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Chapter 30
Sources of the Magnetic Field
Biot-Savart Law
• Biot and Savart arrived at a mathematical
expression that gives the magnetic field at
some point in space due to a current
• The magnetic field is dB at some point P;
the length element is ds; the wire is
carrying a steady current of I
Jean-Baptiste Biot
1774 – 1862
Félix Savart
1791 – 1841
Biot-Savart Law
• Vector dB is perpendicular to both ds and to
the unit vector r̂ directed from ds toward P
• The magnitude of dB is inversely
proportional to r2, where r is the distance
from ds to P
• The magnitude of dB is proportional to the
current and to the magnitude ds of the
length element
Biot-Savart Law
• The magnitude of dB is proportional to sinq,
where q is the angle between the vectors ds
and r̂
• The observations are summarized in the
mathematical equation called the BiotSavart law (magnetic field due to the
current-carrying conductor):
μo I ds  ˆr
dB 
4π r 2
• µo = 4  x 10-7 T.m / A: permeability of free
space
Biot-Savart Law
• To find the total field, sum up the
contributions from all the current elements
μo I
B
4π
ds  ˆr
 r2
μo I ds  ˆr
dB 
4π r 2
Biot-Savart Law
• The magnitude of the magnetic field varies as the
inverse square of the distance from the ds element
• The electric field due to a point charge also varies as
the inverse square of the distance from the charge
• The electric field created by a point charge is radial in
direction
• The magnetic field created by a current element is
perpendicular to both the length element and the unit
vector
• The current element producing a magnetic field is part
of an extended current distribution
A Long, Straight Conductor
• The thin, straight wire is carrying a constant current

ds  rˆ  (dx cos q )kˆ
adq
dx  
2
cos q
x  a tan q
a
r
cos q
 0 I dx cos q
ˆ
dB 
k
2
4
r
0 I ˆ

k cos qdq
4a
q2
0 I
B
cos qdq

4a q
1
A Long, Straight Conductor
• The thin, straight wire is carrying a constant current
q2
0 I
B
cos qdq

4a q
0 I
sin q1  sin q 2 

4a
1
• If the conductor is an infinitely
long, straight wire, θ1 = π/2 and
 θ2 = – π/2 , and the field becomes
0 I
B
2a
q2
0 I
B
cos qdq

4a q
1
A Long, Straight Conductor
• The magnetic field lines are circles
concentric with the wire
• The field lines lie in planes
perpendicular to the wire
• The magnitude of the field is
constant on any circle of radius a
• Right Hand Rule #2: Grasp the wire
in your right hand and point your
thumb in the direction of the current
and your fingers will curl in the
direction of the field
A Curved Wire Segment

Ids
0
• Find the field at point O due to the dB 
2
wire segment (I, a are constants)
4 a
0 I
0 I
0 I
0 I
B
ds 
s
aq 
q
2 
2
2
4a
4a
4a
4a
• The field at the center of the full
circle loop
0 I
B
2
4a
B
0 I
2a
Magnetic Field of a Current Loop
Magnetic Force Between Two Parallel
Conductors
0 I 2
B2 
2a
0 I 2
F1  B2 I1l 
I1l
2a
F1  0 I1 I 2

l
2a
Magnetic Force Between Two Parallel
Conductors
• The force (per unit length) on wire 1
due to the current in wire 1 and the
magnetic field produced by wire 2:
FB  0 I1 I 2

l
2a
• Parallel conductors carrying
currents in the same direction
attract each other
• Parallel conductors carrying
currents in the opposite directions
repel each other
Chapter 30
Problem 19
Two long, parallel wires are attracted to each other by a force per unit
length of 320 μN/m when they are separated by a vertical distance of
0.500 m. The current in the upper wire is 20.0 A to the right. Determine
the location of the line in the plane of the two wires along which the
total magnetic field is zero.
Ampere and Coulomb revisited
• The force between parallel conductors can be used
to define the Ampere (A): If two long, parallel wires 1
m apart carry the same current, and the magnitude
of the magnetic force per unit length is 2 x 10-7 N/m,
then the current is defined to be 1 A
• The SI unit of charge, the Coulomb (C), can be
defined in terms of the Ampere: If a conductor
carries a steady current of 1 A, then the quantity of
charge that flows through any cross section in 1
second is 1 C
Ampère’s Law
• Ampère’s Circuital Law: a procedure
for deriving the relationship between
the current in an arbitrarily shaped
wire and the magnetic field produced
by the wire
• Choose an arbitrary closed path
around the current and sum all the
products of B|| Δℓ around the closed
path (put the thumb of your right
hand in the direction of the current
through the loop and your fingers
curl in the direction you should
integrate around the loop)
 
 B  ds  0 I
Ampère’s Law for a Long Straight Wire
• Use a closed circular path
• The circumference of the circle is 2
 
 B  ds  B  ds  B2r  0 I
0 I
B
2r
r
Ampère’s Law for a Long Straight Wire
I ' r
r
 2
I ' 2 I
I R
R
 


I
'

B
2

r
B

d
s
0

2
r
 0 2 I
R
2
0 I
B
r
2
2R
2
Chapter 30
Problem 24
A long, straight wire lies on a horizontal table and carries a current of
1.20 μA. In a vacuum, a proton moves parallel to the wire (opposite the
direction of the current) with a constant velocity of 2.30 × 104 m/s at a
constant distance d above the wire. Determine the value of d. (You
may ignore the magnetic field due to Earth.)
Magnetic Field of a Solenoid
• If a long straight wire is bent into a coil of
several closely spaced loops, the
resulting device is called a solenoid
• It is also known as an electromagnet since
it acts like a magnet only when it carries a
current
• The field inside the solenoid is nearly
uniform and strong – the field lines are
nearly parallel, uniformly spaced, and
close together
• The exterior field is nonuniform, much
weaker, and in the opposite direction to
the field inside the solenoid
Magnetic Field of a Solenoid
• The field lines of the solenoid resemble
those of a bar magnet
• The magnitude of the field inside a
solenoid is approximately constant at all
points far from its ends
B = µo n I
• n = N / ℓ : the number of turns per unit
length
• This result can be obtained by applying
Ampère’s Law to the solenoid
Magnetic Field of a Solenoid
• A cross-sectional view of a tightly wound
solenoid
• If the solenoid is long compared to its
radius, we assume the field inside is
uniform and outside is zero
• Apply Ampère’s Law to the blue dashed
rectangle
 
 B  ds 
 
 B  ds  B
path1
 ds
 Bl
path1
 0 NI
N
B   0 I  0 nI
l
Magnetic Flux
• Magnetic flux associated with a magnetic field is
defined in a way similar to electric flux
 
 B   B  dA
• SI unit of flux: Weber
• Wb = T. m²
Wilhelm Eduard Weber
1804 – 1891
Magnetic Flux
• For a flat surface with an area A in a
uniform magnetic field, the flux is (θ is the
angle between B and the normal to the
plane):
ΦB = BA = B A cos θ
• When the field is perpendicular to the
plane, θ = 0 and ΦB = ΦB, max = BA
• When the field is parallel to the plane, θ =
90° and ΦB = 0
• The flux can be negative, for example if θ
= 180°
Magnetic Flux
• The value of the magnetic flux is
proportional to the total number of
magnetic field lines passing through
area
• When the area is perpendicular to
the lines, the maximum number of
lines pass through the area and the
flux is a maximum
• When the area is parallel to the lines,
no lines pass through the area and
the flux is 0
Gauss’ Law in Magnetism
• Magnetic fields do not begin or end at any point
• The number of lines entering a surface equals the
number of lines leaving the surface
• Gauss’ law in magnetism says the magnetic flux
through any closed surface is always zero:
 
 B   B  dA  0
Magnetic Effects of Electrons – Orbits
• An individual atom should act like a magnet because of
the motion of the electrons about the nucleus
• Each electron circles the atom once in about every 10-16
seconds; this would produce a current of 1.6 mA and a
magnetic field of about 20 T at the center of the circular
path
• However, the magnetic field produced by one electron
in an atom is often canceled by an oppositely revolving
electron in the same atom
• The net result is that the magnetic effect produced by
electrons orbiting the nucleus is either zero or very
small for most materials
Magnetic Effects of Electrons – Spins
• Electrons also have spin (it is a
quantum effect)
• The classical model is to consider
the electrons to spin like tops
• The field due to the spinning is
generally stronger than the field due
to the orbital motion
• Electrons usually pair up with their
spins opposite each other, so their
fields cancel each other, hence most
materials are not naturally magnetic
Magnetic Effects of Electrons – Domains
• In some materials – ferromagnetic – the spins do not
naturally cancel
• Large groups of atoms in which the spins are aligned
are called domains
• When an external field is applied, it causes the material
to become magnetized: the domains that are aligned
with the field tend to grow at the expense of the others
Domains and Permanent Magnets
• In hard magnetic materials, the domains remain
aligned after the external field is removed
• The result is a permanent magnet
• In soft magnetic materials, once the external field is
removed, thermal agitation causes the materials to
quickly return to an unmagnetized state
• With a core in a loop, the magnetic field is enhanced
since the domains in the core material align,
increasing the magnetic field
Ferromagnetism
• Some substances exhibit strong magnetic effects
called ferromagnetism (e.g., iron, cobalt, nickel,
gadolinium, dysprosium)
• They contain permanent atomic magnetic moments
that tend to align parallel to each other even in a
weak external magnetic field
Paramagnetism
• Paramagnetic substances have small but positive
magnetism, which results from the presence of
atoms that have permanent magnetic moments
• These moments interact weakly with each other
• When placed in an external magnetic field, atomic
moments tend to line up with the field and the
alignment process competes with thermal motion
which randomizes the moment orientations
Diamagnetism
• When an external magnetic field is applied to a
diamagnetic substance, a weak magnetic moment is
induced in the direction opposite the applied field
• Diamagnetic substances are weakly repelled by a
magnet
Earth’s Magnetic Field
• The Earth’s geographic north (south) pole corresponds
to a magnetic south (north) pole – a north (south) pole
should be a “north- (south-) seeking” pole
• The Earth’s magnetic field
resembles that achieved by burying
a huge bar magnet deep in the
Earth’s interior
• The most likely source of the
Earth’s magnetic field – electric
currents in the liquid part of the
core
Earth’s Magnetic Field
• The magnetic and geographic poles are not in the
same exact location – magnetic declination is the
difference between true north (geographic north pole)
and magnetic north pole
• The amount of declination varies
by location on the earth’s
surface
• The direction of the Earth’s
magnetic field reverses every
few million years (the origin of
these reversals is not
understood)
Answers to Even Numbered Problems
Chapter 30:
Problem 2
200 nT
Answers to Even Numbered Problems
Chapter 30:
Problem 18
(a) 10.0 µT out of the page
(b) 80.0 µT toward wire 1
(c) 16.0 µT into the page
(d) 80.0 µT toward wire 2
Answers to Even Numbered Problems
Chapter 30:
Problem 54
1.80 mT