An Electron and a Positron System Surroundings e+ e- Release electron and positron – the electron (system) will gain kinetic energy Conservation of energy  surrounding energy must decrease Does the energy of the positron decrease? - No, it increases Where is the decrease of the energy in the surroundings? - Energy stored in the fields must decrease

An Electron and a Positron System Surroundings e+ e- Energy: Single charge: Dipole: Can in principle integrate: run into problem – E is infinity close to the charge! (far) Energy stored in the E fields decreases as e+ and e- get closer!

Potential Energy and Field Energy A different way to express instead of calculating a work it takes to assemble charges. all space The idea of energy stored in fields is a general one: Magnetic and gravitational fields can also carry energy.

Chapter 18 Magnetic Field

Magnetic Field 5th century BC. Greeks. Some rocks attract pieces of iron. These rocks are plentiful in the district of Magnesia. 1100 AD Chinese. Used needles of magnetite to make compasses. 16th century Gilbert – earth is a giant magnet. 1820 rsted – magnetic needle responds to current in the wire.

Magnetic Field A compass needle turns and points in a particular direction there is something which interacts with it Magnetic field (B): whatever it is that is detected by a compass gravitational field – detected by force on mass, Electrical field –detected by force on charge. Compass: similar to electric dipole

Electron Current Magnetic fields are produced by moving charges Current in a wire: convenient source of magnetic field Static equilibrium: net motion of electrons is zero Can make electric circuit with continuous motion of electrons The electron current (i) is the number of electrons per second that enter a section of a conductor. Counting electrons: complicated Moving charges in a spark – too complicated. Why current heats up a metal? Friction – collide with atomic cores, loose energy – transfer to vibration – rise temperature Indirect methods: measure magnetic field measure heating effect Both are proportional to the electron current

Detecting Magnetic Fields We will use a magnetic compass as a detector of B. How can we be sure that it does not simply respond to electric fields? Compass needle: Interacts with iron, steel – even if they are neutral Unaffected by aluminum, plastic etc., though charged tapes interact with these materials Points toward North pole – electric dipole does not do that

The Magnetic Effects of Currents Make electric circuit: Use short bulb The polarity changes – direction changes. Could it be in principle that there is no dependence on current direction? Consider piece of wire with net current zero – thermal fluctuations, electrons move back a and forth – magnetic field is zero (experiment), but it is only possible if superposition = 0, I.e. e moving in opposite directions produce opposite field What is the effect on the compass needle? What if we switch polarity? What if we run wire under compass? What if there is no current in the wire?

The Magnetic Effects of Currents Conclusions: A wire with no current produces no B B is perpendicular to the direction of current B under the wire is opposite to B over the wire The magnitude of B depends on the amount of current Hans Christian Ørsted (1777 - 1851) Oersted effect: discovered in 1820 by H. Ch. Ørsted How does the field around a wire look?

Magnetic Field Due to Long Current-Carrying Wire

The Magnetic Effects of Currents The moving electrons in a wire create a magnetic field Principle of superposition: What can you say about the magnitudes of BEarth and Bwire? What if BEarth were much larger than Bwire?

Exercise A current-carrying wire is oriented N-S and laid on top of a compass. The compass needle points 27o west. What is the magnitude and direction of the magnetic field created by the wire Bwire if the magnetic field of Earth is BEarth= 210-5 T (tesla).

Biot-Savart Law for a Single Charge Electric field of a point charge: Moving charge makes a curly magnetic field: B units: T (tesla) = kg s-2A-1 Jean-Baptiste Biot (1774-1862) Felix Savart (1791-1841) Nikola Tesla (1856-1943) Short: Biot&Savart – french. Biot: professor of math at age 23, three years later Prof. of math. physics. Interested in applied math primarily. Considered magnetic field as fundamental property. 1/r^2 dependence (not trivial – for wire will be 1/r) Tesla –serbian-american inventor. Discovered rotating magnetic field – basis for alternating machinery. Emigrated to US at age 28 and sold patents for alternating current generator, motor and transformer to George Westinghouse. Later designed Tesla coil used in radio technology. high tension coils. Biot, Jean-Baptiste 21 Apr 1774 - 3 Feb 1862 French Educated at École Polytechnique in Paris, he became professor of mathematics at the University of Beauvais in 1797. Three years later he became professor of mathematical physics at the Collège de France. He studied a wide range of mathematical topics, mostly on the applied mathematics side. Biot made advances in astronomy, elasticity, electricity and magnetism, heat and optics on the applied side while he also did important work in geometry. He collaborated with Arago on refractive properties of gases. He, together with Savart, discovered that the intensity of the magnetic field set up by a current flowing through a wire varies inversely with the distance from the wire. This is now known as Biot-Savart's Law and is fundamental to modern electromagnetic theory. For his work on the polarisation of light passing through chemical solutions he was awarded the Rumford Medal of the Royal Society. Savart, Felix 30 June 1791 - 16 Mar 1841 French He taught at the Collège de France from 1828, becoming a professor there in 1836. He collaborated with Biot on a theory of magnetism. They took magnetism as the fundamental property rather than the Ampère approach which treated it as derived from electric circuits. Savart also carried out experiments on sound which became important for later students of acoustics. Tesla, Nikola (b. July 9/10, 1856, Smiljan, Croatia--d. Jan. 7, 1943, New York City), Serbian-American inventor and researcher who discovered the rotating magnetic field, the basis of most alternating-current machinery. He emigrated to the United States in 1884 and sold the patent rights to his system of alternating-current dynamos, transformers, and motors to George Westinghouse the following year. In 1891 he invented the Tesla coil, an induction coil widely used in radio technology. Tesla was from a family of Serbian origin. His father was an Orthodox priest; his mother was unschooled but highly intelligent. A dreamer with a poetic touch, as he matured Tesla added to these earlier qualities those of self-discipline and a desire for precision. http://www.neuronet.pitt.edu/~bogdan/tesla/bio.htm

The Cross Product Calculate magnitude: Calculate direction: Vector perpendicular to the plane defined by A and B 1) Point fingers of the right hand in direction of first vector v 2) Rotate wrist, if necessary, to make it possible to 3) Curl fingers of right hand through angle theta toward second vector 4) Stick out thumb, which points in direction of cross product Calculate direction: Right-hand rule

Clicker A ) +x W h a t i s t he d ir e ct i o n o f B ) – x C ) +y ) – x C ) +y < , , 3 > x < 0 , 4 , > ? D ) – y E ) z er o m ag n it u d e B

Two-dimensional Projections a vector (arrow) is facing into the screen  a vector (arrow) is facing out of the screen B B B r v B B

Exercise What is B straight ahead? What if the charge is negative?

Moving Charge Sign Dependence B1 v r + Magnetic field depends on qv: Positive and negative charges produce the same B if moving in opposite directions at the same speed B v r - B1 For the purpose of predicting B we can describe current flow in terms of ‘conventional current’ – positive moving charges. v r -

We need to know the number of moving charges. Electron Current A steady flow of charges in one direction will create a magnetic field. How can we cause charges to flow steadily? Need to find a way to produce and sustain E in a wire. Use battery How can we know magnitude and direction of magnetic field produced by wire? We need to know the number of moving charges. Suppose we somehow managed to do that (explain later) – what would the magnetic field due to the flow of multiple charges. No excess charges inside, overall - neutral

Electron Current mobile electron density wire Cross sectional area Average drift speed Electron current:

Conventional Current In some materials current moving charges are positive: Ionic solution “Holes” in some materials (same charge as electron but +) Observing magnetic field around copper wire: Can we tell whether the current consists of electrons or positive ‘holes’? The prediction of the Biot-Savart law is exactly the same in either case.

Conventional Current Metals: current consists of electrons Semiconductors: n-type – electrons p-type – positive holes Most effects are insensitive to the sign of mobile charges: introduce conventional current: André Marie Ampère (1775 - 1836) It has been claimed that Ampère had mastered all known mathematics by the age of twelve years but this seems somewhat of an exaggeration since, by Ampère's own account, he did not start to read elementary mathematics books until he was 13 years old http://www-gap.dcs.st-and.ac.uk/~history/Mathematicians/Ampere.html French scientist. Home-schooled by parents. Ampère read articles from L'Encyclopédie many of which, Arago remarked many years later, he could recite in full in later life. Arago also claims that Ampère read the Encyclopédie starting at volume 1 and reading the articles in alphabetical order. Whether Ampère's later desire for classification in all subjects arose from this education, or whether he enjoyed Buffon and the Encyclopédie because of a natural liking for classifying, is hard to say. While still only 13 years old Ampère submitted his first paper to the Académie de Lyon. This work attempted to solve the problem of constructing a line of the same length as an arc of a circle. His method involves the use of infinitesimals but since Ampère had not studied the calculus the paper was not found worthy of publication. Shortly after writing the article Ampère began to read d'Alembert's article on the differential calculus in the Encyclopédie and realised that he must learn more mathematics. French revolution (storming of the Bastille on 14 July 1789). He was 17 when his sister died. Shortly thereafter his father was guillotine, In the early 1820s, Ampère attempted to give a combined theory of electricity and magnetism after hearing about experimental results by the Danish physicist Hans Christian Orsted. Ampère formulated a circuit force law and treated magnetism by postulating small closed circuits inside the magnetised substance. Ampère's most important publication on electricity and magnetism was also published in 1826. It is called Memoir on the Mathematical Theory of Electrodynamic Phenomena, Uniquely Deduced from Experience and contained a mathematical derivation of the electrodynamic force law and describes four experiments. Maxwell, writing about this Memoir in 1879, says:- We can scarcely believe that Ampère really discovered the law of action by means of the experiments which he describes. We are led to suspect, what, indeed, he tells us himself, that he discovered the law by some process which he has not shown us, and that when he had afterwards built up a perfect demonstration he removed all traces of the scaffolding by which he had raised it. Ampère's theory became fundamental for 19th century developments in electricity and magnetism. Faraday discovered electromagnetic induction in 1831 and, after initially believing that he had himself discovered the effect in 1822, Ampère agreed that full credit for the discovery should go to Faraday. Weber also developed Ampère's ideas as did Thomson and Maxwell. Units: C/s  A (Ampere)

The Biot-Savart Law for Currents Superposition principle is valid Short means same distance from source to observation point for all qi. The Biot-Savart law for a short length of thin wire

Magnetic Field of Current Distributions Four-step approach: Cut up the current distribution into pieces and draw B Write an expression for B due to one piece Add up the contributions of all the pieces Check the result

A Long Straight Wire Step 1: Cut up the current distribution into pieces and draw B. Origin: center of wire Vector r: Magnitude of r:

A Long Straight Wire Step 2: Write an expression for B due to one piece. Unit vector: : B field due to one piece:

A Long Straight Wire need to calculate only z component

A Long Straight Wire Step 3: Add up the contribution of all the pieces.

A Long Straight Wire Special case: x<<L Has same distance dependence as E-field from long uniformly charged wire. What is the meaning of “x”?

A Long Straight Wire Step 4: Check results  direction  far away: r>>L  units:

Right-hand Rule for Wire Conventional Current Direction

Cklicker B1 B2 r v Which of B1 and B2 is larger? B1 is equal B2 45 v Which of B1 and B2 is larger? B1 is equal B2 B1 is larger than B2 B1 is smaller than B2 B1 > B2, B1 > B3.