PHY2049: Chapter 291Chapter 29: Creating Magnetic FieldsPHY2049: Chapter 292Creating Magnetic FieldsÎSources of magnetic fields Spin of elementary particles (mostly electrons) Atomic orbits (L > 0 only) Moving charges (electric current)ÎCurrents generate the most intense magnetic fields Discovered by Oersted in 1819 (deflection of compass needle)ÎThree examples studied here Long wire Wire loop SolenoidPHY2049: Chapter 293B Field Around Very Long WireÎField around wire is circular, intensity falls with distance Direction given by RHR (compass follows field lines)02iBrμπ=70410μπ−=×Right Hand Rule #2PHY2049: Chapter 294Long Wire B Field ExampleÎI = 500 A toward observer. Find B vs r RHR ⇒ field is counterclockwise r = 0.001 m B = 0.10 T = 1000 G r = 0.005 m B = 0.02 T = 200 G r = 0.01 m B = 0.010 T = 100 G r = 0.05 m B = 0.002 T = 20 G r = 0.10 m B = 0.001 T = 10 G r = 0.50 m B = 0.0002 T = 2 G r = 1.0 m B = 0.0001 T = 1 G()704 10 5000.000122iBrrrπμππ−×== =PHY2049: Chapter 295Charged Particle Moving Near WireÎWire carries current of 400 A upwards Proton moving at v = 5 × 106m/s downwards, 4 mm from wire Find magnitude and direction of force on protonÎSolution Direction of force is to left, awayfrom wire Magnitude of force at r = 4 mmIv02IF evB evrμπ⎛⎞==⎜⎟⎝⎠()()719 62 10 4001.6 10 5 100.004F−−⎛⎞××=× ×⎜⎟⎜⎟⎝⎠141.6 10 NF−=×PHY2049: Chapter 296Ampere’s LawÎTake arbitrary path around set of currents Let iencbe total enclosed current (+ up, − down) Let B be magnetic field, and ds be differential length along pathÎOnly currents inside path contribute! 5 currents inside path (included) 1 outside path (not included)0encdiμ⋅=∫BsvNot includedin iencPHY2049: Chapter 297Ampere’s Law For Straight WireÎLet’s try this for long wire. Find B at distance at point P Use circular path passing through P (radius r) From symmetry, B field must be circularÎAn easy derivation()0022dB r iiBrπμμπ⋅= ==∫BsvrPPHY2049: Chapter 298Useful Application of Ampere’s LawÎFind B vs r inside long wire, assuming uniform current Wire radius R, total current i Find B at radius r = R/2ÎKey fact: enclosed current ∝ area()2enc2/24RiiiRππ⎛⎞⎜⎟==⎜⎟⎝⎠00224122RiBiBRπμμπ⎛⎞=⎜⎟⎝⎠=rR02iBRμπ=On surfacePHY2049: Chapter 299Force Between Two Parallel CurrentsÎForce on I2from I1RHR ⇒ Force towards I1ÎForce on I1from I2RHR ⇒ Force towards I2ÎMagnetic forces attracttwo parallel currentsI1I201 012221 222IIIFIBLI L Lrrμμππ⎛⎞== =⎜⎟⎝⎠I1I202 012112 122IIIFIBLI L Lrrμμππ⎛⎞== =⎜⎟⎝⎠PHY2049: Chapter 2910Force Between Two Anti-Parallel CurrentsÎForce on I2from I1RHR ⇒ Force away from I1ÎForce on I1from I2RHR ⇒ Force away from I2ÎMagnetic forces repeltwo antiparallel currentsI1I2I1I201 012221 222IIIFIBLI L Lrrμμππ⎛⎞== =⎜⎟⎝⎠02 012112 122IIIFIBLI L Lrrμμππ⎛⎞== =⎜⎟⎝⎠PHY2049: Chapter 2911Parallel Currents (cont.)ÎLook at them edge on to see B fields more clearlyAntiparallel: repelFFParallel: attractFFBBBBPHY2049: Chapter 2912B Field @ Center of Circular Current LoopÎRadius R and current i: find B field at center of loop Direction: RHR #3 (see picture)ÎIf N turns close together02iBRμ=02NiBRμ=From calculusPHY2049: Chapter 2913Current Loop ExampleÎi = 500 A, r = 5 cm, N=20()()7020 4 10 5001.26T2 2 0.05iBNrπμ−×== =×PHY2049: Chapter 2914Field at Center of Partial LoopÎSuppose loop covers angle φ Calculate B field from proportion of full circleÎUse example where φ = π (half circles) Define direction into page as positive022iBRμφπ⎛⎞=⎜⎟⎝⎠001201222 22114iiBRRiBRRμμππππμ⎛⎞ ⎛⎞=−⎜⎟ ⎜⎟⎝⎠ ⎝⎠⎛⎞=−⎜⎟⎝⎠PHY2049: Chapter 2915Partial Loops (cont.)ÎNote on problems when you have to evaluate a B field at a point from several partial loops Only loop parts contribute, proportional to angle (previous slide) Straight sections aimed at point contribute exactly 0 Be careful about signs, e.g.in (b) fields partially cancel, whereas in (a) and (c) they
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