Magnetic field and trajectory of movig charges

PHY2049: Chapter 28 1
Magnetic Field and Work
Magnetic force is always perpendicular to velocity
Therefore B field does no work!
Why? Because
Consequences
Kinetic energy does not change
Speed does not change
Only direction changes
Particle moves in a circle (if )
( ) 0K F x F v t∆ = ⋅∆ = ⋅ ∆ =
r rr r
v B⊥
rr

x x x x x x x x x x x x x x
Trajectory in a Constant Magnetic Field
A charge q enters B field with velocity v perpendicular to
B. What path will q follow?
Force 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
FFv
R
v
B
qF
v

x x x x x x x x x x x x x x x x x
Circular Motion of Positive Particle
B
qF
v
2
mv
qvB
R
=
mv
R
qB
=

Magnetic Force
Two 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
A B
mv
R
qB
=
Bigger mass means larger
inertia, less acceleration,
thus bigger radius

Work and Energy
A charged particle enters a uniform magnetic field. What
happens to the kinetic energy of the particle?
(1) it increases
(2) it decreases
(3) it stays the same
(4) it depends on the direction of the velocity
(5) it depends on the direction of the magnetic field
Magnetic field does no work, so K is constant

Cosmic Ray Example
Protons with energy 1 MeV move ⊥ earth B field of 0.5
gauss or 5 × 10-5 T. Find radius & frequency of orbit.
21
2
2K
K mv v
m
= ⇒ =
2mv mK
R
eB eB
= =
( )( )6 19 13
27
K 10 1.6 10 =1.6 10 J
1.67 10 kgm
− −
−
= × ×
= ×
( )
1
2 2 / 2
v v eB
f
T R mv eB mπ π π
= = = = 760Hzf =
2900mR =
Frequency is independent of v!

Helical Motion in B Field
Velocity of particle has 2 components
(parallel to B and perp. to B)
Only v⊥ = v sinφ contributes to circular motion
v|| = v cosφ is unchanged
So 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⊥= +
r r r
mv
R
qB
⊥=
B
v
φ
v⊥
v||

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

Mass Spectrometer
Originally developed by physicists, now an important tool in chemistry,
biology, environmental studies, forensics, pharmaceutics, etc.
Sample 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.

Mass Spectrometer (simplified)
Sample 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)
D
V
Brq
m
2
)( 2
=
2r
detector

Torque on Current Loop
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
Equal forces give net torque!
Bottom side up, top side down (RHR)
Rotates around horizontal axis
µ = NiA ⇒ “magnetic dipole moment”
Assuming N turns
τ = µB, true for any shape!!
If plane tilted angle θ to B field
τ = µBsinθ
θ is angle between normal and B
B
a
b
a
b
( )Fd iBa b iBab iBAτ = = = =
Plane normal is ⊥ B
(θ = 90°)

Magnetic Dipole Moment

Torque Example
A 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.
Rotation is always in direction to align µ with B field
30°
( )22 2
3 3 5 3.14 0.03 0.0339A mi rµ π= = × × × = ⋅
sin30 0.0339 2.5 0.5 0.042N mBτ µ= = × × = ⋅

Magnetic Force
A 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
B
x
z
y
Forces cancel on
opposite sides of loop

Electromagnetic Flowmeter
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
E

Hall Effect: Do + or – Charges Carry Current?
+ charges moving counter-clockwise
experience upward force
Upper plate at higher potential
– charges moving clockwise experience
upward force
Upper plate at lower potential
Very quickly, equilibrium between electrostatic & magnetic forces is
established and potential difference stops growing:
up driftF qv B= H
down induced
w
V
F qE q= = H drift "Hall Voltage"V v Bw= =
This type of experiment led to the discovery (E. Hall, 1879) that current in
conductors is carried by negative charges
Hall effect is used to measure moderate to moderately high B (10-4 T – 3 T)
It is also used to measure the speed of computer hard drive

FAQ on Magnetic Field and Work
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
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.
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 non-
uniform B, then this force does work. There is no contradiction.

Magnetic field and trajectory of movig charges

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