Fields and Waves I

Fields and Waves I Lecture 15 Intro to Magnetic Fields K. A. Connor Electrical, Computer, and Systems Engineering Department Rensselaer Polytechnic Institute, Troy, NY Y. Maréchal Power Engineering Department Institut National Polytechnique de Grenoble, France

These Slides Were Prepared by Prof. Kenneth A. Connor Using Original Materials Written Mostly by the Following: • Kenneth A. Connor – ECSE Department, Rensselaer Polytechnic Institute, Troy, NY • J. Darryl Michael – GE Global Research Center, Niskayuna, NY • Thomas P. Crowley – National Institute of Standards and Technology, Boulder, CO • Sheppard J. Salon – ECSE Department, Rensselaer Polytechnic Institute, Troy, NY • Lale Ergene – ITU Informatics Institute, Istanbul, Turkey • Jeffrey Braunstein – Chung-Ang University, Seoul, Korea Materials from other sources are referenced where they are used. Those listed as Ulaby are figures from Ulaby’s textbook. Fields and Waves I

Overview • General considerations on magnetic fields • Magnetostatics & Electrostatics • Similarities • Differences • Ampere’s Law • Magnetic Vector Potential Fields and Waves I

The Earth’s Magnetic Field Fields and Waves I

Navy applications • Compute the magnetic signature of the ship • Reduce magnetic risk in real time Closed loop degaussing Field modulus under the ship http://www.lmn.ensieg.inpg.fr/index/ind_bref.html Fields and Waves I

Experimental means http://www.lmn.ensieg.inpg.fr/index/ind_bref.html Fields and Waves I

Integral Form Differential Form Maxwell’s Equations Magnetostatics 0 0 Fields and Waves I

Introducing B and H fields • Magnetostatic form of Maxwell’s equations • Calculate B and H fields from I and J Integral form Ampere’s Law with m0 as a constant In air, Fields and Waves I

Maxwell’s equations Magnetostatics Electrostatics have curl (rotation) but no divergence (flux) do not have curl (rotation) but have divergence (flux) Fields and Waves I

E-Fields points away from q and towards -q Direction of multiple charges - use superposition http://www.slcc.edu/schools/hum_sci/physics/tutor/2220/e_fields/ Fields and Waves I

Use right-hand rule • thumb along • fingers are in & B-Fields & wraps around Direction of http://encarta.msn.com/media_701504656_761566543_-1_1/Right-Hand_Rule.html multiple wires or segments - use superposition Fields and Waves I

Example 1 Electrostatic or Magnetostatic? Fields and Waves I

Intro to Magnetic Fields Some first applications

Standard Geometries Torus Solenoid http://www.directindustry.fr/prod/lcr-electronics/assemblage-de-cables-electriques-pour-applications-telecom-donnees-35095-214564.html#prod_214564 http://www.magasia.com.tw/inductor.html Fields and Waves I

Standard Geometries http://cbdd.wsu.edu/kewlcontent/cdoutput/tr501/page15.htm http://upload.wikimedia.org/wikipedia/commons/thumb/a/a1/Electronic_component_inductors.jpg/676px-Electronic_component_inductors.jpg http://optical-components.globalspec.com/Industrial-Directory/wave_frontier_toroidal Fields and Waves I

Standard Geometries http://hyperphysics.phy-astr.gsu.edu/hbase/magnetic/solenoid.html http://www.irf.com/technical-info/guide/circuit.html http://www.amethyst-designs.co.uk/Product_Range/Toroidal_transformers.php http://www.cse.iitk.ac.in/users/dheeraj/cs425/lec01.html http://detail.en.china.cn/provide/detail,1065931280.html http://library.thinkquest.org/16600/advanced/ampere.shtml Fields and Waves I

References for Inductors http://hyperphysics.phy-astr.gsu.edu/hbase/magnetic/imgmag/ http://library.thinkquest.org/16600/advanced/ampere.shtml http://neo.lcc.uma.es/cEA-web/population.htm http://www.hut.fi/~then/mytexts/radiohairiot.html Fields and Waves I

Hand Wound Solenoid for Paperclip Launcher Fields and Waves I

Not So Standard Torus International Tokamak Experimental Reactor Planned for nuclear fusion research Note standard person http://www.plasma.inpe.br/LAP_Portal/LAP_Site/Text/Tokamak_Development.htm Fields and Waves I

Intro to Magnetic Fields Calculating the B and H fields with Ampere’s law

Ampere’s law Ampere’s law Fields and Waves I

Example 2 • Three standard geometries for analytical magnetostatic calculations are shown on the next slide. • Use the right hand rule (thumb along the current direction, fingers for B) and determine the direction of B in each case. • All 3 geometries can best be analyzed in cylindrical coordinates. For each, determine whether B is a function of r, φ, and/or z. (Example from electric fields, E of cylindrically symmetric charge is only a function of r.) • Add up B-field for different segments - see what cancels and what adds up - use symmetry Fields and Waves I

Example 2 Add up B-field for different segments - see what cancels and what adds up - use symmetry Fields and Waves I

Example 2: Case of the solenoid (a) Loosely wound solenoid (b) Tightly wound solenoid (c) Infinite tightly wound solenoid Ulaby Fields and Waves I

Example 2 Recall that for the E field, source distributions that only depended on cylindrical radius (r), produced E fields that only depended on r and only had an r component. For the B field, source distributions that only depend on cylindrical r also produce B fields that only depend on r, but B has components in directions perpendicular to r. Fields and Waves I

Ampere’s Law - solve for B & H Approach similar to using Gauss’ Law, use symmetry to get B-field out of integral or Example: Consider an infinite wire solenoid sectional view Fields and Waves I

4 2 3 1 Ampere’s Law - solve for B & H Find Solenoid has current I through n turns/length STEP 1: Choose path for integral - • Chosen paths are 1,2, 3 and 4 - they form a closed loop Fields and Waves I

Ampere’s Law - Infinite Solenoid STEP 2: Evaluate • Segments 2 and 4 have (will show later) • Segment 3 has arbitrary length • Segment 1 has Fields and Waves I

Ampere’s Law - Infinite Solenoid STEP 3: find Inet • current passing through loop : STEP 4: solve for Fields and Waves I

Example 3 – Ampere’s Law Fields and Waves I

Using Ampere’s Law • Just like with Gauss’ Law, a great deal of symmetry is necessary to use Ampere’s Law to find B or H. • Simplify everything before attempting a solution. • Applicability is limited, but this technique is still very useful. • There is an analog to using the electric potential, although for B, it is a bit more complex since it involves a vector potential instead of a scalar potential. It is still easier since the vector potential is in the direction of the current. Fields and Waves I

Intro to Magnetic Fields Magnetic vector potential

, for any • this means that you can find , with possible • more than one • like , can be easier to work with than the field Magnetic Vector Potential, In electrostatics: In magnetostatics: • there is a math theorem that states : Note: Fields and Waves I

Example 4 – Magnetic Vector Potential Fields and Waves I

This surface integral encloses the volume Flux and Vector Potential, Previously we used: Now we will look at the effect of Recall, Fields and Waves I

Physical meaning The flux is conservative: flux coming in = flux going out Fields and Waves I

In contrast, static lines start & end on charges lines close on themselves (no beginning or end) Physical meanings of conservative flux Flux lines of circular coil Fields and Waves I

, enters from left and leaves to the right Define Flux, Concept of Flux Tubes (lines) Field lines Incoming field outgoing field Along sides: Fields and Waves I

Example of flux tubes (lines) cross section The same flux passes through both surfaces since they are in the same flux tube Fields and Waves I

is related to Flux Flux and Vector Potential After some math…. Alternative way to find FLUX Fields and Waves I

Example 5 – Magnetic Vector Potential and flux Fields and Waves I

Fields and Waves I