Chapter 28 lecture

# Chapter 28 lecture - Chapter 28 Magnetic Fields generated...

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Chapter 28 Magnetic Fields generated by Currents

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Sources of B field B fields exert forces on moving charges and currents (ch. 27) Oersted experiment makes it clear that current-carrying wires exert forces on compass needles Moving charges and currents generate B fields (ch.28) B generated by a moving charged particle—read 28.1 B generated by currents—our primary focus for today Demos: Oersted/tangent galvanometer
Biot-Savart law (circa 1820) ( of a current element) 2 0 2 0 ˆ 4 sin 4 r r l Id B d r Idl dB in in × = = π μ ϕ B=0 I dl r (dB out) (dB in) μ 0 = 4π x 10 -7 Tm/A Integrate along wire path to get total contribution to B at point P P B d φ

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Field lines around current element (end/axial view)
B for a finite straight wire segment ) sin (sin 4 cos 4 cos cos cos 4 2 1 0 0 2 2 2 0 1 2 1 2 θ π μ + = = = - - x I B d x I B x xd I B x Idl P θ 1 θ 2 For an ∞ wire, θ 1 and θ 2 go to 90° x I B wire 2 0 = see also text section 28.3 Ampere’s Law will make this calculation simple……. θ r

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B for the ∞ straight wire r I B wire π μ 2 0 = DEMO
Question 1 If we bend an ∞ wire at 90° what is B at point P? P I 1. 0 2. μ 0 I/(2πr) 3. μ 0 I/(4πr) 4. too difficult You can chop an ∞ wire in half to get a semi-infinite wire which has half the B field, by symmetry…. .but you can’t halve it any further!

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B at center of a Circular current loop (a magnetic dipole) a dl = adθ loops N for a NI B a I a I B a Iad B center loop 2 2 2 4 4 90 sin 0 0 0 2 0 2 0 μ π θ = = = = For B along axis see derivation in section 28.5: above result is for x=0 in eq. 28.15
B field of circular loop μ Field lines are more complicated than the circles of an ∞ wire

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Question 2 What is B at point P, at the center of a semi-circular wire of radius=a with ∞ straight leads? P 1. μ 0 I/(4a) + μ 0 I/(2πa) 2. μ 0 I/(2a) 3. μ 0 I/(4a) 4. μ 0 I/(4πa)
Question 3 What is the B field at point P for a circular segment with ∞ leads? P 45° 1. μ 0 Icos45°/2a 2. μ 0 I/8a 3. μ 0 I/8a 4. μ 0 I/12a 5. μ 0 I/16a In general we can chop up a circular current loop as much as we want because each little piece of the arc contributes the same amount to the total B field

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Magnetic force between parallel wires B 1 I 1 B 1 0 I 1 /(2πr) I 2 (out) 2 1 0 2 1 0 1 2 2 2 2 I I r L F L I I r LB I F on π μ = = = Total B fields by superposition DEMOS
Ampere’s Law (Magnetic Materials moved to Ch 29) Time to get serious/mathematical about B = × = = = = × = = = ? or ? 0

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Chapter 28 lecture - Chapter 28 Magnetic Fields generated...

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