Chapter 20
Magnetic Forces and
Magnetic Fields
The Motion of a Charged Particle in a Magnetic Field
The circular trajectory.
Since the magnetic force always remains
perpendicular to the velocity, if a charged
particle moves perpendicular to a uniform
B-field its path will be circular. The magnitude
of the force remains constant and it is directed
toward the center of the circular path, i.e. it is
the centripetal force for the motion.
Centripetal
Force
FM = F c
Solve for r
v2
Fc = m
r
v2
qvB = m
r
mv
r=
qB
Example. Find the radius of curvature for a fast electron with speed
v = 2 x 107 m/s in a) the Earth’s magnetic field at the surface, and b) in a
10 T magnetic field.
a) BEarth ~ 0.5 gauss = 0.5 x 10-4 T
r = mv/(qB) = (9.11 x 10-31)(2 x 107)/[(1.6 x 10-19)(0.5 x 10-4)]
= 2.3 m
b) B = 10 T
r = mv/(qB) = (9.11 x 10-31)(2 x 107)/[(1.6 x 10-19)(10)]
= 1.1 x 10-5 m = 11 µm
The Motion of a Charged Particle in a Magnetic Field
Conceptual Example: Particle Tracks in a Bubble Chamber
The figure shows the bubble-chamber
tracks from an event that begins at point
A. At this point a gamma ray travels in
from the left, spontaneously transforms
into two charged particles. The particles
move away from point A, producing two
spiral tracks. A third charged particle is
knocked out of a hydrogen atom and
moves forward, producing the long track.
The magnetic field is directed out of the
paper. Determine the sign of each particle
and which particle is moving most rapidly.
The Mass Spectrometer
Mass spectrometers can be used to measure the masses and relative abundances
of isotopes (atoms which have nuclei with same number of protons but different
number of neutrons).
mv mv
r=
=
qB eB
1
2
magnitude of
electron charge
mv 2 = eV
KE=PE
& er 2 # 2
!! B
m = $$
% 2V "
Accelerate
Ions to v
can count ions hitting it
Put sample atoms here,
ionize one electron --> q=+e
The Mass Spectrometer
most abundant
Isotope of neon
The mass spectrum of naturally
occurring neon, showing three
isotopes.
For neon,
Z = 10 --> 10 protons
N = 10, 11, 12 --> number of neutrons
for three isotopes
A = Z + N = 20, 21, 22 --> atomic mass
numbers for
neon isotopes -->
The Force on a Current in a Magnetic Field
A magnet force can be exerted on
charged particle moving in a
magnetic field.
èA magnetic force can also be
exerted on a current of charged
particles moving in a magnetic field,
e.g. a current in a wire
The magnetic force on the
moving charges pushes the
wire to the right.
Use a RHR to find the direction
of the force on a wire.
The Force on a Current in a Magnetic Field
Derivation of the force F on a wire of
length L with current I and angle θ
with the B field.
F = qvB sin θ
{
( Δq %
F =&
vΔt )B sin θ
#(
Δ
t$ L
'
I
F = ILB sin θ
Example: Force on a current carrying wire in a B field.
A horizontal straight wire 2.0 m long has a current of 20 A pointed towards the
East. Find the magnitude and direction of the force on the wire due to the
Earth’s magnetic field pointing towards the North with magnitude 0.5 Gauss.
N
B
F (up)
I
E
F = ILBsin θ
= ( 20 A) ( 2 m ) ( 5.0 ×10 T ) sin 90
−5
= 2.0 ×10 −3 N
o
The Force on a Current in a Magnetic Field
Magnetohydrodynamic (MHD) propulsion for ships and submarines.
MHD propulsion uses the principle of forces on charged currents in magnetic
fields. As water is expelled from the rear of the ship by the magnetic force,
Newton’s 3rd law causes the ship to recoil forward with the same force.
è Has promise to be a low-noise, reliable and inexpensive system.
Example calculation for a
MHD-powered boat:
Assume a 1000 kg boat which
generates a 1.0 T B-field, and a
current of 100 A at 120 V
across plates separated by 2.0 m.
Find a) the electric power required, and b) the time it takes to accelerate the
boat to 10 m/s (~20 knots), ignoring the friction due to the water.
a) P = IV = (100 A ) (120 V ) = 12000 W = 12 kW
b) F = ILBsin θ = (100 A ) ( 2.0 m ) (1.0 T ) sin 90 o = 200 N
F 200 N
a= =
= 0.2 m 2
s
m 1000 kg
v 10 m s
∴time to accelerate to 20 kts = =
= 50 s
m
a 0.2 2
s
Magnetic Fields Produced by Currents
Currents in wires produce magnetic fields. The long current-carrying
vertical wire shown will cause a compass needle to deflect in a circular
pattern around the wire in a horizontal plane.
Right-Hand Rule. Curl the
fingers of the right hand into the
shape of a half-circle. Point the
thumb in the direction of the
conventional current, and the tips
of the fingers will point in the
direction of the magnetic field.
Magnetic Fields Produced by Currents
The magnitude of the B-field created from a long, straight wire
carrying current I at a distance r from the wire is found
experimentally to be,
µo I
B=
2π r
µ o = 4π ×10 −7 T ⋅ m A
Why must the wire be long?
è to avoid fringe field effects.
permeability of
free space
Magnetic Fields Produced by Currents
Example: A Current Exerts a Magnetic Force on a Moving Charge
The long straight wire carries a current
of 3.0 A. A particle has a charge of
+6.5x10-6 C and is moving parallel to
the wire at a distance of 0.050 m. The
speed of the particle is 280 m/s.
Determine the magnitude and direction
of the magnetic force on the particle.
Magnetic Fields Produced by Currents
µo I
B=
2π r
Magnetic field at a distance r
from the wire with current I
Force on a moving charge in a B-field,
( µo I %
## sin θ
F = qvB sin θ = qv&&
' 2π r $
= (6.5 x 10-6)(280) {(4π x 10-7)(3.0)/[2π(0.050)]} sin 90o
= 2.2 x 10-8 N
From RHR for B and RHR for F
à F is radially inward.
Magnetic Fields Produced by Currents
Current carrying wires can exert forces on each other.
currents in opposite directions
currents in same direction
Calculate the magnitude of the force on Wire 2 due to Wire 1, F21, if the wires
carry currents I1 and I2, respectively, and are separated by a distance r.
F21 = I2LB21sin θ
where B21 = µ0I1/(2πr) is the B-field at 2 from 1
F21 = I2L{µ0I1/(2πr)}sin θ = I1L{µ0I2/(2πr)}sin θ = F12 , the force on 1 by 2
Using RHR’s, this agrees with Newton’s 3rd law that F12 = - F21
Example: Force between two current-carrying rods.
A 1.0 m long current-carrying rod of mass 0.5 kg is “levitated” 7.0 cm below
another 1.0 m long current carrying rod carrying an equal current. Find the
magnitude of the currents in the rods.
I
FB
I
mg
µ0 I 2 L
∑ F = FB − mg = 0 ⇒ 2π r sinθ − mg = 0
2π ( 0.07) ( 0.5) ( 9.8)
2π rmg
I=
=
= 1300 A
−7
o
µ 0 L sin θ
4π ×10 (1.0 ) sin 90
Magnetic Fields Produced by Currents
Conceptual Example: The Net Force That a Current-Carrying Wire
Exerts on a Current Carrying Coil
Is the coil attracted to, or repelled by the wire? è answer: attracted to (why?)
×
×
×
×
×
×
×
B1 (in)
×
×
×
×
Magnetic Fields Produced by Currents
The magnetic field at the center of a
loop of wire of radius R with current
I is found to be experimentally,
µo I
B=
2R
center of circular loop
Find the direction of the B-field at
the center by using RHR.
Magnetic Fields Produced by Currents
Example: Finding the Net Magnetic Field
A long straight wire carries a current of 8.0 A and a circular loop of
wire carries a current of 2.0 A and has a radius of 0.030 m. The
wires are adjacent to each other without touching. Find the magnitude
and direction of the magnetic field at the center of the loop.
Magnetic Fields Produced by Currents
B = B1(straight wire) + B2(loop)
µ o I1 µ o I 2 µ o & I1 I 2 #
$$
B=
−
=
− !!
2π r 2 R
2 %πr R "
(
4π ×10
B=
−7
)
T ⋅ m A ( 8.0 A
2.0 A %
&&
## = 1.1×10 −5 T
−
2
' π (0.030 m ) 0.030 m $
Since B1 > B2 , the direction of B is along B1, normal to the current plane.