Chapter 29: Magnetic Fields Due to Currents. PHY2049: Chapter 29 1

Transcription

1 Chapter 29: Magnetic Fields Due to Currents PHY2049: Chapter 29 1

2 Law of Magnetism Unlike the law of static electricity, comes in two pieces Piece 1: Effect of B field on moving charge r r F = qv B (Chapt. 28) Piece 2: B field produced by current Equivalent! Biot-Savart Law Ampere s Law Proof of equivalence not in the book (Requires vector calculus and relies on the absence of magnetic monopoles) Reminds you of similar equivalence between Coulomb s law and Gauss law PHY2049: Chapter 29 2

3 Creating Magnetic Fields Sources of magnetic fields Electric current (moving charges) Atomic orbits of electrons (angular momentum L > 0 only) Internal spin of elementary particles (mostly electrons) Magnetic field produced by current is fundamental How about field produced by a bar magnet? Bar magnet magnetic ions orbital motion and spin of electrons in them they are microscopic currents Three examples studied here Long wire Wire loop Solenoid PHY2049: Chapter 29 3

4 B Field Around Very Long Wire Field around wire is circular, intensity falls with distance Direction given by RHR #2 (compass follows field lines) B = µ i 0 2π r µ 0 = 4π 10 7 Right Hand Rule #2 Derived from Ampere s law PHY2049: Chapter 29 4

5 (continued) Why does µ 0 have such a simple value? Magnetism is inseparable from electricity. This allows the units in electricity and magnetism (in particular coulomb and tesla) to be chosen so that only one constant, ε 0, has a non-trivial value. This example illustrates important general property of magnetic fields: Magnetic field lines have no beginning/end, unlike electric field lines. PHY2049: Chapter 29 5

6 Long Wire B Field Example I = 500 A toward observer. Find B vs r RHR #2 field is counterclockwise ( 7 π ) µ i B = = = 2π r 2π r r r = 1 mm B = 0.10 T = 1000 gauss r = 1 cm (~0.4 ) B = T = 100 gauss r = 10 cm (~4 ) B = T = 10 gauss PHY2049: Chapter 29 6

7 Charged Particle Moving Near Wire Wire carries current of 400 A upwards Proton moving at v = m/s downwards, 4 mm from wire Find magnitude and direction of force on proton Solution Direction of force is to left, away from wire Magnitude of force at r = 4 mm µ 0I F = evb = ev 2π r 7 ( 19 )( 6 ) F = F = N v I PHY2049: Chapter 29 7

8 Ampere s Law First (Biot-Savart law later) Take arbitrary closed path around set of currents Let i enc be total enclosed current (signs +/ according to RHR #2) Let B be magnetic field, and ds be differential length along path Not included B d s = µ i 0 enc in i enc Direction of field due to each current element obeys RHR #2 Only currents inside path count! 5 currents inside path (included) 1 outside path (not included) This does not mean that current outside path does not contribute to B (note similarity to Gauss law) PHY2049: Chapter 29 8

9 Ampere s Law For Straight Wire Let s try this for long wire. Find B at distance at point P According to RHR #2, B field has only azimuthal component, no radial component Draw circular path passing through P (radius r) From symmetry, strength of B must be constant along path P B d s = B( 2π r) = µ 0i r µ 0i B = 2π r An easy derivation PHY2049: Chapter 29 9

10 Ampere s Law: More Application Find B vs r inside long wire, assuming uniform current Wire radius R, total current i Draw circular path of radius r Key fact: enclosed current area 2 Inside r i enc = i r 2 R B ( 2πr) = µ 0ienc µ 0ir B = µ 2 0i 2πR B = On surface 2π R Outside: µ 0i B = (derived in previous slide) 2 πr PHY2049: Chapter R

11 Question 10 Figure shows the magnitude of B field inside and outside four long wires. Current is uniformly distributed in each wire. Which wire carries the largest current? (a) 1 (b) 1 and 2 (c) 1 and 3 (d) Insufficient info In which wire is the current density the highest? (a) 1 (b) 1 and 2 (c) 1 and 3 (d) Insufficient info B 2 4 PHY2049: Chapter r

12 Force Between Two Parallel Currents Force on I 2 from I 1 µ 0I1 µ 0I1I2 F2 = I2B1L= I2 L L 2πr = 2πr RHR Force towards I 1 Force on I 1 from I 2 Must be the same and towards I 2 Why? Newton s third law Or view from behind the screen. (I 1 is now on left, and I 2 now on right.) Magnetic forces attrac two parallel currents I 2 I 1 I 2 I 1 PHY2049: Chapter 29 12

13 Force Between Two Anti-Parallel Currents Force on I 2 from I 1 µ 0I1 µ 0I1I2 F2 = I2B1L= I2 L L 2πr = 2πr RHR Force away from I 1 Force on I 1 from I 2 Must be the same and away from I 2 Magnetic forces repel two antiparallel currents I 2 I 1 I 2 I 1 PHY2049: Chapter 29 13

14 Parallel Currents (cont.) Look at them edge on to see B fields more clearly B 2 1 Antiparallel: repel 2 1 B F F B 2 1 Parallel: attract 2 1 B F F PHY2049: Chapter 29 14

15 Question 6 Long wires, carrying equal currents, are parallel to each other and equally spaced. In which arrangement is the net force on the central wire the largest? (a) a (b) b (c) c (d) d (e) Insufficient info Wires are of equal lengths. (a) a (b) b (c) c (d) d OUT IN OUT IN OUT OUT OUT IN IN IN OUT IN OUT OUT IN IN OUT IN IN IN PHY2049: Chapter 29 15