Electricity and Magnetism-Applying Ampere's Circuital Law

1P18-Class 18: Outline Hour 1: Levitation Experiment 8: Magnetic Forces Hour 2: Ampere’s Law 2P18-Review: Right Hand Rules 1. Torque: Thumb = torque, fingers show rotation 2. Feel: Thumb = I, Fingers = B, Palm = F 3. Create: Thumb = I, Fingers (curl) = B 4. Moment: Fingers (curl) = I, Thumb = Moment 3P18-Last Time: Dipoles 4P18-Magnetic Dipole Moments Anµ GG IIA ≡≡ ˆ Generate: Feel: 1) Torque to align with external field 2) Forces as for bar magnets (seek field) -Dipole U= ⋅µB GG 5P18-Some Fun: Magnetic Levitation 6P18-Put a Frog in a 16 T Magnet… For details: http://www.hfml.sci.kun.nl/levitate.html 7P18-How does that work? First a BRIEF intro to magnetic materials 8P18-Para/Ferromagnetism Applied external field B0 tends to align the atomic magnetic moments (unpaired electrons) 9P18-Diamagnetism Everything is slightly diamagnetic. Why? More later. If no magnetic moments (unpaired electrons) then this effect dominates. 10P18-Back to Levitation 11P18-Levitating a Diamagnet 1) Create a strong field (with a field gradient!) 2) Looks like a dipole field 3) Toss in a frog (diamagnet) 4) Looks like a bar magnet pointing opposite the field 5) Seeks lower field (force up) which balances gravity S NN S Most importantly, its stable: Restoring force always towards the center SN 12P18-Using ∇B to Levitate For details: http://www.hfml.ru.nl/levitation-movies.html •Frog •Strawberry •Water Droplets •Tomatoes •Crickets 13P18-Demonstrating: Levitating Magnet over Superconductor 14P18-Perfect Diamagnetism: “Magnetic Mirrors” N S N S 15P18-Perfect Diamagnetism: “Magnetic Mirrors” N S N S No matter what the angle, it floats --STABILITY 16P18-Using ∇B to Levitate For details: http://www.hfml.sci.kun.nl/levitate.html A Sumo Wrestler 17P18-Two PRS Questions Related to Experiment 8: Magnetic Forces 18P18-Experiment 8: Magnetic Forces (Calculating µ0) http://ocw.mit.edu/ans7870/8/8.02T/f04/visualizations/magnetostatics/16-MagneticForceRepel/16-MagForceRepel_f65_320.html 19P18-Experiment Summary: Currents feel fields Currents also create fields Recall… Biot-Savart 20P18-The Biot-Savart Law Current element of length ds carrying current I produces a magnetic field: 2 0 ˆ 4 r dI rsBd× = GGπ µ 21P18-Today: 3rd Maxwell Equation: Ampere’s Law Analogous (in use) to Gauss’s Law 22P18-Gauss’s Law – The Idea The total “flux” of field lines penetrating any of these surfaces is the same and depends only on the amount of charge inside 23P18-Ampere’s Law: The Idea In order to have a B field around a loop, there must be current punching through the loop 24P18-Ampere’s Law: The Equation The line integral is around any closed contour bounding an open surface S. Ienc is current through S: ∫ =⋅ encId 0µsBGG enc S I d= ⋅∫ JA GG 25P18-PRS Question: Ampere’s Law 26P18-Biot-Savart vs. Ampere Biot-Savart Law general current source ex: finite wire wire loop Ampere’s law symmetric current source ex: infinite wire infinite current sheet 0 2 ˆ 4 I d r µ π × = ∫ srB GG ∫ =⋅ encId 0µsB GG 27P18-Applying Ampere’s Law 1. Identify regions in which to calculate B field Get B direction by right hand rule 2. Choose Amperian Loops S: Symmetry B is 0 or constant on the loop! 3. Calculate 4. Calculate current enclosed by loop S 5. Apply Ampere’s Law to solve for B ∫ ⋅ sB GG d ∫ =⋅ encId 0µsB GG 28P18-Always True, Occasionally Useful Like Gauss’s Law, Ampere’s Law is always true However, it is only useful for calculation in certain specific situations, involving highly symmetric currents. Here are examples… 29P18-Example: Infinite Wire I A cylindrical conductor has radius R and a uniform current density with total current I Find B everywhere Two regions: (1) outside wire (r ≥ R) (2) inside wire (r < R) 30P18-Ampere’s Law Example: Infinite Wire I I B Amperian Loop: B is Constant & Parallel I Penetrates 31P18-Example: Wire of Radius R Region 1: Outside wire (r ≥ R) d⋅∫ Bs GG v ckwisecounterclo2 0 r I π µ=B GB d= ∫ s G v( )2B rπ= 0 enc Iµ= 0 Iµ= Cylindrical symmetry ÆAmperian Circle B-field counterclockwise 32P18-Example: Wire of Radius R Region 2: Inside wire (r < R) 2 0 2 rI R π µ π ⎛ ⎞ = ⎜ ⎟ ⎝ ⎠ ckwisecounterclo2 2 0 R Ir π µ=B G Could also say: ( )2 22 ; rR IJAIR I A IJ encenc πππ ==== d⋅∫ Bs G G v B d= ∫ s G v ( )2B rπ= 0 enc Iµ= 33P18-Example: Wire of Radius R 2 0 2 R IrBinπ µ= r IBoutπµ 2 0= 34P18-Group Problem: Non-Uniform Cylindrical Wire I A cylindrical conductor has radius R and a non-uniform current density with total current: Find B everywhere 0 RJ r =J G 35P18-Applying Ampere’s Law In Choosing Amperian Loop: • Study & Follow Symmetry • Determine Field Directions First • Think About Where Field is Zero • Loop Must • Be Parallel to (Constant) Desired Field • Be Perpendicular to Unknown Fields • Or Be Located in Zero Field 36P18-Other Geometries 37P18-Helmholtz Coil 38P18-Closer than Helmholtz Coil 39P18-Multiple Wire Loops 40P18-Multiple Wire Loops – Solenoid 41P18-Magnetic Field of Solenoid loosely wound tightly wound For ideal solenoid, B is uniform inside & zero outside 42P18-Magnetic Field of Ideal Solenoid d d d d d⋅ ⋅ + ⋅ + ⋅ + ⋅∫ ∫ ∫ ∫ ∫ 1 2 3 4 B s = B s B s B s B s G G G G GG G G G G v Using Ampere’s law: Think! along sides 2 and 4 0 along side 3 d⎧ ⊥⎪⎨ =⎪⎩ B s B G G G n: turn density enc I nlI = 0d Bl nlI µ⋅ = =∫ B s G G v 0 0 nlIB nIl µ µ= =/: # turns/unit length nN L= 0 0 0Bl= + + + 43P18-Demonstration: Long Solenoid 44P18-Group Problem: Current Sheet A sheet of current (infinite in the y & z directions, of thickness 2d in the x direction) carries a uniform current density: Find B everywhere ˆ s J=J k G y I 45P18-Ampere’s Law: Infinite Current Sheet Amperian Loops: B is Constant & Parallel OR Perpendicular OR Zero I Penetrates B B 46P18-Solenoid is Two Current Sheets Field outside current sheet should be half of solenoid, with the substitution: 2nI dJ= This is current per unit length (equivalent of λ, but we don’t have a symbol for it) I 47P18-= 2 Current Sheets Ampere’s Law: .∫ =⋅ encId 0µsB GG B B X X X X X X X X X X X X X X X X X X X X X X X X X X X X B Long Circular Symmetry (Infinite) Current Sheet Solenoid Torus 48P18-Brief Review Thus Far… 49P18-Maxwell’s Equations (So Far) 0Ampere's Law: Currents make curling Magnetic Fields enc C d Iµ⋅ =∫ B s G G v Magnetic Gauss's Law: 0 No Magnetic Monopoles! (No diverging B Fields) S d⋅ =∫∫ BA GG w 0 Gauss's Law: Electric charges make diverging Electric Fields in S Qd ε ⋅ =∫∫ EA GG w