Electricity and Magnetism-- Magnetic Field and Ampere's Law

Topics: 􀂄 􀂄 Magnetic force 􀂄 Ampere’s law 8.022 (E&M) – Lecture 10 Magnetic field B acting on charges in motion 2 The Origins of Magnetism 􀂄 􀂄 Coul􀂄 Can attract or repel pi􀂄 By the 12th 􀂄 􀂄 􀂄 􀂄 demo 􀂄 demo 􀂄 Magnetic forces can be pretty strong! Demo G3: N S G. Sciolla – MIT 8.022 – Lecture 10 Ancient Greeks noticed that a piece of a mineral magnetite (an oxide of iron) had very special properties: d attract a piece of iron, but no effect on Au, Ag, Cu, etc ece of magnetite depending on relative orientation century people could build a magnetic compass A small magnetic needle is suspended so it can pivot around vertical axis The needle will always come to rest with one end pointing North By definition we call that end “North” and the other “South” Like poles repel, unlike poles attract: North and South cannot be separated in a magnet: nail on a sting 18.022 – Lecture 10 3 The big step forward 􀂄 In 1820 Oersted 􀂄 􀂄 ’s experiment with parallel wires carrying current 􀂄 If currents are parallel, wires attract 􀂄 􀂄 􀂄 􀂄 Demo I1 I2 I1 I2 G. Sciolla – MIT 8.022 – Lecture 10 realized that current flowing in a wire made the needle of a compass swing The direction depends on the direction of the current Soon after, AmpereIf anti-parallel, wires repel No force on a stationary charge nearby… NB: wires are overall neutral! BIG discovery: proves that Electricity and Magnetism are related! 4 Magnetic force between currents 􀂄 􀂄 F~I1I2 􀃆 􀂄 􀂄 Interpretation 􀂄 􀂄 v x B 􀂄 B: B 􀂄 demo G2 vF q B c = × 􀁇􀁇 􀁇 G. Sciolla – MIT More refined observations followed: F is proportional to velocity of charges in motion Direction of F is perpendicular to velocity Some field (magnetic field B) is created by the charges in motion Magnetic force is proportional to cross product Direction of curls around the current (right hand rule) Iron fillings can be used to visualize B field lines: NB: this is an empirical law so far 25 􀂄 When a charged particle moves in electric (E) and ): 􀂄 􀂄 􀂄 􀂄 􀂄 􀂄 􀂄 4 G vF B c ⎛ ⎞ = + ×⎜ ⎟⎝ ⎠ 􀁇􀁇 􀁇 􀁇 ( )F = + × 􀁇 􀁇 􀁇􀁇 G. Sciolla – MIT 8.022 – Lecture 10 Lorentz force magnetic (B) fields it feels a force (FLorentzThe above formula defines the magnetic field B Units of B in cgs: [B] = [F]/[q] = dyne/esu = Gauss (G) NB: [B] = [E] Units of B in SI: [B] = [F]/[q v] = N s /(m C) = Tesla (T) Conversion: 1 T = 10Lorentz q E Lorentz q E v B 6 Trajectory in magnetic fields 􀂄 v //+x B //+z 􀂄 􀂄 􀃆 circular motion! v B x y 2 F F mvcR qBc R = ⇒ == ⇒ vF q B c = × 􀁇􀁇 􀁇 G. Sciolla – MIT 8.022 – Lecture 10 A particle of charge q and mass m moves with velocity axis in a magnetic field axis (out of the page): What is the trajectory of q in the magnetic field? v, B and F (a) are always perpendicular Lorentz centripetal qvB mv 37 Deflection of electron beam by B 􀂄 􀂄 Velocity of electrons: ve 􀂄 B ve 􀂄 􀂄 Electrons curve according to Lorentz force (Demo G5, G6 TV) B G. Sciolla – MIT 8.022 – Lecture 10 An electron beam is produced by a cathode in a vacuum tube Magnetic field perpendicular to is produced by current in a wire or by permanent magnet What do we expect to happen? 8 J.J. Thompson’s experiment 􀂄 e 􀂄 􀂄 􀂄 e􀃆 FLorentz 􀂄 E and B can be adjusted so FMagnetic = -FElectric 􀂄 Electric fiel􀂄 􀂄 E v c B = 2 22y mv 2 2 2 2 e e q m m ∆ = = x y z vF qE B c ⎛ ⎞ = ⎜ ⎟⎝ ⎠ 􀁇􀁇 􀁇 􀁇 G. Sciolla – MIT 8.022 – Lecture 10 Discovery of electrons and measurement of e/min 1897 The idea: A beam of “cathode rays” crosses a region with E and B present Choosing v//x axis, B//z axis, E//y axis //FElectric so that e will go straight d alone causes a shift: Now turn on B and set it to cancel the shift due to E: Substituting this in the previous equation gives: qEL ∆=− yc E B L Lorentz + × 4G. Sciolla – MIT 8.022 – Lecture 10 109 Application in modern physics 􀂄 􀂄 􀂄 +e-) i􀂄 How can we “see” these particles? 􀂄 􀂄 How can I measure the properties of these particles? 􀂄 􀂄 􀂄 Charged particles will curve according to 􀂄 􀂄 mvcR qB = G. Sciolla – MIT 8.022 – Lecture 10 Tracking detectors in modern particle physics The problem High energy collisions between elementary particles (such as eproduce many particles (protons, electrons, pons, muons,…) Build detectors that can “visualize” the trajectory of charged particles using the fact that particles ionize the material they cross E.g.: measure momentum, energy, mass, etc. Immerse the detector in a very strong magnetic field B ~ 2 T Direction measures the charge Radius of curvature measures momentum p=mv Gold plated event in BaBar detector Zoom on interaction regin 511 Magnetic force and work 􀂄 i􀂄 􀃆 i( ) 0qdW F v c = = = × = 􀁇 􀁇 􀁇􀁇 􀁇 􀁇 􀁇i i i 2 12 1 W q=− ∫ 􀁊􀁇 􀁇i G. Sciolla – MIT 8.022 – Lecture 10 Moving a charge in an electric feld E requires work: How much work does it take to move a charge in a magnetic field? No work is needed to move a particle in a magnetic feld because v and F are always perpendicular! F ds vdt B vdt E ds Force on a current 􀂄 A magnetic field will excerpt a force on a current 􀂄 Since a current is just a stream of moving charges! 􀂄 Current I flowing in a wire can be seen a density of charges λ moving with velocity v: I=λv 􀂄 The force dF exerted on the infinitesimal wire dl is: 􀁇􀁇 dF 􀁇 =(λdl) v ×B c 􀁇􀁇􀁇 I 􀂄 Rewrite this in terms of the current: dF = dl B ×􀁇 􀁇􀁇 c 􀂄 Total force F: F = I ∫dl ×B c wire 􀂄 For a long straight wire in a constant magnetic field: 􀁇 I 􀁇 F = Lnˆ ×B c G. Sciolla –MIT 8.022 – Lecture 10 12 6 Ampere’s law 􀂄 In electrostatics, the electric field E and its sources (charges) are related by Gauss’s law: 􀁇􀁇 EdA = 4πQ∫ i encl Surface 􀂄 Why useful? When symmetry applies, E can be easily computed 􀂄 Similarly, in magnetism the magnetic field B and its sources (currents) are related by Ampere’s law: 􀁇 􀁇 4πBds = Iencl i􀁶∫ cC 􀂄 Why useful? When symmetry applies, E can be easily computed 􀂄 NB: This is a line integral! NB: no demonstration has been given so far for Ampere’s law. G. Sciolla –MIT 8.022 – Lecture 10 13 14 B created by current in a wire 􀂄 􀂄 Calculate magnetic field B created by I I 􀁇 B 􀁇 r 􀂄 Solution: 􀂄 Apply Ampere’s law: 􀂄 􀂄 NB: Bwire 􀂄 􀂄 4 2 ˆ C B r r I c IB cr ϕππ= = ⇒ =∫ 􀁇􀁇 􀁇i􀁶 2()Er r λ = G. Sciolla – MIT 8.022 – Lecture 10 Application of Ampere’s law: Long, straight wire in which flows a current I Direction: right hand rule ~ 1/r. Does this look familiar? Remember E created by a line of charge: Coincidence? Not at all… ()2 encl Bds 7 8.022 – Lecture 10 15 – Force between 2 wires 􀂄 􀂄 􀂄 Total force F: 􀂄 􀂄 Direction? 􀂄 I1 and I2 􀂄 I1 and I2 21 2 2 IIF L cr = 2 2 2 ˆIB cr ϕ= 􀁇 1 1 2ˆIF Ln B c = × 􀁇 􀁇 21 2 2 IIF L = 1 2ˆF I ϕ∝ × 􀁇 G. Sciolla – MIT Force on wire 1 due to magnetic field B created by wire 2: Magnetic field created by wire 2: Usually we quote the force/unit length: Using right hand rule: parallel: attractive anti-parallel: repulsive c r Can we test this experimentally? Demo G8, G9 8.022 16 􀂄 ing in a 􀂄 􀂄 􀂄 B created by sheet of current L x y θ 􀂄 Solution: 􀂄 B from a wi􀂄 Just apply superposition… 􀂄 􀂄 2 ˆIB cr ϕ= 􀁇 2 2(2 )/2I IB BLc πθ θ π= → ⇒ = G. Sciolla – MIT Lecture 10 Calculate the magnetic field B created by current flowsheet of conductor Current //-z axis (into the page) Width of sheet of conductor: L Current in a metal sheet ~ N parallel wires Another application of Ampere’s law: re is know: Direction: for y>0: B //+x; for y<0: B //-x Magnitude: integrate dB = B field from each infinitesimal wire When L>>y, Lc NB: magnitude of B does not depend on y. As for E of sheet of charges 817 Calculation: B created by plane of current L x y θ /2 /2 /2 /2 /2 /2 2 ˆ ) 2 2 2 2 x xL x L xL x L xL x L B dI cr I dxL cr I Lc r yd I x yLc θ θ θ θ θ θ θ θ θ = =− = =− = =− − = ⎛ ⎞ = ⎜ ⎟⎝ ⎠ ⎛ ⎞ ⎜ ⎟⎝ ⎠= = = = ∫ ∫ ∫ ∫ ∫ ; 4 ˆ; yd yytg r IB xLc θθ θ θ θ ⇒ = = ⇒ = ± 􀁇 B 􀁇 B 􀁇 G. Sciolla – MIT 8.022 – Lecture 10 Another application of Ampere’s law: (only component //x survives because of symmetrycos cos cos cos cos cos dB dx cos cos + for y>0-for y<0 dx 18 􀂄 􀂄 􀂄 Does it ring a bell? 􀂄 Yes, ∆E across a plane of charge! 􀂄 More on B from sheet of current L x y θ B 􀁇 B 􀁇 2 ˆKB x c π=± 􀁇 4 KB c π∆= 4E πσ∆= G. Sciolla – MIT 8.022 – Lecture 10 If we define current per unit length K=I/L: What is the change of B across the sheet of current? Another similarity between electric and magnetic fields. …This must be more than a pure coincidence… 919 Ampere’s law in SI 􀂄 􀂄 where µ 0=4 10-7 N/A2 􀂄 􀂄 􀃆 µ 0c/(4π ) 􀂄 􀂄 􀂄 􀂄 in SI 0 C B ds Iµ=∫ 􀁇 􀁇i􀁶 02 ˆ ˆ 2 IIB B cr r µϕ ϕπ = ⇒ = 􀁇 􀁇 02 121 2 2 2 IIIIF F L L r µ π = ⇒ = G. Sciolla – MIT 8.022 – Lecture 10 In SI Ampere’s law takes the form: is the magnetic permeability of free space Be careful not to mix cgs and SI formulae! To convert cgs SI: multiply by Examples: Magnetic field created by a wire: Force between 2 wires: NB: factor 1/c missing in FLorentz encl c r 20 Divergence of B 􀂄 􀂄 􀂄 􀂄 Similar equation for E: 􀂄 􀂄 􀃆 i2 ˆIB cr ϕ= 􀁇 2 2 2 2 2 2 2 2 2 2 ˆ ˆˆ ˆ ˆ-sin -ˆ ˆ2 -xy yxr x y y x x y x y IB x y x y ϕ ϕ ϕ= + = = ⇒ + + ⎛ ⎞ = ⇒⎜ ⎟+ +⎝ ⎠ 􀁇 2 2 2 2 2 2 2 2 2 -( ) ( ) IB x y x y ⎛ ⎞∇ = ⎜ ⎟+ +⎝ ⎠ 􀁇 􀁇 i 0B∇ = 􀁇 􀁇 i 4E π ρ∇ = 􀁇 􀁇 i G. Sciolla – MIT 8.022 – Lecture 10 Consider the B produced by a wire of current: Calculate its divergence in Cartesian coordinates: This is a general property of the magnetic field: The divergence of E is related to the density of electric charges The divergence of B must be related to the density of magnetic charges Magnetic monopole don’t exist (There may be magnetic monopoles leftover from the Early Universe, but never observed expermentally so far) Given and cos xy yx cr = 0 yx xy cr 1021 Thoughts on B 􀂄 􀂄 􀂄 l􀂄 􀂄 􀂄 􀂄 􀂄 􀂄 G. Sciolla – MIT 8.022 – Lecture 10 What exactly is a magnetic field B? Why does it have so much in common with electric field E? Why should there be a fied that acts only on moving charges? Answer: Special Relativity Relativity: the physics must be the same in all reference frames A charge at rest for observer 1 appears in motion to observer 2 that moves with a certain velocity w.r.t. observer 1: Observer 1 will measure an electric field Observer 2 will measure a magnetic field Calculating attractive or repulsive force acting on a test charge in the 2 reference frames will lead to the same conclusion 22 Summary and outlook 􀂄 􀂄 􀂄 Magnetic Force 􀂄 Ampere’s Law 􀂄 Next time: 􀂄 􀂄 Goals: 􀂄 􀂄 G. Sciolla – MIT 8.022 – Lecture 10 Today: Magnetic Field B acting on charges in motion Quick Introduction to Special Relativity Understand how and why Magnetism and Electricity are related Finally play with some really cool physics! 11