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
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.
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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http://solar.gmu.edu/teaching/2008_CSI769/solar_magnetic_field.jpg
The Earth and Sun are magnetic
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.
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
dB = µ o 4 π
Ids × ˆ r r 2
dI=Ids dB
r
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:
The field of straight wire
All current elements produce B out of page
x a
r = x 2 + a 2 r
dB = µ
o4 π
Ids × ˆ r r
2= µ
o4 π
I
r
2sin θ
= µ
o4 π
I r
2a
r = µ
o4 π I
a x
2+ a
2( )
3 / 2B = µ
oIa 4 π
dx x
2+ a
2( )
3 / 2=
−∞
∞
∫ 2 µ π
oI a
Add them all up:
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EXAMPLE 33.4 The magnetic field strength
near a heater wire
EXAMPLE 33.4 The magnetic field strength near a heater wire
Twice B(Earth).
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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.
Magnetic field lines
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 m 2 . The
on-axis field of a magnetic
dipole is
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EXAMPLE 33.7 The field of a magnetic dipole
EXAMPLE 33.7 The field of a magnetic dipole
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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.
Magnetic materials
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 I through 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:
Ampère’s law example
• Could have used Ampere’s law to calculate B
r
I B(r)
B • ds
∫ = Bds = ∫ B ds ∫ = B2 π r = µ o I ⇒ B = 2 µ π o I r
Circular path
Surface bounded by path
B||ds B constant on path
path length =
2πr
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The strength of the uniform magnetic field inside a solenoid is
where n = N/l is the number of turns per unit length.
Gauss’s law for magnetism
• Net magnetic flux through any closed surface is always zero
B • dA
∫ = 0
No magnetic ‘charge’,
so right-hand side=0 in the case of magnet field.
E • dA
∫ = Q enclosed ε
o
Compare to Gauss’ law for
electric field
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General laws of electromagnetism
Gauss’ law Ampere’s law
B • dA
∫ = 0
E • dA
∫ = Q enclosed ε
o
B • ds
∫ = µ o I
Magnetostatics
Electrostatics ∫ E • ds = ? 0
• Integral of E-field around
closed loop is is the change in electric potential going
around = zero.
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
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.
EXAMPLE 33.13 Magnetic Levitation
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EXAMPLE 33.13 Magnetic Levitation
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.
τ = r × F
τ = 2
2 F sin θ
⎛
⎝ ⎜ ⎞
⎠ ⎟
F = IB ⇒ τ = AIBsin θ A = 2 =loop area
B F
B I
r
I
F
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
A. Clockwise
B. Counterclockwise C. Not at all
Does the compass needle rotate clockwise
(cw), counterclockwise (ccw) or 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.
A. Into the page.
B. Up.
C. Down.
D. Out of the page.
The magnetic field at the position P points
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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
A. Left B. Right C. Down D. Up
The positive charge is
moving straight out of the
page. What is the direction
of the magnetic field at the
position of the dot?
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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
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
What is the current
direction in this loop? And
which side of the loop is
the north pole?
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A. Left
B. Into the page C. Out of the page D. Up
E. Down
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
An electron moves perpendicular to a
magnetic field. What is the direction
of ?
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