Magnetic Field and Work
ÎMagnetic
force is always perpendicular to velocity
‹ Therefore
B field does rno work! r
r
r
‹ Why? Because ∆K = F ⋅ ∆x = F ⋅ ( v ∆t ) = 0
ÎConsequences
‹ Kinetic
energy does not change
‹ Speed does not change
‹ Only direction changes
r
‹ Particle moves in a circle (if v ⊥
r
B)
PHY2049: Chapter 28
1
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
2
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
3
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
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
Bigger mass means larger
inertia, less acceleration,
thus bigger radius
PHY2049: Chapter 28
A
B
4
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
5
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
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
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
6
Helical Motion in B Field
ÎVelocity
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 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
7
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
8
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
9
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
2r
D
PHY2049: Chapter 28
detector
10
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
11
Magnetic Dipole Moment
PHY2049: Chapter 28
12
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
13
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
14
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
15
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
16
FAQ on Magnetic Field and Work
Î Magnetic
force does no work. But an electric motor (=a current loop in
B) does work. Where does this work come from?
Magnetic force does no work on a moving charge
‹ Magnetic torque on a current loop does work: ∆W=τ∆θ
‹
Î There
is no net force, only torque, on a current loop (=magnetic dipole
moment) in B. But two bar magnets (=collection of atomic magnetic
dipole moments) attract each other. How come?
There is no net force, only torque, on magnetic dipole moment in uniform B
‹ When B is non-uniform, then there is net force. Can be shown that the
direction of this force is such that magnetic dipole moment is attracted to the
region of high B.
‹
Î Magnetic
force does no work. But when two bar magnets are attracted
to each other, there must be work done by B. Something is not right
here.
Magnetic force does no work on a moving charge
‹ When a magnetic dipole moment moves as a result of force due to nonuniform B, then this force does work. There is no contradiction.
‹
PHY2049: Chapter 28
17