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CH 6. Static Magnetic Fields

CH 6. Static Magnetic Fields. q. 6-1 Introduction. Electric Force. Static Electric Field. Magnetic Force. Fleming’s Left hand rule. CD Pickup Control. “Hunt for Red October” (magneto-hydro-dynamics). Particle accelerator (cyclotron) CRT monitor. Nicola Tesla – extra-terrestrial?.

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CH 6. Static Magnetic Fields

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  1. CH 6. Static Magnetic Fields

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

  3. 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 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. • 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. Nicola Tesla – extra-terrestrial? high voltage discharge experiments, Colorado Springs Laboratory. Dec 31, 1899

  5. General Three-Beam Optical Pickup Organization

  6. There is no constant such as !! • ☞ are the fundamental quantities, not or !! 6-1 Introduction • Elecromagnetic Force Lorentz’s Force Equation 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. N N N S N N S N S N S S N S S N S S 6-2 Fundamental Postulates of Magnetostatics in Free Space The Law of conservation of magnetic flux “There are no magnetic flow sources, (no magnetic monopole) and the magnetic flux lines always close upon themselves” Magnetic poles cannot be isolated.

  8. 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. 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. (6-10) Path C is the contour bounding the surface S, I is the total current through S. Ampere’s Circuital Law Ex.6-1, -2, -3, p228

  9. Function of position!!! Ex.6-1 x b y I Function of position!!!

  10. Bf r b - a b+ a Magnetic Flux Density Inside a Closely Wound Toroidal Coil (Ex. 6-2) B=0 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. B=0 3. Integrate and calculate B.

  11. 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 Infinitely long solenoid can be considered as a part of toroidal coil of infinite radius. • no magnetic field outside • B field inside must be parallel to the axis. Or as a special case of toroid

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

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

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

  15. 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 integralaround any closed path equals the total magnetic flux passing through the area enclosed by the path. 6-3 Vector Magnetic Potential

  16. 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. 6-4 The Biot-Savart Law we use the following identity equal 0 So, Magnetic flux density is Biot-Savart Law Sometimes it is convenient to write above equation in two steps: Ex. 4,5,6 p236

  18. B from a Current-Carrying Straight Wire (Ex. 6-4) (I)

  19. 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? to find E. Why? It is easier to use

  20. B at the Center of a Square Loop (Ex. 6-5) 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), Converting the direction f to z and multiplying 4, However, it takes considerable efforts to calculate B other than center!

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

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