Particle tracks
ÎWhat
field?
is the direction of the uniform magnetic
electron e–
positron e+
electron e–
PHY2049: Chapter 28
1
Cosmic Ray Example
with energy 1 MeV move ⊥ earth B field
of 0.5 gauss or 5 × 10-5 T. Find radius &
frequency of orbit.
ÎProtons
2K
1
K = 2 mv ⇒ v =
m
2
( )(
)
K = 106 1.6 × 10−19 =1.6 × 10−13 J
m = 1.67 × 10−27 kg
mv
2mK
R=
=
eB
eB
R = 2900 m
1
v
v
eB
f = =
=
=
T 2π R 2π ( mv / eB ) 2π m
f = 760 Hz
Frequency is independent of v!
PHY2049: Chapter 28
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Helical Motion in B Field
ÎIf
velocity of particle has 2 components
r r r
‹ v = v|| + v⊥ (parallel to B and perp. to B)
‹Only v⊥ = v sinφ contributes to circular motion v||
‹v|| = v cosφ is unchanged
v
v⊥
ÎSo the particle moves in a helical path
φ
‹v|| is the constant velocity along the B field
‹v⊥ is the velocity around the circle
R=
B
mv⊥
qB
PHY2049: Chapter 28
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Helical Motion in Earth’s B Field
Particles moving along field lines cause Aurora Borealis and
Australis:
http://science.nasa.gov/spaceweather/aurora/gallery_01oct03.html
PHY2049: Chapter 28
4
Magnetic Force on Current-Carrying Wire
of force
F = iBL sin φ
‹Easy to derive from charge, number density & drift
velocity of individual charge carriers
ÎMagnitude
ÎDirection
of force: RHR
PHY2049: Chapter 28
5
Example
ÎA
4 m long wire carries current of 500 A 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
6
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
B
‹(c) no force
‹(d) into the page
‹(e) out of the page
I is parallel to B, so
no magnetic force
I
PHY2049: Chapter 28
7
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
ε
= NiA ⇒ “magnetic dipole 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
8
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
B
‹(c) no force
‹(d) – x
‹(e) – y
Forces cancel on
opposite sides of loop
z
y
x
PHY2049: Chapter 28
9
Magnetic Dipole Moment
PHY2049: Chapter 28
10
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
11
Mass Spectrometer
ÎOriginally
developed by physicists. Now an important tool in chemistry,
biology, environmental studies, forensics, pharmaceutics, etc.
ÎSample
is vaporized, broken into fragments of molecules, which are
positively ionized. Positive ions are first accelerated by a potential
difference V, and then their trajectories are bent by B. Varying B
(sometimes V) allows ions of different masses to reach the detector.
PHY2049: Chapter 28
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Mass Spectrometer (simplified)
ÎSample
is vaporized, broken into fragments of molecules,
which are positively ionized. Positive ions are first
accelerated by a potential difference V, and then their
trajectories are bent by B. Varying B (sometimes V) allows
ions of different masses to reach the detector.
‹Spectrometer determines mass from B
(sometimes from V)
q ( Br ) 2
m=
2V
D
PHY2049: Chapter 28
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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
Very quickly, equilibrium between electrostatic & magnetic forces is
established and potential difference stops growing:
V
VH = vdrift Bw = "Hall Voltage"
Fdown = qEinduced = q H
Fup = qvdrift B
w
¾ This type of experiment led to the discovery (E. Hall, 1879) that current in
conductors is carried by negative charges
¾ Hall effect is used to measure moderate to moderately high B (10-4 T – 3 T)
¾ It is also used to measure the speed of computer hard drive
PHY2049: Chapter 28
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