1P17-

Class 17: Outline

Hour 1:Dipoles & Magnetic Fields

Hour 2:Expt. 7: Dipoles in B Fields

2P17-

Last Time:Biot-Savart

3P17-

The Biot-Savart LawCurrent element of length ds carrying current Iproduces a magnetic field:

20 ˆ

4 rdI rsBd ×=

πµ

2

ˆx4

o qr

µπ

=v rB

Moving charges are currents too…

(http://ocw.mit.edu/ans7870/8/8.02T/f04/visualizations/magnetostatics/03-CurrentElement3d/03-cElement320.html)

http://ocw.mit.edu/ans7870/8/8.02T/f04/visualizations/magnetostatics/03-CurrentElement3d/03-cElement320.htmlhttp://ocw.mit.edu/ans7870/8/8.02T/f04/visualizations/magnetostatics/03-CurrentElement3d/03-cElement320.htmlhttp://ocw.mit.edu/ans7870/8/8.02T/f04/visualizations/magnetostatics/03-CurrentElement3d/03-cElement320.htmlhttp://ocw.mit.edu/ans7870/8/8.02T/f04/visualizations/magnetostatics/03-CurrentElement3d/03-cElement320.htmlhttp://ocw.mit.edu/ans7870/8/8.02T/f04/visualizations/magnetostatics/03-CurrentElement3d/03-cElement320.html

4P17-

PRS Question:Force between wires

5P17-

Magnetic Dipoles:Torque & Force

6P17-

First: Review From Friday

7P17-

Rectangular Current LoopPlace rectangular current loop in uniform B field

( )BLFF I031 ==2

ˆ ˆ ˆ( ) ( )I a B IaB= − × =F j i k

4ˆ ˆ ˆ( ) ( )I a B IaB= × = −F j i k

1 2 3 4 0net = + =F F F + F + F

î

ĵ

k̂ No net force on the loop!!

8P17-

Torque on Rectangular Loop

2 4ˆ ˆ

2 2b b⎛ ⎞ ⎛ ⎞= − × + ×⎜ ⎟ ⎜ ⎟

⎝ ⎠ ⎝ ⎠τ i F i F

Frτ ×=

Torque Direction:Thumb in torque direction, fingers rotate with object

Recall:

( ) ( )ˆ ˆˆ ˆ2 2b bIaB IaB⎛ ⎞ ⎛ ⎞= − × + × −⎜ ⎟ ⎜ ⎟

⎝ ⎠ ⎝ ⎠i k i k

ˆ ˆ ˆ2 2

IabB IabB IabB= + =j j j

îĵk̂

x

9P17-

Torque on Rectangular Loop

jτ ˆIAB=

ˆ ˆ: area vectorA ab= =A n nˆˆˆ , =B= +n k B i

îĵk̂

x

I= ×τ A BFamiliar? No net force but there is a torque

10P17-

Magnetic Dipole Moment

Anµ IIA ≡≡ ˆDefine Magnetic Dipole Moment:

= ×τ µ BThen:

Analogous to = ×τ p E

τ tends to align µ with B

11P17-

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/18-dipNeedle/18-Dip_320.html)

http://ocw.mit.edu/ans7870/8/8.02T/f04/visualizations/magnetostatics/18-dipNeedle/18-Dip_320.htmlhttp://ocw.mit.edu/ans7870/8/8.02T/f04/visualizations/magnetostatics/18-dipNeedle/18-Dip_320.htmlhttp://ocw.mit.edu/ans7870/8/8.02T/f04/visualizations/magnetostatics/18-dipNeedle/18-Dip_320.html

12P17-

Interactive Java Applet:Another Way To Look At Torque

Field of wire connects to field of compass needle and “pulls” the dipole into alignment

http://ocw.mit.edu/ans7870/8/8.02T/f04/visualizations/magnetostatics/35-wireandmagnetapp/35-wirecompass320.html

http://ocw.mit.edu/ans7870/8/8.02T/f04/visualizations/magnetostatics/35-wireandmagnetapp/35-wirecompass320.htmlhttp://ocw.mit.edu/ans7870/8/8.02T/f04/visualizations/magnetostatics/35-wireandmagnetapp/35-wirecompass320.htmlhttp://ocw.mit.edu/ans7870/8/8.02T/f04/visualizations/magnetostatics/35-wireandmagnetapp/35-wirecompass320.html

13P17-

Demonstration:Galvanometer

14P17-

Magnetic Dipole Moment

Anµ IIA ≡≡ ˆ

15P17-

PRS Question:Torque on Dipole in Uniform Field

16P17-

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?

17P17-

PRS Question:Force on Magnetic Dipole

18P17-

Something NewDipoles in Non-Uniform Fields:

Force

19P17-

Force on Magnetic Dipole

-DipoleU = ⋅µ BTo determine force, we need to know energy:

( ) -Dipole DipoleU= ∇ = ∇ ⋅F µ BForce tells how the energy changes with position:

( )(after math) = ⋅∇µ BDipoles only feel force in non-uniform field

20P17-

Force on Magnetic Dipole

( )Dipole zµ∂

= ⋅∇ =∂BF µ B

negative

Force down!z

∂∂B

21P17-

Force on Magnetic Dipole

Alternate ThoughtWhat makes the field pictured?

22P17-

Force on Magnetic Dipoleµ

N

S

N

S

Bar magnet below dipole, with N pole on top It is aligned with the dipole pictured, they attract!

23P17-

Experiment 7:Magnetic Forces on Dipoles

24P17-

Force on Dipole from Dipole:Anti-Parallel Alignment

http://ocw.mit.edu/ans7870/8/8.02T/f04/visualizations/magnetostatics/16-MagneticForceRepel/16-MagForceRepel_f65_320.html

http://ocw.mit.edu/ans7870/8/8.02T/f04/visualizations/magnetostatics/16-MagneticForceRepel/16-MagForceRepel_f65_320.htmlhttp://ocw.mit.edu/ans7870/8/8.02T/f04/visualizations/magnetostatics/16-MagneticForceRepel/16-MagForceRepel_f65_320.html

25P17-

Force on Dipole from Dipole:Parallel Alignment

http://ocw.mit.edu/ans7870/8/8.02T/f04/visualizations/magnetostatics/15-MagneticForceAttract/15-MagForceAtt_f65_320.html

http://ocw.mit.edu/ans7870/8/8.02T/f04/visualizations/magnetostatics/15-MagneticForceAttract/15-MagForceAtt_f65_320.htmlhttp://ocw.mit.edu/ans7870/8/8.02T/f04/visualizations/magnetostatics/15-MagneticForceAttract/15-MagForceAtt_f65_320.html

26P17-

PRS Questions:Force on Magnetic Dipole

Class 17: OutlineLast Time:Biot-SavartThe Biot-Savart LawPRS Question:Force between wiresMagnetic Dipoles:Torque & ForceFirst: Review From FridayRectangular Current LoopTorque on Rectangular LoopTorque on Rectangular LoopMagnetic Dipole MomentAnimation:Another Way To Look At TorqueInteractive Java Applet:Another Way To Look At TorqueDemonstration:GalvanometerMagnetic Dipole MomentPRS Question:Torque on Dipole in Uniform FieldDipoles don’t move???PRS Question:Force on Magnetic DipoleSomething NewDipoles in Non-Uniform Fields: ForceForce on Magnetic DipoleForce on Magnetic DipoleForce on Magnetic DipoleForce on Magnetic DipoleExperiment 7:Magnetic Forces on DipolesForce on Dipole from Dipole:Anti-Parallel AlignmentForce on Dipole from Dipole:Parallel AlignmentPRS Questions:Force on Magnetic Dipole