Magnetic Fields and Magnetic Field Lines

3/5/2008 1 GENERAL PHYSICS II Electromagnetism & Thermal Physics 3/5/2008 2 Chapter X Magnetic Field §1. Magnetic interaction and magnetic field §2. Magnetic forces on a moving charged particle and on a current-carrying conductor §3. Magnetic field of a current – magnetic field calculations §4. Amper’s law and application 3/5/2008 3 §1. Magnetic interaction and magnetic field 1.1 Magnetic phenomena:  Some history:  Magnetic effects from natural magnets have been known for a long time. Recorded observations from the Greeks more than 2500 years ago.  The word magnetism comes from the Greek word for a certain type of stone (lodestone) containing iron oxide found in Magnesia, a district in northern Greece.  Properties of lodestones: could exert forces on similar stones and could impart this property (magnetize) to a piece of iron it touched.  Bar magnet: a bar-shaped permanent magnet. It has two poles: N and S Like poles repel; Unlike poles attract. We say that the magnets can interact each with other. This kind of interaction differs from electric interactions, and is called magnetic interaction 3/5/2008 4  We have known that the means of transfering interactions between electric charges is electric field. By analogy to electric interaction we introduce for magnetic interaction the concept of magnetic field which is the means of transfering magnetic interactions: A magnet sets up a magnetic field in the space around it and the second magnet responds to that field.  The direction of the magnetic field at any point is defined as the direction of the force that the field would exert on a magnetic north pole of compass needle The earth itself is a magnet. Note that for the earth magnet: the geographical pole ≠the magnetic pole N S North geographic pole South magnetic pole South geographic pole North magnetic pole N S 3/5/2008 5 1.2 Magnetic field vector and magnetic field lines: By analogy to electric field vector E we can introduce magnetic field vector B : + The direction of magnetic field vector at each point in the space can be defined experimentally by a compass + The mathematical expression for magnetic field vector (magnitude and direction) will be defined below (the law of Biot and Savart) Magnetic field lines can be drawn in the same manner as electric field lines (direction and density) NS 3/5/2008 6 Magnetic Field Lines of a bar magnet Electric Field Lines of an Electric Dipole NS 3/5/2008 7 Magnetic Monopoles ?  Perhaps there exist magnetic charges, just like electric charges. Such an entity would be called a magnetic monopole (having + or - magnetic charge).  How can you isolate this magnetic charge? Try cutting a bar magnet in half: • Many searches for magnetic monopoles no monopoles have ever been found ! NS Even an individual electron has a magnetic “dipole”! N NS S 3/5/2008 8 Source of Magnetic Fields?  What is the source of magnetic fields, if not magnetic charge?  Answer: electric charge in motion!  e.g., current in wire surrounding cylinder (solenoid) produces very similar field to that of bar magnet.  Therefore, understanding source of field generated by bar magnet lies in understanding currents at atomic level within bulk matter Motions of electrons on orbits and intrinsic motions produce magnetic field. Orbits of electrons about nuclei Intrinsic “spin” of electrons (more important effect) 3/5/2008 9 • The force F on a charge q moving with velocity v through a region of space with magnetic field B is given by: F x x x x Magnetic Fields and Magnetic Field Lines Magnetic Fields and Magnetic Field Lines Bởi: OpenStaxCollege Einstein is said to have been fascinated by a compass as a child, perhaps musing on how the needle felt a force without direct physical contact His ability to think deeply and clearly about action at a distance, particularly for gravitational, electric, and magnetic forces, later enabled him to create his revolutionary theory of relativity Since magnetic forces act at a distance, we define a magnetic field to represent magnetic forces The pictorial representation of magnetic field lines is very useful in visualizing the strength and direction of the magnetic field As shown in [link], the direction of magnetic field lines is defined to be the direction in which the north end of a compass needle points The magnetic field is traditionally called the B-field Magnetic field lines are defined to have the direction that a small compass points when placed at a location (a) If small compasses are used to map the magnetic field around a bar magnet, they will point in the directions shown: away from the north pole of the magnet, toward the south pole of the magnet (Recall that the Earth’s north magnetic pole is really a south pole in terms of definitions of poles on a bar magnet.) (b) Connecting the arrows gives continuous magnetic field lines The strength of the field is proportional to the closeness (or density) of the lines (c) If the interior of the magnet could be probed, the field lines would be found to form continuous closed loops Small compasses used to test a magnetic field will not disturb it (This is analogous to the way we tested electric fields with a small test charge In both cases, the fields represent only the object creating them and not the probe testing them.) [link] shows how the magnetic field appears for a current loop and a long straight wire, as could be 1/3 Magnetic Fields and Magnetic Field Lines explored with small compasses A small compass placed in these fields will align itself parallel to the field line at its location, with its north pole pointing in the direction of B Note the symbols used for field into and out of the paper Small compasses could be used to map the fields shown here (a) The magnetic field of a circular current loop is similar to that of a bar magnet (b) A long and straight wire creates a field with magnetic field lines forming circular loops (c) When the wire is in the plane of the paper, the field is perpendicular to the paper Note that the symbols used for the field pointing inward (like the tail of an arrow) and the field pointing outward (like the tip of an arrow) Making Connections: Concept of a Field A field is a way of mapping forces surrounding any object that can act on another object at a distance without apparent physical connection The field represents the object generating it Gravitational fields map gravitational forces, electric fields map electrical forces, and magnetic fields map magnetic forces Extensive exploration of magnetic fields has revealed a number of hard-and-fast rules We use magnetic field lines to represent the field (the lines are a pictorial tool, not a physical entity in and of themselves) The properties of magnetic field lines can be summarized by these rules: The direction of the magnetic field is tangent to the field line at any point in space A small compass will point in the direction of the field line The strength of the field is proportional to the closeness of the lines It is exactly proportional to the number of lines per unit area perpendicular to the lines (called the areal density) Magnetic field lines can never cross, meaning that the field is unique at any point in space Magnetic field lines are continuous, forming closed loops without beginning or end They go from the north pole to the south pole The last property is related to the fact that the north and south poles cannot be separated It is a distinct difference from electric field lines, which begin and end on the positive 2/3 Magnetic Fields and Magnetic Field Lines and negative charges If magnetic monopoles existed, then magnetic field lines would begin and end on them Section Summary • Magnetic fields can be pictorially represented by magnetic field lines, the properties of which are as follows: The field is tangent to the magnetic field line Field strength is proportional to the line density Field lines cannot cross Field lines are continuous loops Conceptual Questions Explain why the magnetic field would not be unique (that is, not have a single value) at a point in space where magnetic field lines might cross (Consider the direction of the field at such a point.) List the ways in which magnetic field lines and electric field lines are similar For example, the field direction is tangent to the line at any point in space Also list the ways in which they differ For example, electric force is parallel to electric field lines, whereas magnetic force on moving ...RESEARCH ARTIC LE Open Access Detection of breast cancer cells using targeted magnetic nanoparticles and ultra-sensitive magnetic field sensors Helen J Hathaway 1,2*† , Kimberly S Butler 3† , Natalie L Adolphi 2,4 , Debbie M Lovato 3 , Robert Belfon 1 , Danielle Fegan 5 , Todd C Monson 6 , Jason E Trujillo 3,5 , Trace E Tessier 5 , Howard C Bryant 5 , Dale L Huber 7 , Richard S Larson 2,3 and Edward R Flynn 2,5 Abstract Introduction: Breast cancer detection using mammography has improved clinical outcomes for many women, because mammography can detect very small (5 mm) tumors early in the course of the disease. However, mammography fails to detect 10 - 25% of tumors, and the results do not distinguish benign and malignant tumors. Reducing the false positive rate, even by a modest 10%, while improving the sensitivity, will lead to improved screening, and is a desirable and attainable goal. The emerging application of magnetic relaxometry, in particular using superconducting quantum interference device (SQUID) sensors, is fast and potentially more specific than mammography because it is designed to detect tumor-targeted iron oxide magnetic nanoparticles. Furthermore, magnetic relaxometry is theoretically more specific than MRI detection, because only target-bound nanoparticles are detected. Our group is developing antibody-conjugated magnetic nanoparticles targeted to breast cancer cells that can be detected using magnetic relaxometry. Methods: To accomplish this, we identified a series of breast cancer cell lines expressing varying levels of the plasma membrane-expressed human epidermal growth factor-like receptor 2 (Her2) by flow cytometry. Anti-Her2 antibody was then conjugated to superparamagnetic iron oxide nanoparticles using the carbodiimide method. Labeled nanoparticles were incubated with breast cancer cell lines and visualized by confocal microscopy, Prussian blue histochemistry, and magnetic relaxometry. Results: We demonstrated a time- and antigen concentration-dependent increase in the number of antibody- conjugated nanoparticles bound to cells. Next, anti Her2-conjugated nanoparticles injected into highly Her2- expressing tumor xenograft explants yielded a significantly higher SQUID relaxometry signal relative to unconjugated nanoparticles. Finally, labeled cells introduced into breast phantoms were measured by magnetic relaxometry, and as few as 1 million labeled cells were detected at a distance of 4.5 cm using our early prototype system. Conclusions: These results suggest that the antibody-conjugated magnetic nanoparticles are promising reagents to apply to in vivo breast tumor cell detection, and that SQUID-detected magnetic relaxometry is a viable, rapid, and highly sensitive method for in vitro nanoparticle development and eventual in vivo tumor detection. * Correspondence: hhathaway@salud.unm.edu † Contributed equally 1 Department of Cell Biology & Physiology, University of New Mexico School of Medicine, MSC08 4750, 1 University of New Mexico, Albuquerque, NM 87131, USA Full list of author information is available at the end of the article Hathaway et al. Breast Cancer Research 2011, 13:R108 http://breast-cancer-research.com/content/13/5/R108 © 2011 Hathaway et al.; licensee BioMed Central Ltd. This is an open access article distributed under the terms of the Creative Commons Attribution License (http://cre ativecommons.org/licenses/by/2.0), which permits unrestricted use, distribution, and reproduction in any medium, provided the original work Proc Natl Conf Theor Phys 37 (2012), pp 115-120 IMPACT OF THE EXTERNAL MAGNETIC FIELD AND THE CONFINEMENT OF PHONONS ON THE NONLINEAR ABSORPTION COEFFICIENT OF A STRONG ELECTROMAGNETIC WAVE BY CONFINED ELECTRONS IN COMPOSITIONAL SUPERLATTICES Hoang Dinh Trien, Le Thai Hung, Vu Thi Hong Duyen, Nguyen Quang Bau Department of Physics, University of Natural Sciences, Hanoi National University Nguyen Thu Huong, Nguyen Vu Nhan Academy of Air Defense and Air Force, Son Tay, Hanoi, Vietnam Abstract Impact of the external magnetic field and the confinement of phonon on the nonlinear absorption coefficient (NAC) of a strong electromagnetic wave (EMW) by confined electrons in compositional superlattices is theoretically studied by using the quantum transport equation for electrons The formula which shows the dependence of the NAC on the energy ( Ω), the intensity E0 of EMW, the energy ( ΩB ) of external magnetic field and quantum number m characterizing confined phonon is obtained The analytic expressions are numerically evaluated, plotted and discussed for a specific of the GaAs − Al0.3 Ga0.7 As compositional superlattices The results show clearly the difference in the spectrums and values of the NAC in this case from those in the case without the impact of the external magnetic field and the confinement of phonon I INTRODUCTION Recently, there are more and more interests in studying the behavior of low-dimensional system, such as compositional superlattices, doped superlattices, compositional superlattices, quantum wires and quantum dots The confinement of electrons and phonons in low-dimensional systems considerably enhances the electron mobility and leads to unusual behaviors under external stimuli Many attempts have been conducted dealing with these behaviors, for examples, electron-phonon interaction effects in two-dimensional electron gases (graphene, surfaces, quantum wells) [1, 2, 3] The dc electrical conductivity [4, 5], the electronic structure [6], the wavefunction distribution [7] and the electron subband [8] in quantum wells have been calculated and analyzed The problems of the absorption coefficient for a weak electromagnetic wave in quantum wells [9], in doped superlattices [10] have also been investigated by using Kubo-Mori method The nonlinear absorption of a strong electromagnetic wave in low-dimensional systems have been studied by using the quantum transport equation for electrons [11] However, the nonlinear absorption of a strong electromagnetic wave in compositional superlattices in the presence of an external magnetic field with influences of confined phonons is still open question In this paper, we consider quantum theories of the nonlinear absorption of a strong electromagnetic wave caused by confined electrons in the presence of an external magnetic field in low dimensional systems taking into account the effect of confined phonons The problem is considered for the case of electron-optical phonon scattering Analytical expressions of 116 HOANG DINH TRIEN, LE THAI HUNG, the nonlinear absorption coefficient of a strong electromagnetic wave caused by confined electrons in the presence of an external magnetic field in low-dimensional systems are obtained The analytical expressions are numerically calculated and discussed to show the differences in comparison with the case of absence of an external magnetic with a specific of the GaAs − Al0.3 Ga0.7 As compositional superlattices II CALCULATIONS OF THE NONLINEAR ABSORPTION COEFFICIENT OF A STRONG ELECTROMAGNETIC WAVE BY CONFINED ELECTRONS IN A COMPOSITIONAL SUPERLATTICE IN THE PRESENCE OF A MAGNETIC FIELD IN CASE OF CONFINED PHONONS It is well known that in the compositional superlattices, the motion of electrons is restricted in one dimension, so that they can flow freely in two dimensions In Proc Natl Conf Theor Phys 37 (2012), pp 86-91 TRANSPORT PROPERTIES OF A QUASI-TWO-DIMENSIONAL ELECTRON GAS IN A Si/SiGe HETEROSTRUCTURE: TEMPERATURE AND MAGNETIC FIELD EFFECTS NGUYEN QUOC KHANH, NGUYEN NGOC THANH NAM Department of Theoretical Physics, National University in Ho Chi Minh City, 227-Nguyen Van Cu Street, 5th District, Ho Chi Minh City, Vietnam Abstract We investigate the mobility and resistivity of a quasi-two-dimensional electron gas (Q2DEG) in a Si/SiGe heterostructure (HS) at arbitrary temperatures for two cases: with and without in-plane magnetic field We consider two scattering mechanisms: charged impurity and interface-roughness scattering We study the dependence of the mobility on the temperature, magnetic field, carrier density, impurity concentration and position At low temperatures our results reduce to those of Gold (Semicond Sci Technol 26, 045017 (2011)) Our results and new measurements of transport properties can be used to obtain information about the scattering mechanisms in the Si/SiGe heterostructures I INTRODUCTION Recently Gold has calculated the zero temperature mobility of the two-dimensional electron gas in Si/SiGe heterostructures for charged-impurity scattering, and interfaceroughness scattering [1] He has included the exchange-correlation and multiple-scattering effects (MSE) and obtained good agreement with experiment on a metal-insulator transition (MIT)[2, 3, 4, 5] In this paper we generalize Gold’s work to the finite temperature case We show that even at low temperature (T > 0.1TF ) the temperature effect is considerable II THEORY We consider a Q2DEG in the xy-plane with parabolic dispersion We describe extension effects of the Q2DEG perpendicular to the Si/SiGe interface by the triangular potential well using the Howard-Stern expression for the envelope wave function [1, 2] ς0 (z) = 1/2 b3 z exp − bz (1) When the in-plane magnetic field B is applied to the system, the carrier densities for spin up/down are not equal [7, 8] At T = we have n B 1± Bs n+ = n, n− = 0, n± = , B < Bs B ≥ Bs (2) TRANSPORT PROPERTIES OF A Q2D ELECTRON GAS IN A Si/SiGe HETEROSTRUCTURE 87 Here n = n+ + n− is the total density and Bs is the so-called saturation field given by Bs = 2EF /(gµB ), where g is the electron spin g-factor and µB is the Bohr magneton For T > 0, n± is determined using the Fermi distribution function and given by n+ = − e2x/t + n tln (e2x/t − 1)2 + 4e(2+2x)/t n− = n − n+ (3) where x = B/Bs and t = T /TF with TF is the Fermi temperature The energy averaged transport relaxation time for the ± components are given in the Boltzmann theory by [7] +∞ τ( +∞ τ± ( ) = m∗ = τ (k) 2π k (q) = + 2k ± ) [− ∂f∂ ( ) ]d (4) ± [− ∂f∂ ( ) ]d |U (q)|2 [ (q)]2 q dq (5) − (q/2k)2 2πe2 FC (q)[1 − G(q)]Π(q, T ) q L (6) Π(q, T ) = Π+ (q, T ) + Π− (q, T ) Π± (q, T ) = β ∞ dµ  ∗ g m ν 1 − Π0± (q, EF ± ) = π FC (q) = with f ± (ε) = 2 q 1+ b 1+exp(β[ε−µ± (T )]) , Here, m∗ is the −3 (7) Π0± (q, µ ) (8) cosh2 ( β2 (µ± − µ )) 1− 2kF ± q q q 8+9 +3 b b β = (kB T )−1 , µ =  θ(q − 2kF ± ) (9) β ln(exp[βEF ± ] (10) − 1) , EF ± = k2 F± 2m∗ and ε = 2mk∗ effective mass in xy-plane and mz is the effective mass perpendicular to the xy-plane, G(q) is the local field correction (LFC) describing the exchange-correlation effects and U (q) is the random potential which depends on the scattering mechanism [1] For charged-impurities of density Ni located on the plane with z = zi we have |UR (q)|2 = Ni 2πe2 Lq FR2 (q, zi ) (11) 88 NGUYEN QUOC KHANH, NGUYEN NGOC THANH NAM ∗ −q|z | 1/3 i 33πnmz e e with qs = 2gνLm 2e , F (q, zi ) = (1+ Here L is the background q and b = 16 L ) b static dielectric constant and gν is the valley degeneracy For the interface-roughness scattering (IRS) the random potential is given by |US (q)|2 = 4π e4 (∆ΛN )2 e−q Λ2 /4 (12) L where ∆ represents the average height of the roughness perpendicular to the Q2DEG and Λ represents the correlation length parameter of the roughness in the plane of the Q2DEG The Electric and magnetic field calculations with finite-element methods Stanley Humphries President, Field Precision LLC Professor Emeritus, University of New Mexico Published by Field Precision LLC PO Box 13595, Albuquerque, NM 87192 U.S.A Telephone: +1-505-220-3975 E mail: techinfo@fieldp.com Internet: http://www.fieldp.com Copyright c 2015 by Field Precision LLC All rights reserved This electronic book may be distributed freely if (and only if) it is distributed in its entirety as the file humphriessfem.pdf Sections of the file, text excerpts and figures may not be reproduced, distributed or posted for download on Internet sites without written permission of the publisher Contents Introduction Installing 2D electric-field software First 2D electrostatic solution 11 Electrostatic application: building the mesh 19 Electrostatic application: calculating and analyzing fields 26 Electrostatic application: meshing and accuracy 30 Magnetostatic solution: simple coil with boundaries 38 Magnetostatic solution: boundary effects and automatic operation 45 Magnetostatic solution: the role of steel 50 10 Magnetostatic solution: when steel gets complicated 56 11 Magnetostatic solution: permanent magnets 64 12 Adding 3D software 70 13 3D electrostatic example: STL input 73 14 3D electrostatic example: mesh generation and solution 77 15 3D electrostatic application: getting started 81 16 3D electrostatic application: extrusions 87 17 3D electrostatic application: mutual capacitance 94 18 3D magnetic fields: defining coil currents 102 19 3D magnetic fields: free-space calculations 109 20 3D magnetic fields: iron and permanent magnets 117 Introduction Field Precision finite-element programs covers a broad spectrum of physics and engineering applications, including charged particle accelerators and X-ray imaging The core underlying most of our software packages is the calculation of electric and magnetic fields over threedimensional volumes To use our electric and magnetic fields software effectively, researchers should have a background in electromagnetism and should be able to make informed decisions about solution strategies First-time users of finite-element software may feel intimidated by these requirements My motivation in writing this book is to share my experience in field calculations I hope to build users knowledge and experience in steps so they can apply finiteelement programs confidently In the end, readers will be able to solve real-world problems with the following programs: • EStat (2D electrostatics) • HiPhi (3D electrostatics) • PerMag (2D magnetostatics) • Magnum (3D magnetostatics) To begin, its important to recognize the difference between 2D and 3D programs All finiteelement programs solve fields in three-dimensions, but often systems have geometric symmetries that can be utilized to reduce the amount of work The term 2D applies to the following cases: • Cylindrical systems with variations in r and z but no variation in θ (azimuth) • Planar systems with variations in x and y and a long length in z Which brings us to the first directive of finite-element calculations: never use a 3D code for a calculation that could be handled by a 2D code The 3D calculation would increase the complexity and run time with no payback in accuracy We need to clarify the meaning of static in electrostatics and magnetostatics The implication is that the fields are constant or vary slowly in time The criterion of a slow variation is that the systems not emit electromagnetic radiation Examples of electrostatic applications are power lines, insulator design, paint coating, ink-jet printing and biological sorting Magnetostatic applications include MRI magnets, ... the north and south poles cannot be separated It is a distinct difference from electric field lines, which begin and end on the positive 2/3 Magnetic Fields and Magnetic Field Lines and negative... negative charges If magnetic monopoles existed, then magnetic field lines would begin and end on them Section Summary • Magnetic fields can be pictorially represented by magnetic field lines, the properties... electric field lines, whereas magnetic force on moving charges is perpendicular to magnetic field lines Noting that the magnetic field lines of a bar magnet resemble the electric field lines of