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Chapter 29 Chapter 29 -- Magnetic FieldsMagnetic Fields
A PowerPoint Presentation byA PowerPoint Presentation by
Paul E. Tippens, Professor of PhysicsPaul E. Tippens, Professor of Physics
Southern Polytechnic State UniversitySouthern Polytechnic State University
A PowerPoint Presentation byA PowerPoint Presentation by
Paul E. Tippens, Professor of PhysicsPaul E. Tippens, Professor of Physics
Southern Polytechnic State UniversitySouthern Polytechnic State University
© 2007
Objectives: Objectives: After completing this After completing this module, you should be able to:module, you should be able to:
•• Define the Define the magnetic field,magnetic field, discussing discussing magnetic polesmagnetic poles and flux lines.and flux lines.
•• Solve problems involving the Solve problems involving the magnitude and direction of magnitude and direction of forces on forces on chargescharges moving in a magnetic field.moving in a magnetic field.
•• Solve problems involving the magnitude Solve problems involving the magnitude and direction of and direction of forces on currentforces on currentcarrying conductorscarrying conductors in a Bin a B--field.field.
MagnetismMagnetism
Since ancient times, certain materials, called Since ancient times, certain materials, called magnetsmagnets, have been known to have the property of , have been known to have the property of
attracting tiny pieces of metal. This attractive attracting tiny pieces of metal. This attractive
property is called property is called magnetismmagnetism..
NS
Bar Magnet
N
S
Magnetic PolesMagnetic Poles
The The strengthstrength of a magnet is of a magnet is concentrated at the ends, concentrated at the ends,
called north and south called north and south
““polespoles” of the magnet.” of the magnet.
A suspended magnet: A suspended magnet: NN--seeking end and seeking end and
SS--seeking end are seeking end are NN
and and SS polespoles..NNSS
N
E
W
SNN
CompassCompassBar magnetBar magnet
S
N
Iron filings
Magnetic AttractionMagnetic Attraction--RepulsionRepulsion
N
SN
N
S
S
NSNS
Magnetic Forces: Magnetic Forces: Like Poles RepelLike Poles Repel Unlike Poles AttractUnlike Poles Attract
Magnetic Field LinesMagnetic Field Lines
N S
We can describe We can describe magnetic field linesmagnetic field lines
by imagining a tiny by imagining a tiny
compass placed at compass placed at nearby points.nearby points.
The The directiondirection of the of the magnetic field magnetic field BB at at
any point is the same any point is the same
as the direction as the direction indicated by this indicated by this
compass. compass.
Field Field BB is is strong strong where where lines are lines are densedense and weak and weak
where lines are sparse.where lines are sparse.
Field Lines Between MagnetsField Lines Between Magnets
N S
N N
Unlike poles
Like poles
Leave N and enter S
Attraction
Repulsion
The Density of Field LinesThe Density of Field Lines
Magnetic Field B is sometimes called the flux
density in Webers per square meter (Wb/m2).
∆N
NE
A
∆∝
∆Line density
∆A
Electric field
∆φB
A
∆Φ∝
∆
Line density
∆A
Magnetic field flux lines φ
NS
Magnetic Flux DensityMagnetic Flux Density∆φ
Magnetic Flux
density:
∆ABA
Φ=
•• Magnetic flux lines are Magnetic flux lines are continuous and closed.continuous and closed.
•• Direction is that of the B Direction is that of the B vector at any point.vector at any point.
•• Flux lines are Flux lines are NOTNOT in in direction of force but direction of force but ⊥..
; = B BAA
Φ= Φ
When area A is perpendicular to flux:
The unit of flux density is the The unit of flux density is the Weber per square meterWeber per square meter..
Calculating Flux Density When Calculating Flux Density When Area is Not PerpendicularArea is Not Perpendicular
The flux penetrating the The flux penetrating the area area AA when the normal when the normal vector vector nn makes an angle makes an angle of of θθ with the with the BB--fieldfield is:is:
The angle The angle θ θ is the complement of the angle a that the is the complement of the angle a that the plane of the area makes with the B field.plane of the area makes with the B field. (Cos (Cos θθ = Sin = Sin α)α)
n
A θ
α
B
Origin of Magnetic FieldsOrigin of Magnetic Fields
Recall that the strength of an Recall that the strength of an electric field Eelectric field E was was defined as the electric force per unit charge.defined as the electric force per unit charge.
Since Since no isolated magnetic poleno isolated magnetic pole has ever been has ever been foundfound, we can’t define the magnetic field , we can’t define the magnetic field B B in in
terms of the terms of the magnetic force per unit north polemagnetic force per unit north pole..
We will see instead that magnetic fields result from charges in motion—not from stationary charge or poles. This fact will be covered later.
+E
+ B vv
⊥⊥⊥⊥
Magnetic Force on Moving ChargeMagnetic Force on Moving Charge
N S
B
N
Imagine a tube that Imagine a tube that projects charge projects charge +q+qwith velocity with velocity vv into into
perpendicular perpendicular BB field.field.
Upward magnetic force F on charge moving in B field.
vv
FF
Experiment shows:Experiment shows:
Each of the following results in a greater magnetic Each of the following results in a greater magnetic force Fforce F: an increase in : an increase in velocityvelocity vv, an increase in , an increase in
chargecharge qq, and a larger , and a larger magnetic field Bmagnetic field B..
Direction of Magnetic ForceDirection of Magnetic Force
B
vv
FF
N SN
The right hand ruleThe right hand rule::
With a flat With a flat rightright hand, hand, point point thumbthumb in direction in direction
of velocity of velocity vv, , fingersfingers in in
direction of direction of BB field. The field. The flat flat handhand pushes in the pushes in the
direction of direction of force Fforce F..
The force is greatest when the velocity v is perpendicular to the B field. The deflection
decreases to zero for parallel motion.
B
vv
FF
Force and Angle of PathForce and Angle of Path
SNN
SNN
SNN
Deflection force greatest Deflection force greatest when path perpendicular when path perpendicular
to field. Least at parallel.to field. Least at parallel.
sinF v θ∝
B
vv
FF
v sin v sin θθvv
θθ
Definition of BDefinition of B--fieldfieldExperimental observations show the following:Experimental observations show the following:
sin or constantsin
FF qv
qvθ
θ∝ =
By choosing appropriate units for the constant of By choosing appropriate units for the constant of proportionality, we can now define the proportionality, we can now define the BB--fieldfield as:as:
or sinsin
FB F qvB
qvθ
θ= =Magnetic Field
Intensity B:
A A magnetic field intensitymagnetic field intensity of one of one tesla (T)tesla (T) exists in a exists in a region of space where a charge of region of space where a charge of one coulombone coulomb (C)(C)moving at moving at 1 m/s1 m/s perpendicular to the Bperpendicular to the B--field will field will experience a force of one experience a force of one newton (N).newton (N).
Example 1.Example 1. A A 22--nCnC charge is projected with charge is projected with velocity velocity 5 x 105 x 1044 m/sm/s at an angle of at an angle of 303000 with a with a 3 3 mTmT magnetic field as shown. What are the magnetic field as shown. What are the magnitude and direction of the resulting force? magnitude and direction of the resulting force?
v sin v sin φφvv
303000
B
vv
FFDraw a rough sketch.Draw a rough sketch.
qq = 2 x 10= 2 x 10--99 C C vv = 5 x 10= 5 x 1044 m/s m/s B B = 3 x 10= 3 x 10--33 T T θθ = 30= 3000
Using rightUsing right--hand rule, the force is seen to behand rule, the force is seen to be upwardupward..
-9 4 -3 0sin (2 x 10 C)(5 x 10 m/s)(3 x 10 T)sin30F qvB θ= =
Resultant Magnetic Force: F = 1.50 x 10-7 N, upward
B
Forces on Negative ChargesForces on Negative ChargesForces onForces on negativenegative charges are opposite to those on charges are opposite to those on positive charges. The force on the negative charge positive charges. The force on the negative charge requires a requires a leftleft--hand rulehand rule to show to show downwarddownward force force FF..
N SN N SN
B
vv
FFRight-hand rule for
positive q FF
B
vvLeft-hand rule for
negative q
Indicating Direction of BIndicating Direction of B--fieldsfieldsOne way of indicating the directions of fields perpenOne way of indicating the directions of fields perpen--dicular to a plane is to use crosses dicular to a plane is to use crosses X X and dots and dots • :
X X X X X X X X X X X X X X X X X X X X X X X X X X X XX X X X
• • • •
• • • •
• • • •
• • • •
A field directed into the paper is denoted by a cross “X” like the tail feathers of an arrow.
A field directed out of the paper is denoted by a dot “ ” like the front tip end of an arrow.
•
Practice With Directions:Practice With Directions:
X X X X X X X X X X X X X X X X X X X X X X X X X X X XX X X X
• • • •
• • • •
• • • •
• • • •
X X X X X X X X X X X X X X X X X X X X X X X X X X X XX X X X
• • • •
• • • •
• • • •
• • • •
What is the direction of the force F on the charge in each of the examples described below?
-vv
-
vv
+
vv
vv+
Up
FF
LeftFF
FFRight
Up
FF
negative qnegative q
Crossed E and B FieldsCrossed E and B Fields
The motion of charged particles, such as electrons, can The motion of charged particles, such as electrons, can be controlled by combined electric and magnetic fields.be controlled by combined electric and magnetic fields.
x x x x x x x x
+
-
e-
v
Note:Note: FFEE on electron on electron is is upwardupward and and opposite Eopposite E--field.field.
But, But, FFBB on electron is on electron is downdown (left(left--hand rule).hand rule).
Zero deflection Zero deflection when when FFBB = F= FEE
B
vv
FFEE
E e--
B
vvFFBB
-
The Velocity SelectorThe Velocity SelectorThis device uses crossed fields to select only those This device uses crossed fields to select only those velocities for which Fvelocities for which FBB = F= FEE. (Verify directions for +q). (Verify directions for +q)
x x x x x x x x
+
-
+q
v
Source of +q
Velocity selector
When FWhen FBB = F= FE E ::
qvB qE=
Ev
B=
By adjusting the E and/or B-fields, a person can select only those ions with the desired velocity.
Example 2.Example 2. A lithium ion, A lithium ion, qq = +1.6 x 10= +1.6 x 10--1616 CC, , is projected through a velocity selector where is projected through a velocity selector where B = 20 mTB = 20 mT. The E. The E--field is adjusted to select a field is adjusted to select a velocity of velocity of 1.5 x 101.5 x 1066 m/sm/s. What is the electric . What is the electric field E?field E?
x x x x x x x x
+
-
+q
v
Source of +q
VV
Ev
B=
E = vBE = vB
E = E = (1.5 x 10(1.5 x 1066 m/s)(20 x 10m/s)(20 x 10--33 T);T); E = 3.00 x 104 V/m
Circular Motion in BCircular Motion in B--fieldfieldThe magnetic force F on a moving charge is alwaysperpendicular to its velocity v. Thus, a charge moving in a B-field will experience a centripetal force.
X X X X X X X X X X X X
X X X X X X X X X X X X
X X X X X X X X X X X X
X X X X X X X X X X X X
X X X X X XX X X X X X+
+
+
+
Centripetal FCentripetal Fcc = F= FBB
RR
FFcc
2
; ;C B
mvF F qvB
R= =
2mvqvB
R=C BF F=
The radius of path is:
mvR
qB=
Mass SpectrometerMass Spectrometer
+q
R
Ev
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
Photographic plate
m1
m2
slit
Ions passed through a Ions passed through a velocity selector at velocity selector at known velocity emerge known velocity emerge into a magnetic field as into a magnetic field as shown. The radius is:shown. The radius is:
The mass is found by The mass is found by measuring the radius R:measuring the radius R:
mvR
qB=
qBRm
v=
2mv
qvBR
=
Example 3.Example 3. A Neon ion, A Neon ion, q = 1.6 x 10q = 1.6 x 10--19 19 CC, follows , follows a path of radius a path of radius 7.28 cm7.28 cm. Upper and lower . Upper and lower B = B = 0.5 T0.5 T and and E = 1000 V/mE = 1000 V/m. What is its mass?. What is its mass?
mvR
qB=
qBRm
v=
1000 V/m
0.5 T
Ev
B= =
v = v = 2000 m/s2000 m/s
-19(1.6 x 10 C)(0.5 T)(0.0728 m)
2000 m/sm = m = 2.91 x 10-24 kg
+q
R
EvB
=+-
x x x x x x x x
Photographic plate
m
slitx x x x x x x x x x x x x x x x x x x x x x x x x x x
x x x x
Summary Summary
N SN
B
vv
FFRight-hand rule for
positive q
N SN
FF
B
vvLeft-hand rule for
negative q
The direction of forces on a charge moving in an electric The direction of forces on a charge moving in an electric field can be determined by the rightfield can be determined by the right--hand rule for positive hand rule for positive charges and by the leftcharges and by the left--hand rule for negative charges.hand rule for negative charges.
Summary (Continued)Summary (Continued)
B
vv
FF
v sin v sin θθvv
θθ
For a charge moving in a For a charge moving in a BB--field, the magnitude of field, the magnitude of the force is given by:the force is given by:
F = qvB sin θ
Summary (Continued)Summary (Continued)
mvR
qB=
qBRm
v=
x x x x x x x x
+
-
+qv
VV
Ev
B=
The velocity The velocity selector:selector:
+q
R
EvB
=+-x x x x x x x x
m
slitx x x x x x x x x x x x x x x x x x x x x
x x x x x
The mass The mass spectrometer:spectrometer: