1. Today’s Lecture
Forces and Torques
Magnetism in Materials
Magnetic BCs

2. Forces and Torques F = q(v x B) Work done by B fields is zero !!
This is why classical mechanics doesn’t allow magnetism,
because magnetic fields don’t change energy of charges!
Net force 0

3. Different orientation
F = q(v x B)
= I∫(dl x B)
Net force 0
Torque non 0
T = NIA x B = m x B

4. Same as electric fields
T = NIA x B = m x B
T = (d/2 x F+) + (d/2 x F+)
= qd x E = p x E
F+ = qE + -
F- = -qE

5. Could Force be non-zero on a dipole?
Fx = qEx(dx/2,dy/2)
q[Ex(0,0) +(d/2.)Ex]
Similarly for Fy
Yes – in a spatially varying field
(In fact, field gradients are used to collect polar molecules) - +
Fx = -qEx(-dx/2,-dy/2) 
-q[Ex(0,0) - (d/2.)Ex]
Similarly for Fy
Net force F = (p.)E
For magnets F = (m.)B

6. So far, we explored the force of an external
field on a dipole.
What about the force between two dipoles?

7. How do currents interact? (Magnetostatics)
dl1 I1 I2
dl2 R
i12
i21
Another current senses it
Force ~ i2 x H

8. I1 I2 B =(m0I1/2pD)f F = I2 ∫dl x B F/l = -r(m0I1I2/2pD)
Force Between Wires I1 I2
B =(m0I1/2pD)f
F = I2 ∫dl x B
F/l = -r(m0I1I2/2pD)
Parallel currents
attract

9. + - + - Force Between Dipoles
From the basic notion that
Like charges repel but Parallel currents attract
we can derive that
El. Dip. Mom. p=qd
Magn. Dip. Mom. M=Ia2 + -
Parallel, non-collinear dipoles (arrows) repel + -
Parallel, collinear dipoles (arrows) attract

10. Images X X + -
Opposite charges
Attract
Parallel currents
Attract

11. Fields in material media

12. Dipoles Screen field D E E=(D-P)/e0 rtotal = rfree + rbound
Attracts opposite charges
to reduce field E
E=(D-P)/e0
Opposing
Polarization
field
Total
field
External
field
rtotal =
rfree +
rbound
.D = rfree
-.P = rbound
.E = rtotal/e0

13. What happens near a magnet?
Unmagnetized iron
Magnetizing iron

14. What happens near a magnet?
Sets up little current loops

15. What happens near a magnet?
Only surface currents
survive, but they aid
rather than oppose the
incident H field
(because currents attract
Parallel currents) M
x M = Jbound H

16. What happens near a magnet?
So the net field
gets augmented B
B=m0(H+M)
Aiding
magnetization
field
Total
field
External
field H

17. Magnetic dipoles augment flux
B=m0(H+M)
Aiding
magnetization
field
Total
field
External
field
Jtotal =
Jfree +
Jbound
xH = Jfree
xB = m0Jtotal
x M = Jbound

18. Electric vs magnetic fields
E field weaker inside
B field stronger inside
E = (D-4pP/3)/e0
B = m0(H+2pM/3)
Ignore constants, but they Look opposite!

19. Effect of material on C and L- +
Q increases to make
up for weakening E
C goes up
C = eA/d
Flux increases as field
aligns magnetic domains
L goes up
L = mN2A/l

20. Magnetostatic Boundary Condns
Maxwell equations for E
Supplement with constitutive relation
B=mE
.B = 0
 x H = J
B1n
 B.dS=0
 H.dl = Ifree
B2n
Use Gauss’ law for a short cylinder
Only caps matter (edges are short!)
B1n-B2n = 0
Perpendicular B continuous

21. Magnetostatic Boundary Condns
Maxwell equations for E
Supplement with constitutive relation
B=mE
.B = 0
 x H = J
 B.dS=0
 H.dl = Ifree
H1t
H2t
No net circulation on small loop
Only long edges matter (heights are short!)
H1t-H2t = 0
Parallel H continuous

22. Summary: Main equations
.B = 0
 x H = J
Differential eqns
 B.dA = 0
 H.dl = I
Integral eqns
1st tells us to look for loops
2nd tells us to choose loops
cleverly (Ampere’s Law)
1st defines Vector potential B =  x A
2nd gives Magnetic Poisson’s equation
(.A) - 2A = m0J
Choosing reference
For A
F = q(v x B)
Need moving charges to create
and detect fields !
B1n-B2n = 0
H1t-H2t = Js
Magnetic bcs