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Magnetism (Chapter 33) (Hans Christian Oersted: Danish scientist who discovered relationship between electricity and magnetism in 1820) Brief History of Magnetism. • Magnetism like electricity has been known for several thousand years – First mention of static electricity and it s forces: ~600 B.C. by Thales of Miletus (Greek). (Word for electron is from the Greek word for amber.) – According to Aristotle, the same Thales was the first to mention of magnetism. – In ancient China in the 4th century B.C., Book of the Devil Valley Master (鬼谷子): "The lodestone makes iron come or it attracts it. – The ancient Chinese scientist Shen Kuo (1031-1095 AD) was the first person to write of the magnetic needle compass and that it improved the accuracy of navigation. – Alexander Neckham in 1187 was the first in Europe to describe the compass and its use for navigation. – In 1600, William Gilbert concludes that the Earth is itself a magnetic and this explains compasses. – In 1820, Hans Christian Oersted in Copenhagen discovered that an electric current produces a magnetic field. What Do We Already Know About Magnetism From Childhood? • Magnetism is a long range force • Not all objects materials are attracted to a magnet. Only certain metals (iron, cobalt, nickel) are magnetic materials and respond to a magnet. • Magnets have two poles, north (N) and south (S). There is an attractive force between opposite poles and repulsive force between like poles. • A compass is a tiny magnet: N pole of a bar magnet attracts one end of a compass while S pole attracts other end. The Magnetic Field The magnetic field is defined similarly to the electric field: 1. The magnetic field (B)exists at all points in space surrounding a magnet (or current carrying wire) 2. It is a vector field: At each point it has both a magnitude and direction. 3. The magnetic field exerts forces on magnetic poles: • A N pole feels a force in the direction (parallel) of the field. • A S pole feels a force opposite to the direction (anti-parallel) of the field. A compass needle can be used to find the direction of the magnetic field. Superposition • The magnetic field just like the electric field obeys the principle of superposition. • If one has N magnets, the total magnetic field is the sum of the fields from each magnet ! ! ! ! ! Btotal = B1 + B2 + B3 + ... + BN Magnetic Field of a Bar Magnet The magnetic field lines point away from the N pole and towards the S pole. The magnetic field of a bar magnet looks just like the E field of an electric dipole Monopoles and Dipoles: What Happens When You Cut a Magnet ? If you cut a magnet into two pieces you get two smaller magnets each of which has it s own N and S poles. You DO NOT get an isolated N pole and S pole when you cut a magnet. A magnet is a magnetic dipole. Since a magnet and E-dipole have same fields, what if you cut an E-dipole ? THERE ARE NO MAGNETIC MONOPOLES ! There is no Coulomb s Law for Magnetism Discovering the Link Between Electricity and Magnetism In 1820, Hans Christian Oersted noticed in the middle of class that a wire with an electric current deflected a compass needle Oersted discovered that magnetic fields are created by electric currents ! Right hand rule! Two Kinds of Magnetism ?? • Textbook is not exactly correct: There is one kind of magnetic field but two ways to create a magnetic field • All magnetic fields are created by one of two means: – Electric currents – Spin • All fundamental particles (electron, proton, neutron) act like tiny magnets. This is an intrinsic property known as spin. – An electron has a N and a S pole ! • Spin is fundamental property of the universe (no one knows why it exists) and is not the result of a particle spinning. • Permanent magnets are a result of the spin of valence electrons. • The spin of the electron is the basis of new technology called spintronics - digital information processing and storage using electron spin instead of charge. The Source of the Magnetic Field: Moving Charges The magnetic field of a charged particle q moving with velocity v is given by the Biot-Savart law: where r is the distance from the charge and θ is the angle between v and r. The Biot-Savart law can be written in terms of the cross product as: The Magnetic Field of a Moving Charge: Direction of B from Right Hand Rule ! ! ! Vector Cross Product: A = C ! D 1)Curl fingers of right hand in the direction that moves the vector C towards the vector D while passing through the smallest angle. 2)Thumb will point in the direction of the cross product (red arrow). Evaluating Cross Product for Unit Vectors ! ! ! A = C ! D = (C1iˆ + C2 ĵ + C3 k̂) ! (D1iˆ + D2 ĵ + D3 k̂) ! A = (C2 D3 " C3 D2 )iˆ + (C3 D1 " C1 D3 ) ĵ + (C1 D2 " C2 D1 )k̂ iˆ ! iˆ = ĵ ! ĵ = k̂ ! k̂ = 0 Tesla: Unit of Magnetic Field 1 Tesla= 1T= 1N/Am is the fundamental unit of the magnetic field strengt. 1 tesla is an enormous magnetic field ! Some countries really know how to honor their engineers ! Nikola Tesla was an electrical engineer: 1)AC Electric Motor 2)AC Power Transmission 3)Radio Communication 4)Remote Control 5)Logic circuits 6)Wireless energy transmission 7)Tesla coil Biot-Savart Law for Currents • Biot-Savart law for currents can be used for any current geometry although usually integral is very difficult. • It adds up the magnetic fields produced by each length segment ds, which has a current I Biot-Savart Law: From a Moving Charge to a Current (Derivation Review). A current is a collection of moving charges. Each moving charge produces it's own B-field according to Biot-Savart Law. ! ! q v µ j j " r̂ Bj = 0 4! r 2 ! ! ! q j v j " r̂ µ0 B = # Bj = # r2 4! j Treat each charge is as being contained in an infinitesimal line segment ds: ! ds dq ! ! ! q j v j $ dq = ds = Ids dt dt ! µ0 Ids! " r̂ B= 4! % r 2 ! µ0 Ids! " r̂ dB = 4! r 2 The Magnetic Field of a Long Current Carrying Wire We already know the direction of the B-field but what is the magnitude? ! dl ! r̂ = dysin(" / 2 + # ) x x = dy cos # = dy = dy r x 2 + y2 ( 1/2 ) The Magnetic Field of a Current Loop Many practical devices such as transformers, inductors, and solenoid utilize the magnetic field produced by current loops. The magnetic fields of the connecting wires do not contribute since their B-fields cancel out (assuming they lie side by side). Current segments on opposite sides of loop produce magnetic fields with opposite x and y components on axis. On axis, the only magnetic field is in the axis direction. Deriving the Magnetic Field On Axis for Loop ! !sk " r̂ = !sk ( Bk )z = µ0 I !sk µ0 I !sk R cos $ = 4# r 2 4# ( z 2 + R 2 ) ( z 2 + R 2 )1/2 2# Bz = % ( Bk ) & k z 2# ' dB = 0 µ IR Bz = 0 4# ( z 2 + R 2 )3/2 ' R d( 0 µ0 IR 4# ( z 2 + R 2 )3/2 A coil of N current loops: 2# ' R d( 0 µ0 IR 2 Bz = 2 ( z 2 + R 2 )3/2 We can 'amplify' this magnetic field by stacking together N current loops to make a coil. Assuming the length of the coil is negligible (wires stacked essentially on top of each other), we have µ0 NIR 2 Bz = 2 ( z 2 + R 2 )3/2 Ampère s law Whenever total current I passes through an area bounded by a closed curve, the line integral of the magnetic field around the curve is given by Ampère s law: ! ! B ! d s = µ I " 0 Closed line integral around the current Ampere s Law for a Wire With Current I ! ! "! Bidl = µ0 I enc ! ! ! "! Bidl = B "! dl =B2" r µ0 I enc B= 2" r Very easy calculation in comparison to Biot-Savart law For Ampere s Law: What Sign to Use for Current??? Use right hand rule (again!) to determine sign of current. Curl right fingers around integration path. If thumb points in direction of current, I is positive in Ampere s law. If thumb points opposite to direction of current, I is negative in Ampere s law. I enc = I 2 + I 4 ! I 3 Magnetic Field Inside Wire Current Density: J = I / ! R 2 Solenoid A solenoid is a long coil of wire with a current running through it. Inside, away from the edges, the magnetic field is uniform and along the axis. Has many uses in technology: 1)Electromechanical switches (relays) 2)Automobile ignition systems 3)Hydraulic/pneumatic valves for controlling fluid/gas flow 4)Magnetic Resonance Imaging 5)Inductors (Chapters 34+36) Real (Finite Length) Solenoids Have External Magnetic Fields Like a Bar Magnet An Electromechanical Switch Relay (Example of Solenoid) Starter Solenoid of Automobile The Interior Magnetic Field of a Solenoid ! ! Use Ampere's Law: " ! Bidl = µ0 I enc ! ! ! ! ! ! ! ! ! ! "! Bidl = ! Bidl + ! Bidl + ! Bidl + ! Bidl 1 1 1 3 2 ! ! ! Bidl = 2 4 2 ! ! ! Bidl = 0 3 4 ! ! ! ! ! Bidl = 0 B " dl 3 ! ! ! Bidl = B ! dl = Bl 4 4 I enc = NI & N ( I = µ nI B= µ 0 l (' 0 # % % $ n = N/l is the number of turns per unit length. Magnetic Dipoles The external magnetic field produced by a current loop or finite solenoid looks just like the magnetic field of bar magnet. Current loops form magnetic dipoles. Magnetic Dipoles: B-Field Current loops are magnetic dipoles! Bloop µ0 IR 2 µ0 2AI = ! 2 ( z 2 + R 2 )3/2 4" z 3 z!R Define the magnetic dipole moment: " µ = AIn̂ Use right hand rule for normal vector ! When the size of the current loop is much smaller than the distance at which the B field is measured, " µ0 3( µ" ir̂)r̂ # µ" B= 4" r3 The magnetic field has the exact same form as the electric field of an electric dipole ! The Magnetic Force on a Moving Charge The magnetic force on a charge q as it moves through a magnetic field B with velocity v is ! ! ! F = qv ! B If there is a magnetic and electric field, the total force on the charge is: ! ! ! ! Fon q = q E + v ! B ( ) This is equation is known as the Lorentz force Behavior of the Magnetic Force 1)The magnetic field exerts no force on stationary charges !! Moving charges create magnetic fields and only moving charges feel a force from magnetic fields !!! 2)Motion parallel or anti-parallel to the magnetic field produces no force ! B field is pointing into the page Magnetic Fields Do No Work Recall the definition of the work done by a force on a particle in going from position i to f: sf ! ! W (i ! f ) = " Fids si Substitute in the magnetic force: ! ! ! $ ds! ' sf ! sf ! WB = " v # B ids = " v # B i& dt ) si si % dt ( ! ! sf ! WB = " v # B iv dt si ! ! ! v#B * v ! ! ! v # B i v = 0 ! WB = 0 ( ( ) ( ) ( ) ) The magnetic field does no work on the particle ! W=+K= change in kinetic energy The magnetic field can not accelerate a particle ! Cathode Ray Tube (Older TVs and Computer Monitors) Magnetic Field: Uniform Circular Motion (Cyclotron Motion) • Force is always perpendicular to velocity. • Such a force is constantly deflecting the particle sideways. • This causes particle to move in a circle with a constant velocity. • Motion perpendicular to B-field is unaffected. Motion in a Plane Perpendicular to Magnetic Field F = qvB magnetic force = centripetal force towards center mv 2 qvB = r v (speed) is unchanged by B. Only r, radius of orbit is variable: mv rcyc = qB Frequency of orbit: v = !r qB ! cyc = 2" f = m (! cyc is orbital frequency in radians per second) The Hall Effect The Hall effect was the first experiment to prove metals have negative charge carriers. It is used for: 1)Determining the sign and type of charge carriers in conducting materials (particularly in new types of exotic materials). 2)As a sensor for measuring magnetic fields 3)Hall sensors used for motor/engine tachometers and internal combustion engine timing. Hall Effect: Magnetic Field Induces a Transverse Voltage Configuration: Current in conducting bar with B field perpendicular to bar. Lorentz force causes moving charges to be deflected towards sides where they accumulate. Positive Charge Carriers: Negative Charge Carriers: Hall Voltage Determines Sign of Charge Carriers and Magnetic Field Strength Charges are deflected to the side. The charge separation leads to an electric field and, hence, potential difference between the sides. Steady state: Electric Force Between Sides=Magnetic Force FB = FE FE = qE = q!VH / w FB = qvd B !VH = vd wB However it is the current and not the drift velocity that one measures in the lab: vd = I / ( Anq ) !VH = IB " w % $ ' nq # A & w = width of bar A= cross sectional area Magnetic Forces on Current-Carrying Wires Consider a segment of wire of length l carrying current I in the direction of the vector l. The wire exists in a constant magnetic field B. The magnetic force on the wire is ! ! ! F = !Qv " B The charge passing through a length l of the wire is: !Q = I !t !x l v= = !t !t ! ! ! F = Il " B ! l points in direction of current where α is the angle between the direction of the current and the magnetic field. l Magnetic Force on Current Carrying Wire: Jumping Wires Magnetic Force Between Two Parallel Wires µ0 lI1 I 2 F2 || wires = 2! d d = separation l = length of wires Currents in same direction : Attractive force Currents in opposite directions: Repulsive force Torques on Current Loops (Magnetic Dipoles) The Lorentz force on the top and bottom segments are opposite and rotate the current loop. Therefore there is a torque on the current loop: ! ! ! ! =µ"B ! µ = IAn̂ = Il 2 n̂ Potential energy of a magnetic dipole in B field: ! ! U B = # µi B This equations are valid for current loop dipoles and permanent magnets (lik a compass needle)! Electric Motor