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Magnetism
I. History of Magnetism Studies
A. 13th Century B.C. – Chinese are using compasses with magnetic needles that were
probably invented in Arabia or India
B. 500 B.C. – Greeks are using magnetite (Fe3O4) to attract iron filings. The name of
the stone was probably derived from the mythical shepherd Magnes whose shoes and
the tip of whose staff stuck to magnetite
C. 1269 A.D. – Frenchman Pierre de Maricourt mapped out magnetic field lines and
magnetic poles
1. poles were named north and south because when freely suspended, they lined
up with those directions on Earth
2. magnetic poles are always found in pairs – magnetic monopoles have never
been isolated
D. 1600 – William Gilbert proposes that Earth is itself a magnet with a magnetic field
E. 1750 – Experiments show that like magnetic poles repel; unlike magnetic poles
attract; and magnetic force is inversely proportional to the square of the distance
between interacting poles.
F. 1819 – Oersted discovers the relationship between electricity and magnetism during a
lecture demonstration (Romognosi actually observed this in 1802 but failed to
publish his results!)
G. 1820’s – André Ampère formulated quantitative laws for calculating the magnetic
force between current-carrying wires. He also suggests that on an atomic level,
electric current loops produce all magnetism
H. 1820’s - Michael Faraday and Joseph Henry showed that moving magnets could
produce electric currents
I. 1880’s James Clerk Maxwell showed that changing electric fields create magnetic
fields
J. Common Electromagnetic Phenomena
1. auroras
2. magnetic conduction (magnetizing unmagnetized material with a magnet)
3. magnetic induction (moving a magnet to produce a changing electric field
which in turn induces another magnetic field)
4. electromagnets
II. Magnetic Field
A. Exists due to the presence of magnetic poles
B. Definitions
1. Given the symbol B
2. Direction is specified by the orientation of the
magnetic poles
3. lines always leave N poles and terminate on S
poles
4. magnetic field demo with iron filings tracing out
magnetic fields (see Figure 32.1)
C. Magnetic Force
1. proportional to magnitude of moving charge q
2. proportional to particle velocity v
3. proportional to magnitude of B
4. depends upon direction of v and B
5. FB = 0 when v is parallel to B
6. FB is perpendicular to both v and B
7. + and – charges produce magnetic forces in
opposite directions
8. proportional to sin 
9. FB = qv x B
a. || FB ||=|q|vB sin 
b. direction of force is direction of v x B
if q is + and oppositely directed if q is
negative
c. direction is given by right-hand rule.
Point fingers of right hand in the
direction of v, curl them in the
direction of B, and then thumb points in direction of FB for + charges.
Reverse thumb direction for – charges.
d. Fmax = |q|vB
D. Differences between Electric and Magnetic Forces
1. FE is parallel to E; FB is perpendicular to B
2. FE acts on any charges whether the charges are stationary or moving; FB acts
only on moving charges
3. FE does work in displacing a charged particle; FB does no work in displacing a
charged particle
E. Magnetic Field and Kinetic Energy
1. B cannot change speed of charged particle (||v||), but it can change the
direction of v
1
1 2
2. KE  m(v v)  mv which is a scalar
2
2
3. W = KE = 0 so B does no work
F. Units of B
F
N
N
 B has units of

1. B 
qv
C(m/s) Am
N
2. 1 Tesla (T) = 1
Am
3. I T = 104 G where 1 G = 1 Gauss
G. Table 29.1 contains typical magnetic field values
1. superconducting magnet = 30 T
2. conventional magnet = 2 T
3. MRI unit = 1.5 T
4. bar magnet = 10-2 T
5. surface of the sun = 10-2 T
6. surface of Earth = 0.5 x 10-4 T
7. human brain = 10-13 T
H. Example 29.1 in class
I. Magnetic Field of Earth
1. N magnetic pole coincides with S
rotational pole and vice-versa. Magnetic
field lines saturated with radiation
produce Van Allen radiation belts
2. Horizontal component of Earth’s
magnetic field currently points toward
Hudson Bay in Canada
3. At Hudson Bay, compass needles point
straight down
4. The difference between the magnetic
pole and the rotational pole is called the
magnetic declination. The magnetic
declination varies with time; but its average position coincides with the
rotational poles over long time scales (see Figure 32.4).
5. Convection currents in Earth’s outer core generate magnetic fields. Due to
high temperatures, the magnet is not permanent.
6. Strength of magnetic field depends upon rotation rate which partially explains
Jupiter’s strong field and Venus’ absent field
7. Magnetic field reverses polarity ~100,000 years as recorded by magnetic
stripes on the seafloor
III. Magnetic Force Acting on Current-Carrying Conductors
A. A magnetic force is exerted on a charge when it passes through a magnetic field. The
magnetic force on a current-carrying wire is the vector sum of the magnetic force on
all of the individual charges.
B. Let n = # charge carriers/volume and let
q = avg charge per charge carrier in
straight wire
1. FB  qv d  B
2. FB   FB  N FB
FB  (nAL) FB
FB  nAL(qv d  B)
3. Recall that I  (nqA) v d
4. FB  L(I  B) or FB  I (L  B)
where L and I are defined to be in the same direction.
C. Arbitrary Shape
1. dFB  I (ds  B)
b
2. FB  I  (ds  B)
a
D. Two Special Cases in Uniform B
1. curved wire of length L but displacement L’
b
a. vector sum of
 ds = L’
a
b
b. FB  I (  ds)  B
a
c. FB  I (L' B)
2. closed loop of length L but displacement 0
b
a. vector sum of
 ds = 0
a
b
b. FB  I (  ds)  B
a
c. FB  0
d. net magnetic force acting on a closed loop in uniform B field is 0.
E. Example 29.6
F. Biot-Savart Law
1. used to define strength of magnetic field
2. experimental observations made by Jean-Baptiste Biot and Felix Savart
a. dB  ds and dB  rˆ
1
b. dB  2
r
c. dB  I and dB  ds
dB  sin  where  is the angle between ds and r̂
I ds  rˆ
3. dB  K m
(Biot-Savart law)
r2

4. K m  0 where 0 = 4 x 10-7 T m/A is the magnetic permeability of free
4
space
 I ds  rˆ
5. B  0 
(Biot-Savart Law in integral form)
4
r2
6. valid for any current either in a wire or flowing through space
7. comparisons between Coulomb’s law and the Biot-Savart law
a. current element produces a magnetic field whereas a point charge
produces an electric field
b. both electric and magnetic fields obey an inverse-square law
c. E is radial from a point charge, but B is perpendicular to both r̂ and ds
d. E is created by a point charge, but B requires an integration over an
extended current distribution composed of many moving (flowing)
charges
8. Biot-Savart law is only valid for current-carrying conductors, and it is
independent of any external B-fields
9. Examples 30.1 and 30.2 done in class
d.
10. Using right-hand rule and Biot-Savart law, direction of B can be determined
by grasping a wire with the right hand such that the thumb points in the
direction of the current. Then B is composed of concentric circles curling in
the direction of the fingers.
G. Magnetic Field around an Infinitely Long CurrentCarrying Wire
 I ds  rˆ
B 0 
4
r2
 I ds sin 
B 0  2

4 ( s  R 2 )
 I
ds
R
B 0  2

2
4 ( s  R ) ( s 2  R 2 ) 12
B
 0 IR 
ds


4  ( s 2  R 2 ) 3 2
 IR
s
B 0
4 R 2 ( s 2  R 2 ) 12
B
0 I 
1

4R   1  ( R ) 2
s

0 I
[1  (1)] 
4R
 I
B 0 
2R





 

 
B
H. Magnetic Force between Two Parallel Wires
1. current in wire 1 senses magnetic field
from wire 2 (B2)
2. current in wire 2 senses magnetic field
from wire 1 (B1)
3. since wire 1 || wire 2, I1 || I2. Because
B2  I2, B2  I1
4. F1  L(I1  B 2 )
F1  LI1 B2 (zˆ )  LI1 B2 zˆ
5. F2  L(I 2  B1 )
F2  LI 2 B1 (zˆ )   LI 2 B1 zˆ
 I
6. B  0 from Biot-Savart Law
2 a
 I
 II L
7. F1  LI1 0 2 (zˆ )   0 1 2 zˆ
2 a
2 a
F2  LI 2
 0 I1
 II L
(zˆ )   0 1 2 zˆ
2 a
2 a
8. F1 = - F2 as required by Newton’s 3rd law
9. Parallel currents in the same direction are attractive. Parallel currents in
opposite direction are repulsive.
10. units of Amperes and Coulombs are defined by the force between two parallel
wires
F
 2 x 10-7 N/m
a. current in each wire = 1 A if
L
b. current balances like those used in lab can be used to calculate
magnetic forces and electric currents
c. 1 C = (1A) x (1 sec)
11. edge effects have been neglected, so at least one wire must have L >> a for
this to be valid
IV. Torque on a current Loop in a Uniform B-Field
A.  FB  0 on a current loop in a B-field
B. τ  r  F on a wire
1. F  L(I  B i )
a. F1  L1 IBi sin  
F1  bIBi sin  
b. F2  L2 IBi sin( 90   ) Θ
F2  aIBi cos Θ
c. F3  L3 IBi sin  Θ
F3  bIBi sin  Θ
d. F4  L4 IBi sin( 90   ) 
F4  aIBi cos 
e.
F  F  F  F  F
 F  bIB sin   aIB cos
F  0
1
2
3
i
2. τ  r  F where r  F
a. τ1  r1F1rˆ2
a
τ 1  ( )(bIBi sin  ) rˆ2
2
τ1  12 abIBi sin  rˆ2
b. τ 2  r2 F2 r̂1
b
τ 2  ( )( aIBi cos ) rˆ1
2
1
τ 2  2 abIBi cos rˆ1
4
i
Θ + bIBi sin  Θ + aIBi cos 
c. τ 3  r3 F3rˆ2
a
τ 3  ( )(bIBi sin  ) rˆ2
2
1
τ 3  2 abIBi sin  rˆ2
d. τ 4  r4 F4 r̂1
b
τ 4  ( )( aIBi sin  ) rˆ1
2
τ 4  12 abIBi sin  rˆ1
3.
τ  τ  τ  τ  τ
 τ  abIB sin  rˆ  abiB cos rˆ
1
2
i
3
2
4
i
1
τ  IabBi  IABi
4. B  B i  B j  B k
B j  0 by definition
B  B i  B k  B is in the ik-plane
A is defined in the k̂ direction since loop is in the ij-plane
Bi  B sin 
τ  IabBi  IAB sin 
5. τ  IA  B
a. Define μ  IA to be the magnetic dipole moment of the loop
b. τ  IA  B  μ  B
c. τ E  p  E = torque on an electric dipole
τ B  μ  B = torque on a magnetic dipole
d. For N loops
τ B  Nμ loop  B
C.
D.
E.
F.
τ B  μ coil  B where coil = Nloop
In an E-field, U E  p  E ; and by analogy, in a B-field, U B  μ  B
Loop at some angle to B will rotate such that  points in same direction as B to
minimize PE
Current-carrying loop is basis for galvanometer
Examples 29.7 and 29.8
V. Motion of a Charged Particle in Uniform B-Field
A. F  v  work done on a charged particle by a magnetic
field is zero
B. FB = qv x B
1. FB = centripetal force (circular motion)
2. If v  B, FB = qvB
3. If v  B, then v produces circular
motion and v|| causes helical
progression
a. v  v cos vˆ ||  v sin  vˆ 
b. FB = q (v sin )B
c. v||  v cos  and v   v sin 
 x  v||  t and v  v y2  v z2
C. Fc = mar
mv2
r
2. Fc  FB  qv B
1. Fc 
mv2
3.
 qv  B
r
mv
4. r 
qB
v
qB
5.    
r
m
q
q
    B where the quantity
is called the charge to mass ratio.  is
m
m
called the cyclotron frequency.
2 2 m
6. T 


qB
7. Examples 29.3 and 29.4
D. If B is non-uniform, the motion of a particle can be quite complex and would involve
calculus of variations or even numerical techniques to solve the problem.
1. A magnetic bottle is a B-field that is strong at the ends and weak in the
middle. The strong B-fields at the end can redirect the particle’s motion back
to the center, effectively trapping the particle.
2. charged plasmas can be trapped in magnetic bottles until magnetic fields are
saturated.
3. Van Allen radiation belts trap cosmic rays and solar wind particles producing
auroras at the poles and at lower latitudes during solar flares
E. Applications of Moving Charges in E and B fields
1. Ftot  FE  FB  qE  qv  B
2. velocity selector (+q)
a. E is uniform and directed
downward in a plane
b. B is applied  to E and is
directed into the page.
c. When qv  B balances qE,
then particles move in a
straight line.
d. The velocity at which particles move in a straight line is determined
E
from v 
B
E
E
e. If v  , particles are deflected upwards. If v  , particles are
B
B
deflected downwards.
f. Gravity can be neglected because it is very much weaker than the
electric and the magnetic forces.
3. Bainbridge Mass Spectrometer
a. velocity selector is
attached to a region
where E = 0 and a new
magnetic field
(different in strength
from the one in the
velocity selector) of B0 is present.
b. In the region where E = 0,
mv 2
qvB0 
r
mv mv
qB0 

d
r
2
q
2v

m B0 d
E
c. v 
from the velocity solution, so
B
E
q 2( B)
2E


m
B0 d
BB0 d
d. J. J. Thompson used a technique similar to this to determine the mass
of the electron
4. Cyclotrons
a. accelerators push charge at very high speeds
b. E and B are manipulated to produce helical motion in a set of
concentric cylinders
c. q V gives additional energy to the particle so that
1
KE  mv 2
2
qBR
v
where v is the ejection velocity from the cylinder and R is
m
the max. radius of the cylinder
2
1  qBR 
1 q2B2R2
KE  m
 
2  m 
2
m
d. KE expression is valid to 20 MeV before relativistic effects come into
play
e. Example 29.5
F. Hall Effect
1. current-carrying conductor in magnetic field has V  I and V  B.
2. gives info about sign of charge carriers and their density
3. Let I be in the + î direction, B in the + ĵ direction, then vd of electrons is in the
– î direction and electrons are deflected by FB = qvd x B toward the
+ k̂ direction.
4. accumulation of electrons on upper surface causes accumulation of + charge
on lower surface. This creates an E-field that produces an FE opposed to
deflection.
5. Hall voltage is VH  EH d where d = conductor height
FB  FE
qv d B  qE H
E H  vd B
6.
7.
8.
9.
V H  v d Bd
From conduction model,
I
where A is the cross-sectional area of the conductor.
vd 
nqA
IBd
d
VH 
 RH IB
nqA
A
1
RH = Hall coefficient =
nq
a. sign of RH gives sign of charge carriers
b. magnitude gives # density
works well for Li, Na, Cu, and Ag; but not for Fe, Bi, Cd, or superconductors
Example 29.2
VI. Ampere’s Law
A. When no current flows through a
wire, compasses point in
direction of B  . When current
flows, compasses point in
direction of B.
B. B  I
1
B
r
B || ds everywhere on path so
B  ds  B ds
B  cnst 
o I
2 r
o I
 B  ds  B  ds  2 r (2 r )  
o
I
C. For any constant current, Ampere’s Law says that  B  ds   o I , but it is only useful
for highly symmetric situations.
D. Example 30.3 in class
E. General Form of Ampere’s Law
dq
is conduction current carried by the wire
dt
2. If E varies in time, a new Binduced is created that produces a displacement
current Id. Ampere’s Law (  B  ds   o I ) is only valid if E is constant.
1. currents are time-varying so I 
3.  E   E  dA 
qd   0  E
qd
0
 Electric Flux (Gauss’ Law)
dq d
d E
 0
dt
dt
d E
Id  0
dt
d E
(Ampere-Maxwell Law)
dt
5. Magnetic fields are thus produced by both conduction currents and by timevarying E-fields.
6. Examples 32.3 and 32.4
F. Magnetic Field of a Solenoid
1. A solenoid is a long wire wound in a helix.
2. interior of a solenoid contains an approximately uniform B-field if wire
carries a steady I
3. closely-spaced turns can be approximated as circular loops, and the net
magnetic field is the vector sum of fields from all turns.
4. interior field lines are parallel and closely-spaced indicating a uniform field.
External fields are weak because current elements on right side of turn cancel
fields from the left sides of turn.
5. with more and closely-spaced turns, a solenoid becomes more ideal and
approaches that of a bar magnet. For a perfectly ideal solenoid, the interior
field is uniform over a great volume (changing only as you get near the turns
of wire or the end of the solenoid), and the external field is zero.
6.  B  ds   B  ds   B  ds   B  ds   B  ds   0 IN
4.
 B  ds  
loop
o
1
( I  I d )   o I   o 0
2
3
 B  ds  BL  0  0  0  BL  
loop
4
0
IN
 B  ds   B  ds  0 because B  ds
2
4
B = 0 along wire 3 so  B  ds  0
3
7. BL = 0NI
N
B  0 I  
L
B   0 nI
# turns
length
8. This equation is only truly valid for B near the middle of the solenoid; at the
ends and at the walls, edge effects and proximity to the wire come into play.
1
9. At the end of a long solenoid, B end  B center
2
10. Example 30.4
n
VII. Magnetic Flux & Gauss’ Law for Magnetism
A. Magnetic Flux
1. similar to electric flux
2.  B   B  dA
3.  B  BA cos for uniform B
4. the unit of magnetic flux is the weber where 1 Wb =
1 T-m2
5.  B  0 if B  dA
6.  B  BA if B || dA
B. Gauss’ Law for Magnetism
1. Gauss’ Law for E works because # lines through a surface is proportional to
the enclosed charge
2. situation is very different for magnetism because magnetic fields are
continuous and form closed loops
3.
 B  dA  0 because any magnetic field lines that enter a closed surface also
closed
surface
leave a closed surface! Electric field lines do not necessarily return to a
surface that they leave.
4. A result of gauss’ Law for magnetism is that isolated magnetic monopoles
cannot theoretically exist.
5. Experimentally, magnetic monopoles have never been detected, but many
other physical theories suggest their possible existence. This is a fundamental
area of new research.
VIII. Magnetism in Matter
A. Every current loop has both a magnetic field and a dipole moment.
B. Magnetism in matter arises from currents at the atomic level due to an atomic
property called “magnetic spin.”
C. Magnetic Moments of Atoms
1. model atom as a positive nucleus by a negative electron
q q
qv

2. if e- has speed v and orbital radius r, then I  
T 2 2 r
qv
1
)( r 2 )  qvr
3. magnetic moment is   IA  I ( r 2 )  (
2 r
2
4.  for most atoms are randomly oriented so net = 0
5. orbital angular momentum
a. L  r  mv
L  rmv sin( 90)
L
L  rm e v  v 
rm e
 orb 
1
1
L
qvr  q (
)r
2
2 rm e
 orb 
1 q
L
2 me
h
= 1.05 x 10-34 J s is Planck’s constant
2
c. can only measure quantized component of L along one axis
Lorb, z  ml  where ml = 0,  1,  2,  is the orbital magnetic
b.  
quantum number
1 q
L
d.  orb 
2 me
1 (e)
 orb 
ml 
2 me
e
 orb 
ml 
2me
e. U orb  μ orb  B ext is potential energy of orbiting electron in an
external B field
f. magnetic moment of e- is proportional to L and oppositely directed
because e- is negative. Both L and  are perpendicular to plane of
electron’s orbit.
6. Spin angular momentum
e
S
a. μ s  
me
b. μ s cannot be measured directly. Only its quantized component along
one axis can be determined.
S z  ms  where ms =  12 is the spin magnetic quantum number.
Positive values of spin are considered to be spin “up” and negative
values of spin are considered to be spin “down.”
e
Sz
c. μ s , z  
me
e
μs,z  
ms 
me
1 e
μs,z  

2 me
d. B = Bohr magneton = 9.27 x 10-24 J/T
e. U spin  μ s  B ext is potential energy of spinning electron in an external
B field
7. In most atoms, electrons usually pair up to cancel spin, but some atoms have
an odd # of electrons. Thus, some atoms have net = 0 while others have net =
angular momentum + spin.
8. magnetic moments of protons and neutrons are ~1000 times smaller than those
of electrons and can generally be neglected.
9. Loop Model for Electron Orbits
a.  orb  IA
q
 orb  (r 2 )
t
e
 orb 
(r 2 )
2

r
(
)
v
 orb   12 evr
b. L orb  r  (me v)
Lorb  me vr sin 90
Lorb  me vr
c.  orb   12 evr
Lorb  me vr
Take the ratio of the magnetic moment and the angular momentum
 orb  12 evr

Lorb
me vr
1 e
μ orb  
L orb
2m
d. In a non-uniform field, dF  I (dL  B) and sensitivity of atomic
orbital angular momentum to the external magnetic field determines
the magnetic properties of a material.
D. Magnetization Vector and Magnetic Field Strength
μ
1. magnetization vector = M =
volume
2. B = Bint + B0 where B0 is the external magnetic field
Bint = 0M (0 is the magnetic permeability, not the magnetic moment)
B = B0 + 0M
3. magnetic field strength (H) represents the effects of conduction currents. H =
magnetic field strength and B is referred to as the magnetic flux density or
magnetic induction
B
B
4. H  0 
M
0 0
B   0 ( H  M)
H has units of A/m
5. Example of torus in vacuum
a. M = 0 in vacuum because no substance is present
b. B = Bint + B0
B = B0 = 0nI
B
H  0  nI
μ0
c. H depends only upon current and # turns of wire
E. Classification of magnetic Substances
1. paramagnetic substances display permanent but weak magnetic moments
2. diamagnetic substances display no permanent magnetization
3. ferromagnetic substances display strong and permanent magnetic dipole
moments
4. For paramagnetic and diamagnetic materials, M = H where  is defined to be
the magnetic susceptibility of a material
a. If  > 0, M and H lie in the same direction (paramagnetic)
b. If  < 0, M and H lie in opposite directions (diamagnetic)
c. B   0 (H  M)   0 (H   H)   0 (1   )H
d.  m   0 (1   ) is the magnetic permeability of the material.
 = 0 for vacuum
m > 0 is paramagnetism
m < 0 is diamagnetism
e.  is very small for both paramagnetic and diamagnetic substances.
This implies that a linear relationship exists between M and H, and
that m  0.
5. In ferromagnetic substances,  is very large and m >> 0. M and H also
depend upon the magnetic history of the substance.
F. Ferromagnetism
1. Examples include iron, cobalt, nickel, gadolinium, and dysprosium
2. magnetic moments of these materials align in a
weak magnetic field and stay aligned after the field
is removed
3. magnetic domains are microscopic regions typically
10-12 to 10-8 m3 in volume and containing 1017 to
1021 atoms whose magnetic moments are roughly aligned with each other.
4. domain walls separate magnetic domains of different orientations.
5. magnetic domains are randomly oriented in the absence of an external
magnetic field, so average M = 0.
6. In a magnetic field, magnetic domains aligned with B0 grow at the expense of
non-aligned domains. When the field is removed, ferromagnetic materials
retain their alignment unless they are thermally agitated (heated). Heating will
randomize magnetic moments.
7. Seafloor striping occurs because the cooling magma at mid-ocean ridges
freezes in the magnetic field direction of Earth at the time.
8. Rowland rings (toroids connected to galvanometers) can measure
ferromagnetic properties of a
substance by determining B and H.
9. Hysteresis
a. magnetic hysteresis is the
ability to use a past magnetic
state to achieve a new
magnetization state of a
ferromagnetic material
b. hysteresis means “lagging
behind,” and this is visible as B
lags H on the magnetization
curves abc and def. In both cases H reaches 0 before B reaches 0.
c. magnetization that remains after B = 0 at points c and f is B = Bm =
0M. This is called remanent magnetization.
d. Size of loop represents the work done in the hysteresis cycle. Shape of
loop represents the ability to demagnetize a substance with an external
magnetic field. Large area implies more work. Thickness of loop is
proportional to magnitude of demagnetization field.
e. work done goes from transformation of magnetic energy to internal
energy. Devices subjected to alternating fields are made of “soft”
ferromagnetic materials to minimize loop thickness and energy losses.
f. Devices that use hysteresis include floppy disks, audio tapes, and
videotapes to store information.
G. Paramagnetism
1. Paramagnetic materials align with external B0, but when B0 is removed,
magnetic domains compete with thermal agitation for alignment.
B
2. Pierre Curie found experimentally that M   c 0 (Curie’s Law) where c =
T
Curie’s constant for a particular material, M is the magnetic moment, and T is
the temperature
3. Above a critical temperature (called the Curie temperature, Tc), magnetic
domains are randomized. Below Tc, they are aligned with the external
magnetic field B0.
H. Diamagnetism
1. H and M are in opposite directions
2. diamagnetic materials tend to be repelled by magnets
3. present in all substances, but much weaker than both diamagnetism and
ferromagnetism.
I. Example 32.2
J. Superconductors and magnets repel each other because superconductors expel applied
magnetic fields in order to keep B and E equal to 0 in their interiors. Thus magnets
can be levitated above superconductors. This is called the Meissner effect.