Electricity and Magnetism- Magnetic field and Biot-Savart law

Topics: 8.022 (E&M) – Lecture 13 B’s role in Maxwell’s equations Vector potential Biot-Savart law and its applications What we learned about magnetism so far… Magnetic Field B Special RelativivF q B c = × 􀁇􀁇 􀁇 4 C B ds I c π =∫ 􀁇 􀁇i􀁶 2 ˆϕ= 􀁇 IB cr Experiments: currents in wires generate forces on charges in motion Force exerted on charge q with velocity v: Explanation: there must exist a magnetic field B ty: B is just E seen from another reference frame… Ampere’s Law: Application: B generated by current in a wire: encl 13 Divergence of B Similar equation for E: Æ experimentally so far) 2 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 πρ∇ = 􀁇􀁇 i G. Sciolla – MIT 8.022 – Lecture 13 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 Given and cos xy yx cr = 0 yx xy cr Ampere’s law in differential form Apply Stoke’s theorem to Ampere’s law: 􀁇􀁇4πBds = Iencli􀁶∫ cC 􀁇􀁇 􀁇􀁇􀁇4π 􀁇􀁇 Bds =∇× B dS = J dS ii􀁶∫ i ∫Sc ∫SC⎛ 􀁇􀁇4π􀁇⎞ 􀁇 ∫S ⎝⎜∇×B − J ⎟idS =0 for any surface c ⎠ 􀁇􀁇4π􀁇 ∇×B= JÆ Ampere’s law in differential form: c G. Sciolla –MIT 8.022 – Lecture 13 4 2 Toward Maxwell’s equations Let’s collect all the equations in differential form that we found so far: 􀁇􀁇⎧∇i = 4πρ Å Relates E and charge density (ρ) -Gauss⎪􀁇􀁇⎪∇i = 0 Å No magnetic monopoles! ⎪􀁇 􀁇⎨∇× = 0 Å E is a conservative field ⎪􀁇 􀁇 4π 􀁇⎪∇× = Å Relates B and its sources (J) -Ampere ⎪⎩ Not complete Maxwell’s equations yet, but we are getting closer… G. Sciolla –MIT 8.022 – Lecture 13 5 JcEB EB 6 Vector potential A φφ and E: iGoal: enforce div B=0 φ φ= −∇ 􀁇 􀁇 E B A≡∇× 􀁇 􀁇􀁇 Sif = 􀁇 􀁇􀁇􀁇 i ( ) 0Eφ φ φ≡ = ∀ 􀁇 􀁇 G. Sciolla – MIT 8.022 – Lecture 13 Definition of potential for electric field: (P) = work needed to move a unit charge from reference to P Relationship between Hidden advantage: Can we introduce something simlar for B? A is called “vector potential” in analogy with A is not connected to work or energy (but to angular momentum) nce 0 for any f, we define ∇∇× If E=-0 because ∇⇒∇× ∇× ∇ 3 7 Non Uniqueness Electrostatics: Æ potential φyz 0x 0 x z y 0 yx z 0 0 iAA ˆB= 0 y ˆA AB = 0 liˆ ˆ( )2AAB= init ˆie ot f h ield. A yz A xBz x A y B x xy Bz B Ay x ∂⎧ ∂ ⎧ =−− =⎪ ⎪∂ ∂⎪ =⎪⎪ ∂ ∂ ⎪− = ⇒⎨ ⎨∂ ∂ = − +⎪ ⎪ ⎪ ⎪∂∂ − =⎪ =⎩∂ ∂⎩ = 􀁇 􀁇 􀁇 􀁇 􀁇 􀁇 􀁇 ⎪ G. Sciolla – MIT 8.022 – Lecture 13 given a charge distribution and boundary conditions is uniquely identified Magnetism: does it work the same for A? No, there are infinite number of A corresponding to a single B Example: We are given one “coupon” to simplify equations when needed Requrements: Possibe solutons: ...inf. Find A that creates ths B ers! Bx By Q: what current creates this B? 8 Poisson’s equation for A Electrostatics: 2 2 4 4( )4 i( )B A A J A A J c cB J c A A A π π π ⎧⎪ = ⇒ =⎨∇× =⎪⎩ 􀁇 􀁇􀁇 􀁇 􀁇 􀁇 􀁇 􀁇 i􀁇 􀁇􀁇 􀁇 􀁇 􀁇 i 2 4i4 E E φ φ πρ πρ ⎧⎪ ⇒∇⎨ ∇ =⎪⎩ 􀁇 􀁇 􀁇􀁇 i /2i4A J c π∇⇒ =−∇ 􀁇􀁇 􀁇􀁇 i G. Sciolla – MIT 8.022 – Lecture 13 Magnetism: Use your coupon now! We used the identty: (Pset#7) =∇× ⇒∇×∇× ∇ ∇ −∇ ∇×∇× =∇ ∇ −∇ 􀁇 􀁇 􀁇􀁇 􀁇 􀁇 􀁇􀁇 Poisson's equaton =−∇ =− Choosng A=0 49 Solving Poisson’s equation for A 2 2 2 2 2 4l ? 4 4iit iii4 ii4lix x Y Y Z Z V A J c A J c A J c A J c dV r π π π π ρφ πρ φ ∇ ⎧∇⎪ ⎪ ⎪∇⎨ ⎪ ⎪∇⎪⎩ ∇ = ∫ 􀁇 􀁇 1i ili iiV wire J JA c c r IA c r φ ρ→ → ⇒ = = ∫ ∫ 􀁇 􀁇􀁇 􀁇 􀁇􀁇 G. Sciolla – MIT 8.022 – Lecture 13 How do you soveThnk of n cartesan coordnates: Rem em ber Posson's equaton and its souton Sam e as our new e = − = − = − = − = − quatonf replace A and For current fow ngn a w re: dV dl 10 Biot-Savart Law i. B= 1 1 1 1 = ( ) ( ) wire wire wire wire wire I dlA c r I dl I dl c r c r I Idl dl dl d c r r c r r = ∇× = ∇× ∇× ∇ × ⎡ ⎤∇× × = ∇× ×⎢ ⎥ ⎣ ⎦ ∫ ∫ ∫ ∫ ∫ 􀁇􀁇 􀁇 􀁇 􀁇􀁇 􀁇􀁇 􀁇 􀁇 􀁇 􀁇 􀁇􀁇 􀁇 􀁇 􀁇􀁇 􀁇 􀁇 􀁇 2 2 ˆ1l: ˆB wire l r r I rdl r c r ∇ ∇ = − ×⇒ = ∫ 􀁇􀁇 􀁇 􀁇􀁇 􀁇 G. Sciolla – MIT 8.022 – Lecture 13 Find B produced from current knowng that A= Using the fact that (ab)=a( b)+( a) b: ∇× ∇× +∇ +∇ Since ×d =0 and 5G. Sciolla – MIT 8.022 – Lecture 13 12 11 Biot-Savart Law: illustration 2 ˆdB I rdl c r = × 􀁇􀁇 Biot-Savart: E.g.: iÆ B//z ˆr dB 􀁇 dl 􀁇 G. Sciolla – MIT 8.022 – Lecture 13 dB is perpendicular to current and to radial direction f you have dl //x, r //y ( ) 2z 2 2 22 2 2 20 ˆB= sin ˆ ; sin /; 2ˆ ˆz z 2 ˆ ( ) ente re r wi I dl r cr dl r dl R r r R h IB R d cr c RR cz IB z π π θ ϕ θ πθ ϕ = × × = = = = + = = ⇒ + = ∫ ∫ ∫ 􀁇􀁇 􀁇 􀁇 􀁇 􀁇 90-θ x y z B from loop of current Calculate B created by a loop of current Solution on axis Determine direction of dB Symmetry Æ wire 3 /2 dB sin loop cRd IR Application of Biot-Savart: Radius: R Distance from center of the loop: z Apply Biot-Savart only component //z survives 613 B from solenoid What if we stack a N rings over a length L? For L>>R: 2 2 2 2 2/2 /2 2 2 2 2 2 2 2Singlil riiili2 2 ( ) ( ) 2 2 4 L L L L R dI c R nI zB cR z c R z nI L c L R nIdz π π π π = = = + + = + ∫ ∫ 􀁇 􀁇 4 nIB c π = 􀁇 G. Sciolla – MIT 8.022 – Lecture 13 Application of Biot-Savart: Use result of single loop + superposition: 3/2 3/2 3/2 -/2 -/2 e rng: dB (R +z ) Integrate on alngs (n the mdde of the solenod) RdWith n=N/L 14 Solenoid and Ampere’s law ile 4 4 4 4 enclBdl I c NNI I nI c c π π π π = ⇒ = = ∫ 􀁇􀁇 i 􀁇 􀁶 G. Sciolla – MIT 8.022 – Lecture 13 One can prove that B outside the solenod is =0 Ampere can be used to simply prove that B does not depend on r: rectangSince B is //z and present only inside the solenoid: B(r)L= ( ) no dependence on R Br c L 715 Solenoid’s magnetic field: demos Expected: imentally? G13: B from 2 wires G16: B inside solenoid G. Sciolla – MIT 8.022 – Lecture 13 Can we test this experG12: B from a single wire using iron filings 16 More demos on magnetic fields More demos: G18: Long solenoid turn(R=10 Ω8 10 4 4 4.5B nI c π π −= = = i i G. Sciolla – MIT 8.022 – Lecture 13 G14: map B around a wire using a compass G9a: collapsing solenoid Can you explain what’s happening? Long solenoid with N=2760, I=4.5 mA, length = 46 cm , L=128 mH) What is B? Verify with Hall probe 2760 230 10 Gauss ??? 310 50 8I B = nI 4p c IThompson’s experiment: variation Variation on a theme: instead of canceling effects of E and B, one could tune the fields and measure the radius of curvature of the electron beam. Parameters of the problem: V= 300 V I= 1.4 A R= 5 cm Solution: e/m= 2.02 x 1011 C/Kg (cfr: 1.76 x 1011 C/Kg) G. Sciolla –MIT 8.022 – Lecture 13 17 18 Summary and outlook Today: Vector Potential: Biot-Savart Law: Next time: 40π∇ = = 􀁇 􀁇 􀁇 􀁇 􀁇 iB B J c B A≡ ∇× 􀁇 􀁇􀁇 2 ˆdB I rdl c r = × 􀁇􀁇 G. Sciolla – MIT 8.022 – Lecture 13 Toward Maxwell’s equations: What happens when B varies in time? Faraday’s and Lenz’s laws and their applications and ∇× 9