Chapter 28: Magnetic Fields
PHY2049: Chapter 28
1
Magnetic Fields
ÎMagnetic
field (units, field lines)
‹ Magnetic
ÎEffects
field of the earth and other astronomical objects
of magnetic fields on charges and currents
‹ Force
on a moving charge
‹ Force on a current
‹ Torque on a current loop
‹ Path followed by particle in magnetic field
ÎGenerating
magnetic fields
‹ Long
wire
‹ Current loop
‹ Solenoid
ÎInstruments
‹ Mass
spectrometers
‹ Cyclotrons and synchrotrons
PHY2049: Chapter 28
2
Reading Quiz
ÎThe
magnetic force on a moving charged particle is:
‹ (1)
‹ (2)
‹ (3)
‹ (4)
‹ (5)
Perpendicular to the velocity
Parallel to the velocity
Parallel to the B field
Independent of the velocity
None of the above
PHY2049: Chapter 28
3
Reading Quiz
Î
Consider +q moving relative to a B field as shown
Force
‹ Force
‹ Force
‹ Force
‹
is
is
is
is
parallel to v
parallel to B
into the page
out of the page
B
+q
PHY2049: Chapter 28
4
Reading Quiz
ÎWhen
I cut a magnet into two pieces I get:
‹ An
isolated north and south magnetic pole
‹ Two smaller magnets
‹ The two pieces are no longer magnets
PHY2049: Chapter 28
5
Bar Magnets
ÎTwo
poles: “north” and “south”
ÎLike
poles repel
ÎUnlike
poles attract
ÎMagnetic
S
poles cannot be isolated
N
Similar to dipole field from electrostatics
PHY2049: Chapter 28
6
Magnetic Monopoles?
ÎCan
any isolated magnetic charge exist?
‹ We
would call this a “magnetic monopole”
‹ It would have a + or – magnetic charge
ÎHow
can we isolate this magnetic charge?
‹ Cut
a bar magnet in half? NO!
What you get
is a bunch of
little magnets!
No one has ever found magnetic
monopoles in nature
PHY2049: Chapter 28
7
Searches for Magnetic Monopoles
PHY2049: Chapter 28
8
Earth is a big magnet!!
The North pole of a small magnet
(compass) points towards
geographic North because Earth’s
magnetic South pole is up there!!
Particles moving along field lines cause Aurora Borealis.
http://science.nasa.gov/spaceweather/aurora/gallery_01oct03.html
PHY2049: Chapter 28
9
What Causes Magnetism?
ÎWhat
is the origin of magnetic fields?
‹ Electric
charge in motion!
‹ For example, a current in a wire loop produces a field very similar
to that of a bar magnet (as we shall see).
ÎUnderstanding
the source of bar magnet field lies in
understanding currents at the atomic level within matter
Orbits of electrons about nuclei
Intrinsic “spin” of
electrons (more
important effect)
PHY2049: Chapter 28
10
Magnetic Field Units
ÎFrom
the expression for force on a current-carrying wire:
‹B
= Fmax / I L
‹ Units:
Newtons/A⋅m ≡ Tesla (SI unit)
‹ Another unit: gauss = 10-4 Tesla
ÎSome
sample magnetic field strengths:
‹ Earth:
B = 0.5 gauss = 0.5 x 10-4 T
‹ Galaxy: B ∼ 10-6 gauss = 10-10 T
‹ Bar magnet: B ∼ 100 – 200 gauss
‹ Strong electromagnet: B = 2 T
‹ Superconducting magnet: B = 5 – 10 T
‹ Pulse magnet: B ∼ 100 T
‹ Neutron star: B ∼ 108 – 109 T
‹ Magnetar: B ∼ 1011 T
PHY2049: Chapter 28
11
Pulsars
Rapidly Rotating Neutron Stars
Enormous Magnetic Fields
Beam off
Beam on
Crab Pulsar
R = 10 km
M = 1.4 solar mass
B ≈ 108 T
Period = 1/30 sec
PHY2049: Chapter 28
12
Magnetic Field B
ÎMagnetic
field defined by magnetic force on a test charge
v
G
G G
F = qv × B
F = qvB sin φ
B
+q
F (into page)
ÎForce
magnitude depends on direction of v relative to B
‹ v is parallel to B
⇒ sinφ = 0
F =0
‹ v is perpendicular to B
⇒ sinφ = 1
F = qvB
‹ v is at angle 45° to B
⇒ sinφ = 0.71 F = qvB sin 45
ÎForce direction is perpendicular to both B and v
‹ Right
hand rule (next slide)
PHY2049: Chapter 28
13
Right Hand Rule
ÎFirst
point fingers in direction of velocity
‹ Curl
fingers toward B field
‹ ⇒ Thumb points toward force
F
v
B
PHY2049: Chapter 28
14
Example
with m = 1.5 g, q = −2μC moves with velocity
2,000 m/s through a magnetic field of 2.5 T at an angle of
30° to the field.
ÎParticle
‹ Magnitude
of force
(
)
F = qBv sin φ = 2 × 10−6 ( 2.5)( 2000 )( 0.5) = 0.005 N
‹ Direction
of force: up out of the page, from RHR
v
B
−q
F (up)
PHY2049: Chapter 28
15
A charged particle moves in a straight line through some
region of space. Can you conclude that B = 0 here?
1.
2.
Yes
No
A B field can exist since if v || B
there is no magnetic force
B
v
q
PHY2049: Chapter 28
16
Magnetic Force
ÎA
negative particle enters a magnetic field region. What
path will it follow?
‹ (1)
‹ (2)
‹ (3)
‹ (4)
‹ (5)
A
B
C
D
E
x x x x x x x x x x x x
A
x x x x x x x x x x x x
B
x x x x x x x x x x x x
x x x x x x x x x x x x
x x x x x x x x x x x x
x x x x x x x x x x x x
x x x x x x x x x x x x
C
D
E
(1) RHR says it bends down (− charge)
(2) But force cannot instantaneously change v
(3) So the answer is D, not E
PHY2049: Chapter 28
17
Magnetic Force on Current-Carrying Wire
ÎMagnitude
of force F = iBL sin φ
‹ Easy
to derive from charge, number density & drift velocity of
individual charge carriers
ÎDirection
of force: RHR
PHY2049: Chapter 28
18
Example
ÎA
4 m long wire carries current of 500A in NE direction
‹ Magnitude
of force (B = 0.5 gauss = 5 × 10-5 T, pointing N)
(
)
F = iBL sin φ = ( 500 ) 5 ×10−5 ( 4 )( 0.71) = 0.071N
‹ Direction
of force: Upwards, from RHR
ÎCan
adjust current in wire to balance against gravity
iBL sin φ = mg
‹ Calculate
mass from density, length and cross-sectional area
m = ρ LA
‹ Good
exam problem!
PHY2049: Chapter 28
19
Magnetic Force
ÎA
vertical wire carries a current in a vertical magnetic
field. What is the direction of the force on the wire?
‹ (a)
left
‹ (b) right
‹ (c) zero
‹ (d) into the page
‹ (e) out of the page
B
I is parallel to B, so
no magnetic force
I
PHY2049: Chapter 28
20
Magnetic Field and Work
ÎMagnetic
force is always perpendicular to velocity
‹ Therefore
B field does Gno work! G
G
G
‹ Why? Because ΔK = F ⋅ Δx = F ⋅ ( v Δt ) = 0
ÎConsequences
‹ Kinetic
energy does not change
‹ Speed does not change
‹ Only direction changes
G
‹ Particle moves in a circle (if v ⊥
G
B)
PHY2049: Chapter 28
21
Trajectory in a Constant Magnetic Field
ÎA
charge q enters B field with velocity v perpendicular to
B. What path will q follow?
is always ⊥ velocity and ⊥ B
‹ Path will be a circle. F is the centripetal force needed to keep the
charge in its circular orbit. Let’s calculate radius R
‹ Force
x x x x x x x x x x x x x x
B
x x x x x x x x x x x x x x
x x x x x x x x x x x x x x
x x x x x x x x x x x x vx x
F
F
v
v
F
q
R
PHY2049: Chapter 28
22
Circular Motion of Positive Particle
x x x x x x x x x x x x x x x x x
x x x x x x x x x x x x x x x x x
x x x x x x x x x x x x x x x x x
x x x x x x x x x x x x x x x x x
v
B
F
q
x x x x x x x x x x x x x x x x x
x x x x x x x x x x x x x x x x x
x x x x x x x x x x x x x x x x x
x x x x x x x x x x x x x x x x x
2
mv
= qvB
R
mv
R=
qB
PHY2049: Chapter 28
23
Cosmic Ray Example
with energy 1 MeV move ⊥ earth B field of 0.5
Gauss or B = 5 × 10-5 T. Find radius & frequency of orbit.
ÎProtons
K=
1 mv 2
2
2K
⇒ v=
m
( )(
)
K = 106 1.6 × 10−19 =1.6 × 10−13 J
m = 1.67 × 10−27 kg
R = 2900 m
mv
2mK
R=
=
eB
eB
1
v
v
eB
f = =
=
=
T 2π R 2π ( mv / eB ) 2π m
f = 760 Hz
Frequency is independent of v!
PHY2049: Chapter 28
24
Helical Motion in B Field
ÎVelocity
of particle has 2 components
G G G
‹ v = v& + v⊥ (parallel to B and perp. to B)
‹ Only
v⊥ = v sinφ contributes to circular motion
‹ v|| = v cosφ is unchanged
ÎSo
the particle moves in a helical path
‹ v||
is the constant velocity along the B field
‹ v⊥ is the velocity around the circle
v||
v
v⊥
B
φ
mv⊥
R=
qB
PHY2049: Chapter 28
25
Helical Motion in Earth’s B Field
PHY2049: Chapter 28
26
Magnetic Force
ÎTwo
particles of the same charge enter a magnetic field
with the same speed. Which one has the bigger mass?
‹A
‹B
‹ Both
masses are equal
‹ Cannot tell without more info
x x x x x x x x x x x x
x x x x x x x x x x x x
x x x x x x x x x x x x
x x x x x x x x x x x x
mv
R=
qB
x x x x x x x x x x x x
x x x x x x x x x x x x
A
Bigger mass means
bigger radius
PHY2049: Chapter 28
B
27
Mass Spectrometer
ions first enter a “velocity selector” where E ⊥ B
and values are adjusted to allow only undeflected particles
to enter mass spectrometer.
ÎPositive
‹ Magnetic
force is down, electric force is up
‹ Balance forces
qE = qvB
v= E/B
‹ Spectrometer
determines mass using
measured radius r and velocity v
qBr
m=
v
PHY2049: Chapter 28
28
Work and Energy
ÎA
charged particle enters a uniform magnetic field. What
happens to the kinetic energy of the particle?
‹ (1)
‹ (2)
‹ (3)
‹ (4)
‹ (5)
it
it
it
it
it
increases
decreases
stays the same
depends on the direction of the velocity
depends on the direction of the magnetic field
Magnetic field does no work, so K is constant
PHY2049: Chapter 28
29
Mass Spectrometer
ÎA
beam of electrons travels right at v = 5 x 105 m/s
‹ What
value of magnetic field would make electrons go undeflected
through a region where E = 100,000 V/m pointing up vertically?
eE = evB
B = E / v = 105 / 5 × 105 = 0.2T
‹ What
is the frequency of the circular orbit of the electrons if the
electric field is turned off?
2
mv
mv
= evB ⇒ R =
=
R
eB
(
)( ) = 1.4 ×10
) ( 0.2)
9.1× 10−31 5 × 105
(1.6 ×10
−19
−5
m
v
1
5 × 105
f = =
=
= 6.8 × 109 Hz
T 2π R ( 6.28 ) 1.4 × 10−5
(
)
PHY2049: Chapter 28
30
Torque on Current Loop
a
Î Rectangular
current loop in uniform
magnetic field (lengths a & b)
Forces in left & right branches are 0
‹ Force in top branch is into plane
‹ Force in bottom branch is out of plane
‹
Î Equal
b
forces give net torque!
Bottom side up, top side down (RHR)
‹ Rotates around horizontal axis
‹
τ = Fd = ( iBa ) b = iBab = iBA
Îμ
= NiA ⇒ “magnetic moment”
B
b
a
Plane normal is ⊥ B
(θ = 90°)
Assuming N turns
‹ τ = μB, true for any shape!!
‹
Î If
plane tilted angle θ to B field
τ = μBsinθ
‹ θ is angle between normal and B
‹
PHY2049: Chapter 28
31
Torque Example
ÎA
3-turn circular loop of radius 3 cm carries 5A current in
a B field of 2.5 T. Loop is tilted 30° to B field.
30°
2
2
Î μ = 3iπ r = 3 × 5 × 3.14 × ( 0.03 ) = 0.0339 A ⋅ m
2
Îτ
= μ B sin 30 = 0.0339 × 2.5 × 0.5 = 0.042 N ⋅ m
ÎRotation
is always in direction to align μ with B field
PHY2049: Chapter 28
32
Magnetic Force
ÎA
rectangular current loop is in a uniform magnetic field.
What direction is the net force on the loop?
‹ (a)
+x
‹ (b) + y
‹ (c) zero
‹ (d) – x
‹ (e) – y
Forces cancel on
opposite sides of loop
B
z
y
x
PHY2049: Chapter 28
33
Electromagnetic Flowmeter
E
¾
¾
¾
¾
¾
Moving ions in the blood are deflected by magnetic force
Positive ions deflected down, negative ions deflected up
This separation of charge creates an electric field E pointing up
E field creates potential difference V = Ed between the electrodes
The velocity of blood flow is measured by v = E/B
PHY2049: Chapter 28
34
Hall Effect: Do + or – Charges Carry Current?
Î
+ charges moving counter-clockwise
experience upward force
Î
– charges moving clockwise experience
upward force
Î
Upper plate at higher potential
Î
Upper plate at lower potential
Equilibrium between electrostatic & magnetic forces:
Fup = qvdrift B
Fdown = qEinduced = q
VH
w
VH = vdrift Bw = "Hall Voltage"
This type of experiment led to the discovery (E. Hall, 1879) that current
in conductors is carried by negative charges
PHY2049: Chapter 28
35
Partial Loops (cont.)
ÎNote
on problems when you have to evaluate a B field at
a point from several partial loops
‹ Only
loop parts contribute, proportional to angle (previous slide)
‹ Straight sections aimed at point contribute exactly 0
‹ Be careful about signs, e.g.in (b) fields partially cancel, whereas in
(a) and (c) they add
PHY2049: Chapter 28
36