Class 17: Outline Hour 1: Dipoles & Magnetic Fields Hour 2:
advertisement
Class 17: Outline Hour 1: Dipoles & Magnetic Fields Hour 2: Expt. 7: Dipoles in B Fields P17- 1 Last Time: Biot-Savart P17- 2 The Biot-Savart Law Current element of length ds carrying current I produces a magnetic field: G G µ 0 I d s × rˆ dB = 2 4π r G G µo q v x rˆ B= 2 4π r Moving charges are currents too… (http://ocw.mit.edu/ans7870/8/8.02T/f 04/visualizations/magnetostatics/03CurrentElement3d/03cElement320.html) P17- 3 PRS Question: Force between wires P17- 4 Magnetic Dipoles: Torque & Force P17- 5 First: Review From Friday P17- 6 Rectangular Current Loop Place rectangular current loop in uniform B field ( G G G G F1 = F3 = 0 IL B ) G F2 = I (− aˆj) × ( B ˆi ) = IaB kˆ G F4 = I (aˆj) × ( B ˆi ) = − IaB kˆ G G G G G Fnet = F1 + F2 + F3 + F4 = 0 ĵ k̂ î No net force on the loop!! P17- 7 Torque on Rectangular Loop Recall: G G G τ = r×F k̂ ĵ x î G ⎛ b ˆ⎞ G ⎛ b ˆ⎞ G τ = ⎜ − i ⎟ × F2 + ⎜ i ⎟ × F4 ⎝ 2 ⎠ ⎝2 ⎠ ( ) ( ) ⎛ b ˆ⎞ ⎛ b ˆ⎞ ˆ = ⎜ − i ⎟ × IaBk + ⎜ i ⎟ × − IaBkˆ ⎝ 2 ⎠ ⎝2 ⎠ Torque Direction: IabB ˆ IabB ˆ = j+ j = IabBˆj Thumb in torque direction, 2 2 fingers rotate with object P17- 8 Torque on Rectangular Loop G ˆ τ = IABj G A = A nˆ = ab nˆ : area vector G nˆ = +kˆ , B =B ˆi k̂ ĵ x î G G G τ = IA × B Familiar? No net force but there is a torque P17- 9 Magnetic Dipole Moment Define Magnetic Dipole Moment: G G µ ≡ IAnˆ ≡ IA Then: G G G τ = µ×B Analogous to G G G τ = p×E τ tends to align µ with B P17- 10 Animation: Another Way To Look At Torque External field connects to field of compass needle and “pulls” the dipole into alignment (http://ocw.mit.edu/ans7870/8/8.02T/f04/visualizations/magnetostatics/18dipNeedle/18-Dip_320.html) P17- 11 Interactive Java Applet: Another Way To Look At Torque http://ocw.mit.edu/ans7870/8/8.02T/f04/visualizatio ns/magnetostatics/35-wireandmagnetapp/35wirecompass320.html Field of wire connects to field of compass needle and “pulls” the dipole into alignment P17- 12 Demonstration: Galvanometer P17- 13 G G µ ≡ IAnˆ ≡ IA Magnetic Dipole Moment P17- 14 PRS Question: Torque on Dipole in Uniform Field P17- 15 Dipoles don’t move??? This dipole rotates but doesn’t feel a net force But dipoles CAN feel force due to B. What’s up? P17- 16 PRS Question: Force on Magnetic Dipole P17- 17 Something New Dipoles in Non-Uniform Fields: Force P17- 18 Force on Magnetic Dipole To determine force, we need to know energy: U Dipole G G = -µ ⋅ B Force tells how the energy changes with position: G G G FDipole = -∇U Dipole = ∇ G G (after math) = µ ⋅∇ ( ( ) G G µ⋅B G B ) Dipoles only feel force in non-uniform field P17- 19 Force on Magnetic Dipole G FDipole G ∂B negative ∂z Force down! G ∂B G G G = µ ⋅∇ B = µ ∂z ( ) P17- 20 Force on Magnetic Dipole Alternate Thought What makes the field pictured? P17- 21 Force on Magnetic Dipole µ N S N S Bar magnet below dipole, with N pole on top It is aligned with the dipole pictured, they attract! P17- 22 Experiment 7: Magnetic Forces on Dipoles P17- 23 Force on Dipole from Dipole: Anti-Parallel Alignment http://ocw.mit.edu/ans7870/8/8.02T/f04/visualizations/magnetostatics/16MagneticForceRepel/16-MagForceRepel_f65_320.html P17- 24 Force on Dipole from Dipole: Parallel Alignment http://ocw.mit.edu/ans7870/8/8.02T/f04/visualizations/magnetostatics/15MagneticForceAttract/15-MagForceAtt_f65_320.html P17- 25 PRS Questions: Force on Magnetic Dipole P17- 26
![[Answer Sheet] Theoretical Question 2](http://s3.studylib.net/store/data/007403021_1-89bc836a6d5cab10e5fd6b236172420d-300x300.png)