loop