Download Chapter 33. The Magnetic Field

Chapter 33. The Magnetic Field
Digital information is stored
on a hard disk as
microscopic patches of
magnetism. Just what is
magnetism? How are
magnetic fields created?
What are their properties?
These are the questions we
will address.
Chapter Goal: To learn how
to calculate and use the
magnetic field.
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What is the shape of the trajectory that a
charged particle follows in a uniform
magnetic field?
A.  Helix
B.  Parabola
C.  Circle
D.  Ellipse
E.  Hyperbola
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What is the shape of the trajectory that a
charged particle follows in a uniform
magnetic field?
A.  Helix
B.  Parabola
C.  Circle
D.  Ellipse
E.  Hyperbola
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The magnetic field of a straight,
current-carrying wire is
A.  parallel to the wire.
B.  inside the wire.
C.  perpendicular to the wire.
D.  around the wire.
E.  zero.
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The magnetic field of a straight,
current-carrying wire is
A.  parallel to the wire.
B.  inside the wire.
C.  perpendicular to the wire.
D.  around the wire.
E.  zero.
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The Earth and Sun are magnetic
http://solar.gmu.edu/teaching/2008_CSI769/solar_magnetic_field.jpg
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Connections to current
A magnetic field can be sensed with a magnetic
material (compass) and is associated with a
current. The earth’s field results from large scale
internal currents. The field of a permanent
magnet results from atomic scale currents.
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The field appears only if there is
current. It is associated with moving
charge.
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The Source of the Magnetic Field: Moving
Charges
The magnetic field of a charged particle q moving with
velocity v is given by the Biot-Savart law:
where r is the distance from the charge and θ is the angle
between v and r. (Valid if v<<c.)
The Biot-Savart law can be written in terms of the cross
product as
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The field of an element of circuit
The average B field dB due to the moving charge in an element
of circuit of vector length ds carrying current I follows from
superposing the fields of the moving charges:
dB
r
dI=Ids
µo Ids × rˆ
dB =
2
4π r
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The field of straight wire
 
All current elements produce B out of page
µ
dB = o
4π
µ
= o
4π
µ
= o
4π
Ids × rˆ
r2
I
sin θ
2
r
I a µo
a
=
I
r 2 r 4π ( x 2 + a2 )3 / 2
Add them all up:
µo Ia ∞
dx
µo I
B=
=
∫
4 π −∞ ( x 2 + a 2 ) 3 / 2 2πa
r
a
r = x 2 + a2

x
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11
EXAMPLE 33.4 The magnetic field strength
near a heater wire
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EXAMPLE 33.4 The magnetic field strength
near a heater wire
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Twice B(Earth).
Magnetic field lines
Magnetic field lines close upon themselves – they circulate rather
them emanate from charges. Although a current loop can appear
like a dipole pair of charges, there is no magnetic charge.
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Magnetic Dipoles
The magnetic dipole moment
of a current loop enclosing an
area A is defined as
The SI units of the magnetic
dipole moment are A m2. The
on-axis field of a magnetic
dipole is
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EXAMPLE 33.7 The field of a magnetic dipole
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EXAMPLE 33.7 The field of a magnetic dipole
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Magnetic materials
The superposition of the fields
of a stack of loops is a field like
that of a bar magnet.
The bar magnetic field results
from alignment of many atomic
scale electronic currents/
magnetic dipoles.
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Line integrals
Magnetic field lines close upon themselves – they circulate rather
them emanate from poles. The line integral of B around a closed
loop is a measure of the strength in circulation.
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Ampère’s law
Whenever total current Ithrough
passes through an area bounded
by a closed curve, the line
integral of the magnetic field
around the curve is given by
Ampère’s law:
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Ampère’s law example
•  Could have used Ampere’s law to calculate B
µo I
∫ B • ds = ∫ Bds = B ∫ ds = B2π r = µoI ⇒ B = 2π r
B||ds
path length =
2πr
B constant
on path
Circular path
r
B(r)
Surface bounded
by path
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I
The strength of the uniform magnetic field inside a solenoid
is
where n = N/l is the number of turns per unit length.
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Gauss’s law for magnetism
•  Net magnetic flux through any closed surface is
always zero
∫ B • dA = 0
Compare to Gauss’ law for
electric field
∫
Qenclosed
E • dA =
εo
No magnetic ‘charge’,
so right-hand side=0 in the case of
magnet field.
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General laws of electromagnetism
Gauss’ law
Magnetostatics
Electrostatics
Ampere’s law
∫ B • dA = 0
∫ B • ds = µ I
Qenclosed
∫ E • dA = ε
o
∫ E • ds = 0?
•  Integral of E-field around
closed loop is is the change in
electric potential going
around = zero.
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o
x
The Magnetic Force on a Moving Charge
The magnetic force on a charge q
as it moves through a magnetic
field B with velocity v is
where α is the angle between v
and B.
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Motion in a uniform constant magnetic field
The magnetic force on a moving
charge q is perpendicular to B and
to v and results in helical motion.
(Circular motion if there is no
velocity component along the
field.)
Beam of electrons moving in a circle.
Lighting is caused by excitation of atoms
of gas in a bulb. http://en.wikipedia.org/
wiki/Magnetic_field
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Derive force
by adding
forces on
charges
constituting
the current.
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Magnetic Forces on Current-Carrying Wires
Consider a segment of wire of length l carrying current I in
the direction of the vector l. The wire exists in a constant
magnetic field B. The magnetic force on the wire is
where α is the angle between the direction of the current
and the magnetic field.
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EXAMPLE 33.13 Magnetic Levitation
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EXAMPLE 33.13 Magnetic Levitation
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EXAMPLE 33.13 Magnetic Levitation
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Interaction between a field and a dipole
An current loop in a magnetic
field is subject to a net torque
aligning the dipole moment
with the field.
F
I
  
τ =r×F
⎞
 ⎛
τ = 2⎜ F sin θ ⎟
⎝2
⎠
F = IB ⇒ τ = AIBsin θ
A=
2
=loop area
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B

r

I
F
B
Interaction between electromagnet and a
magnetic substance
An electromagnet can be used to pick
up a ferromagnetic material. The field
of the electromagnet induces an
alignment of the atomic scale dipoles
resulting in a net force of attraction.
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Does the compass needle rotate clockwise
(cw), counterclockwise (ccw) or not at all?
A. Clockwise
B. Counterclockwise
C. Not at all
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Does the compass needle rotate clockwise
(cw), counterclockwise (ccw) or not at all?
A. Clockwise
B. Counterclockwise
C. Not at all
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The magnetic field at the position P points
A. Into the page.
B. Up.
C. Down.
D. Out of the page.
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The magnetic field at the position P points
A. Into the page.
B. Up.
C. Down.
D. Out of the page.
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The positive charge is
moving straight out of the
page. What is the direction
of the magnetic field at the
position of the dot?
A. Left
B. Right
C. Down
D. Up
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The positive charge is
moving straight out of the
page. What is the direction
of the magnetic field at the
position of the dot?
A. Left
B. Right
C. Down
D. Up
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What is the current
direction in this loop? And
which side of the loop is
the north pole?
A. Current counterclockwise, north pole on bottom
B. Current clockwise; north pole on bottom
C. Current counterclockwise, north pole on top
D. Current clockwise; north pole on top
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What is the current
direction in this loop? And
which side of the loop is
the north pole?
A. Current counterclockwise, north pole on bottom
B.  Current clockwise; north pole on bottom
C. Current counterclockwise, north pole on top
D. Current clockwise; north pole on top
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An electron moves perpendicular to a
magnetic field. What is the direction
of ?
A. Left
B. Into the page
C. Out of the page
D. Up
E.  Down
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An electron moves perpendicular to a
magnetic field. What is the direction
of ?
A. Left
B.  Into the page
C. Out of the page
D. Up
E.  Down
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Which magnet or magnets
produced this induced
magnetic dipole?
A.  a or d
B.  a or c
C.  b or d
D.  b or c
E.  any of a, b, c or d
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Which magnet or magnets
produced this induced
magnetic dipole?
A.  a or d
B.  a or c
C.  b or d
D.  b or c
E.  any of a, b, c or d
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