Magnetic Forces and Fields (Chapters 32)people.morrisville.edu/~freamamv/Secondary/old... ·...
Transcript of Magnetic Forces and Fields (Chapters 32)people.morrisville.edu/~freamamv/Secondary/old... ·...
Magnetic Forces and Fields
(Chapters 32)
• Magnetism – Magnetic Materials and Sources
• Magnetic Field, B
• Magnetic Force
• Force on Moving Electric Charges – Lorentz Force
• Force on Current Carrying Wires
• Applications • Applications
• Electromagnetic motors
• Torques on magnetic dipole moments µ
• Sources of Magnetic Field
• Magnetic field of a moving charge
• Current Carrying Wires – Biot-Savart law
• Loops, Coils and Solenoids – Ampère’s Law
• Microscopic Nature of Magnetism
How can one magnetize objects?
• Magnetism can be induced: either by stroking an unmagnetized piece of
• Magnets are objects exhibiting magnetic behavior or magnetism
• A magnet exhibits the strongest magnetism at extremities called
magnetic poles: any magnet has two poles, conventionally
dubbed north and south
• Like poles repel each other and unlike poles attract each other
• Unlike charges, magnetic poles cannot be isolated into
monopoles: if a permanent magnetic is cut in half repeatedly, the
parts will still have a north and a south pole
Magnetism – Magnets and relation to electricity
• Magnetism can be induced: either by stroking an unmagnetized piece of
magnetizable material with a magnet, or by placing it near a strong permanent magnet
• Soft magnetic materials, such as iron, are easily magnetized
- They also tend to lose their magnetism easily
• Hard magnetic materials, such as cobalt and nickel, are difficult to magnetize
- They tend to retain their magnetism
• The region of space surrounding a moving charge includes a magnetic field as well
as by an electric field: so, magnetism and electricity cannot be separated
• They are interrelated into the integrated field of electromagnetism: the first
breakthrough in the great effort of developing unified theories about fundamental
interactions (Maxwell, beginning of the XIX-th century).
Magnetic Field – Operational definition and field lines
• Like the sources of electric field, any magnetic material produces a magnetic field
that surrounds it and extends to infinity.
• The symbol used to represent this vector is
• Let’s first describe this vector using an operational definition:
B�
Def: The magnetic field in each location in the surroundings of a magnetic source
is a vector with the direction given by the direction of the north pole of a compass
needle placed in the respective location
• Similar to the electric field, a magnetic field can be patterned using field lines: the • Similar to the electric field, a magnetic field can be patterned using field lines: the
vector B in a point is tangent to the line passing through that point, and the density of
lines represents the strength of the field
• However, while electric field lines start and end on
electric charges (electric monopoles), the magnetic
field lines form closed loops (since there are no
magnetic monopoles)
• Thus, the magnetic field lines should be seen as
closing the loops through the body of the magnet:
that is, the magnetic field inside magnets is not zero
Ex: A compass can be used to
trace the magnetic field lines
• A compass can be used to probe the magnetic field lines produced by various
source, and they will always form closed loops
• Later we’ll look at these sources more systematically:
Magnetic Field – Magnetic field lines for various magnetic sources
#otice the similar
pattern
Magnetic Field – Example: Earth’s Magnetic Field
• The Earth’s geographic north pole is closed to
a (slowly migrating) magnetic south pole
• The Earth’s magnetic field resembles that of a
huge bar magnet deep in the Earth’s interior
slightly tilted with respect to the axis of
rotation of the planet
• The mechanism of Earth’s magnetism is not
very well understood
• There cannot be large masses of permanently • There cannot be large masses of permanently
magnetized materials since the high
temperatures of the core prevent materials from
retaining permanent magnetization
• The most likely source is believed to be electric currents in the liquid part of the
planetary core
• The direction of the Earth’s magnetic field reverses every few million years
• The origin of the reversals is not well understood in detail, albeit there are models
describing how it may happen
Magnetic Force – On a moving charge
•Magnetic fields act on moving charges with magnetic forces. We’ll study this effect
in two (related) cases:
1. Moving Charged Particles
2. Current Carrying Wires
1. Magnetic Force on a Moving Point Charge
• Consider a test charge q moving in a field B with velocity
v making an angle θ with B: the particle will be acted by a
magnetic force F (sometimes called Lorentz force) of
F�
v�
B�
θq>0
� ��
Magnitude:
Direction: given by a right hand rule (let’s call it #1):
B
F�
q>0
F�
q<0
sinBF vq θ=
vF Bq= � ��
• The expression for the magnetic force leads to a
definition for magnetic field unit called Tesla (T)
• A popular alternative is the cgs unit, Gauss (G) (useful
for small fields): 1T = 104 Gauss
SI
NTesla (T)
C m sB = =
⋅
v�
B�
#otation: Vectors perpendicular on page/board/slide:
Outward Inward
Exercises:
1. Force direction: Find the direction of the force on an electron moving through the magnetic
fields represented below.
Problem:
1. Charge moving in a magnetic field: What velocity would a proton need to circle Earth 800
km above the magnetic equator, where Earth's magnetic field is directed horizontally north and
has a magnitude of 4.00×10-8 T?
2. Field direction: Find the direction of the magnetic
field acting on a proton moving as represented by the
adjacent velocity and force vectors. (Assume that the
velocity is perpendicular on the magnetic field.)
• Any moving charge not only that is acted by a magnetic field but it also produces a
magnetic field that surrounds it and extends to infinity
• A test charge q moving in an electric field E and a magnetic field B, with velocity
making an angle θ with B will be acted by a net electromagnetic force (sometimes
called Lorentz force):
Magnetic Force – Charge in an electromagnetic field
( )electric magneticF BF q vEF= + = + �� � � ��
parallel to the
direction of E
perpendicular on
the direction of B +
Ex: One type of velocity selector
• Consider an electric field perpendicular on a magnetic field
• Then only the particles entering the fields with velocity perpendicular
of both will be allowed to pass, which corresponds to the following
condition that the particles are supposed to obey:
+
0E
qE qvB vB
− + = ⇒ =
Magnetic Force – Trajectory of a point charge in a magnetic field
• Let’s look at two particular trajectories that a charged
particle may have in a magnetic field
1. Consider a particle moving into an external magnetic field
so that its velocity is perpendicular to the field
• In this case, the particle will move in a circle, with the
magnetic force always directed toward the center of the
circular path
• Equating the magnetic and centripetal forces, we can find the radius of the circle:
2v mv
+
+
+
v�
v�
v�B
�
F�
F�
F�
2vF qvB m
r= = ⇒
mr
B
v
q= : called cyclotron equation
2. If the particle’s velocity is not perpendicular to
the field, the path followed by the particle is a spiral
called a helix
• The helix spirals along the direction of the field
with a velocity given by the component of the
velocity parallel with B
+
v�
• A current is a collection of many drifting charged particles, such that a magnetic
force is expected to act on a current-carrying wire placed in a magnetic field
• This magnetic force is the resultant of the forces acted on the individual microscopic
electric carriers, but it makes more sense to integrate its effects into a unique
magnetic force acted on the macroscopic current
I = 0 ⇒ F = 0 I ↑ ⇒ F ← I ↓ ⇒ F→
Magnetic Force – Currents in magnetic field
Ex: Experimental observations:
A current carrying vertical wire
placed in a magnetic field pointing
perpendicular into the slide, will be
acted by a magnetic force
perpendicular on the current and
magnetic field: either to the left, or
to the right, depending on the
direction of the current
2. Magnetic Force on Current Carrying Wire
• Consider a straight current carrying wire of length ℓ
immersed in field B, making an angle θ with B: the
portion dℓ of wire will be acted by a magnetic force dF
Magnitude:
F�
B�
θ
�sindF dBI θ= ℓ
Magnetic Force – On a current carrying wire
dF d BI= ×��ℓ�
I
d�ℓ
Direction: Given by right hand rule #1, but instead of
aligning the fingers with the velocity, one aligns the
fingers with the direction of the current
• Since the current flows in the direction of the
positive carriers, the thumbs always indicates the
direction of the force
• If the wire is straight, and the force is the same for
each cross-section, the force on a length L of wire is
F�
I
sindF dBI θ= ℓ
sinF LIB θ= B�
Problems:
2. Basics of a rail gun: A rail gun looks (very)schematically
as in the figure. Evaluate the speed that the projectile of
mass m would achieve after traveling from rest a distance d
on the rails spaced by ℓ with a driving current I with a
magnetic field B.
3. Force on a semicircular current: A semicircular thin
y
I
B�
I
Projectile
Rail
ℓ
m
3. Force on a semicircular current: A semicircular thin
conductor of radius a carries a time dependent current
,
where I0 and τ are positive constant. The wire is allowed to
move vertically through a uniform magnetic field B, as in the
figure. Find the acceleration of the conductor as a time
dependent function.
0
ti I e τ−=R
B�☼
a
x
θ
a
i
Applications – Torque on a Current Loop
• The magnetic force can be used to make electromagnetic motors by using it to rotate
a current carrying loops
• In order to see the principles of such an arrangement, consider
a loop carrying a current I in an external magnetic field B
• The two sides perpendicular on B will be acted by forces
opposite in direction creating a torque that will rotate the loop:
F F
11 1 2
sinF aτ θ=angle between the magnetic
field and the perpendicular to
I I
I I
I
1 1 12 2 2
sin sin sinF a F a F aτ θ θ θ= + =F BIb=
• We see that the torque is maximum when the magnetic field B is parallel with the
plane of area A (θ = 90°), and zero when B is perpendicular on A (θ = 0°)
sin sinBIba BIAτ θ θ⇒ = =
• We can immediately find an expression for the net torque τ = τ1 + τ2:
12 2 2
sinF aτ θ=
field and the perpendicular to
the surface of the loop I I
I
Magnetic Moment, µ
• The net torque exerted by a magnetic field on + current carrying loops is
sin#IABτ θ=
IAµ =��
• Any loop of electric current can be associated with a
magnetic moment pointing perpendicular on the plane
of the loop
• So, we see that a magnetic dipole in a magnetic field
• This magnetic torque exerted on the loop of current can be
written in terms of a vector quantity called magnetic moment:
sinB Bτ µ τ µ θ= × ⇒ =�� �
Ex: Electrons in an atom have an orbital moment due to the
their orbital motion about the nucleus
• So, we see that a magnetic dipole in a magnetic field
will have the tendency to rotate either in a position with
µ parallel with B – stable equilibrium – or anti-parallel
with B – unstable equilibrium
• The “current” doesn’t have to be carried by a wire:
any closed loop of moving charges will have a moment:
as we shall see later, these moments explain magnetism
at a microscopic scale +
Applications – Electromagnetic Motors
• An electric motor converts electrical energy into mechanical energy in the form of
rotational kinetic energy
• As described on the previous slides, the simplest electric motor consists of a rigid
current-carrying loop that rotates when placed in a magnetic field
• The torque acting on the loop will tend to rotate
the loop to smaller values of θ until the torque
becomes 0 at θ = 0°
• If the loop turns past this point and the current
remains in the same direction, the torque reverses remains in the same direction, the torque reverses
and turns the loop in the opposite direction
• To provide continuous rotation in one direction,
the current in the loop must periodically reverse,
such that dc-motors must use split-ring
commutators and brushes
• Actual motors would contain many current loops
and commutators
Sources of Magnetic Field – Moving Charge
• Consider a point charge moving with constant velocity
v: then, the magnetic field B at a position r from the
particle making an angle θ with v is
where µ0= 4 π × 10-7 Tm/A is the magnetic permeability
0
2
ˆ
4
q rvB
r
µπ
×=
��
v�
θ
• We’ve seen that magnetic fields act on moving charges (point-like and currents), so
it is just natural to expect that moving charges also produce magnetic fields: a fact
first discovered serendipitously by Hans Oersted in 1819
r�
B�
+
Magnitude:
Direction: perpendicular on the plane determined by
r and v. Use the following right hand rule (#2): grab
the velocity in your right hand with the thumb in its
direction. Then the curl of the fingers will show the
direction of the field around v: clockwise for positive
charges and anticlockwise for negative charges
where µ0= 4 π × 10-7 Tm/A is the magnetic permeability
of free spacev�
0
2
sin
4
vqB
r
θµπ
=
v�
+
Larger field in
the plane
Weaker field
behind
Weaker fields
ahead
B�
+
B�
B�
Problem:
4. Moving charges interacting electrically and magnetically: Two protons move with
uniform speed v along parallel paths at distance r from each other.
a) Find a symbolical expression for the magnetic force exerted by one proton on the other
one: is it attractive or repulsive? Is this always the case?
b) Calculate the electric force between the charges and compare to the magnetic force.
+
+
• Consider a current I carried along a wire. Then, the
magnetic field produced by a segment dℓ of the current
at a position r from the segment making an angle φ
with dℓ is given by
0
2
ˆ
4
IddB
r
rµπ
×=
�ℓ�
Sources of Magnetic Field – Element of current
φ
r�
dB�
I
d�ℓ
Biot-Savart Law:
Magnitude:0
2
sin
4
IddB
r
µ ϕπ
=ℓ
0
24
ˆId rB
r
µπ
×= ∫
�ℓ�
I
Magnitude:
Direction: perpendicular on the plane determined by r and
v. Use the same right hand rule as for moving positively
charged particles, but curl your right hand fingers around
the current.
• Hence, for a certain finite length of wire
dB�
dB�
dB�
dB�
Problems:
5. Straight Current: A straight wire of length 2a centered in y = 0, carries a current I in
positive y-direction. Calculate the magnetic field at distance r along x-axis.
Useful integral:
6. Circular Current: A circular wire loop of radius a lays in the yz-plane and is traveled by
counterclockwise current I.
( ) ( ) ( )3 2 1 2 1 22 2 2 2 2 2
2
a
a
aa
dy y a
x y x x y x x a−−
= =+ + +
∫
counterclockwise current I.
a) Calculate the magnetic field produced at a distance x along the axis of the loop.
b) Using the result, find the field in the center of the ring.
Sources of Magnetic Fields – Long straight wire
• Consider a long straight wire carrying a current I, the magnetic field at a distance r
perpendicular on the wire is given by:
Magnitude: using the result for Problem 5:
Direction: Given by the right hand rule #2
Comments:
• The magnetic field has cylindrical
0
2B
r
Iµπ
=0
2 2
1
2 1a
IB
r r a
µπ →∞
= →+
I
• The magnetic field has cylindrical
symmetry around the wire
• It gets weaker and weaker as the
circles are larger and larger
a) Half as strong, same direction.
b) Half as strong, opposite direction.
c) One-quarter as strong, same direction.
d) One-quarter as strong, opposite direction.
Quiz 1: A long wire carries a current I as in the figure. Compared to
the magnetic field at point A, the magnetic field at point B is
I
Magnetic Force Between Two Parallel Conductors
• If two long current carrying wires are placed parallel with
each other, they will interact via magnetic forces
• The force on wire 1 is due to magnetic field produced by
wire 2 onto the current in 1, so the force per unit length L
is:
Comments:
• Parallel currents attract each other whereas anti-parallel
conductors repel each other
0
2
F II
L r
µπ
′=0
2
IB
rF I L I L
µπ
′ ′= = ⇒
conductors repel each other
• The force between parallel conductors can be used to redefine the Ampere (A)
• Then the Coulomb (C) can be also defined in terms of the Ampere
Def: If a conductor carries a steady current of 1 A, then the quantity of charge
that flows through any cross section in 1 second is 1 C
Def: If two long, parallel wires 1 m apart carry the same current, and the
magnitude of the magnetic force per unit length is 2 x 10-7 N/m, then the current
is defined to be 1 A
Problems:
7. Force on a moving particle by a current carrying wire: A
proton moves with speed v = 0.25 m/s parallel with a long
wire carrying a current I = 2.0 A, at distance r = 1.0 mm.
Calculate the magnetic force on the proton.
8. Superposition of aligned magnetic fields: Two long I I
r+I
v�
e+
8. Superposition of aligned magnetic fields: Two long
parallel wires carry currents I1 and I2 in opposite directions.
The figure is an end view of the conductors. Calculate the
magnitudes of the magnetic field in points A, B and C located
at given equal distances a from the closest wires.
a a a a
A B CI1 I2
:
S
B� B
�
I
µ�
• If we allow x → 0 in the result of Problem 6, the magnetic field in the center of a
circular loop is
( )
2
0
3 2 02 22 x
I aB
x a
µ=
= →+
Sources of Magnetic Fields – Circular loop of current. Coils
The magnetic field B inside
has the same direction as
the magnetic moment µ
I• Notice that a current loop can be seen as a
magnet with magnetic field lines that remind
of the equipotential lines of an electric dipole:
so the loop behaves like a magnetic dipole
0
2B
a
Iµ=
• # loops form a coil
with the maximum field
in the middle of the coil:
so the loop behaves like a magnetic dipole
• The field produced by a loop or a coil is related to the
respective magnetic moment µ
Ex: Magnetic field and moment: the magnetic field in the
center of a circular loop is related to its magnetic moment
as given by 2 0
32Ba I
aµ
µµ π
π= ⇒ =
� �
0max
2B #
a
Iµ=
Magnetism in Materials – Magnetic moments of electrons
• Now we are prepared to see that the magnetism of materials is microscopically
mainly determined by the alignment of elementary electronic magnetic dipoles
• Notice first that atoms should act like magnets because of
the orbital motion of the electrons about the nucleus
• Since each electron circles the atom once in about every
10-16 seconds, it produces a current of 1.6 mA and a
magnetic field of about 20 T at the center of the orbit
• However, the magnetic field produced by one electron in
an atom is often canceled by an oppositely revolving
+
249.27 10 J TBµ−= ×
an atom is often canceled by an oppositely revolving
electron in the same atom, so the net result is that the
magnetic effect produced by electrons orbiting the nucleus
is either zero or very small for most materials
• The classical model is to consider the electrons to spin like tops but
it is actually a quantum effect.
• The magnetic moment of an electron is given by Bohr magneton:
• Most materials are not naturally magnetic since electrons usually pair up with their
spins opposite each other
spinµ�
Magnetism in Materials – Types of magnetism
Paramagnets Ex: aluminum, uranium
• Moments point in random directions in zero external field but
in an external field Bext they rotate so the net field increases
• The increment in field is small and paramagnetism competes
with thermal motion
Ferromagnets Ex: iron, nickel
• In some materials, large groups of atoms in which the spins are
• So, since the alignment of elementary magnetic dipoles associated with the electronic spins
depends on the microscopic structure, various materials are classified depending on how they
behave in external magnetic fields:
extB�
• In some materials, large groups of atoms in which the spins are
aligned form ferromagnetic domains
• When an external field Bext is applied, the domains that are
aligned with the field tend to grow at the expense of the others
• This causes the material to become magnetized by amounts
larger than in the paramagnetic case
Diamagnets Ex: mercury, superconductors, animal bodies
• In these materials an external magnetic field induces opposite magnetic
• In these cases the internal magnetic field is less than the external one
• A diamagnet placed in an external magnetic field will have the tendency to float
• The magnetization is slowly disappearing after removing the external field
extB�
B�
I3
d�ℓ
B⊥
�
B�
�
Ampère’s Law
• André-Marie Ampère found a procedure for deriving the relationship between the
current in an arbitrarily shaped wire and the magnetic field produced by the wire:
Ampère’s Circuital Law: If a net current Iencl is
enclosed by an arbitrary closed path, the integral of all
products B|| dℓ (where B|| is the component of the
magnetic field along each elementary step dℓ of the
path) is proportional to Iencl:
0 enclB d Iµ⋅ =∫��ℓ�
Amperian
loop
I1I2
I4
Iencl = I2 + I3 + I4
⊥
#et current inside
the path
Line integral around a closed
path called an Amperian loop
Ex: Ampere’s law can be used to demonstrate the result that we
obtained previously for a closed circular path around a long
straight current I: since the circumference of the path is 2π r,
and by symmetry the field around the Amperian is expected to
be everywhere constant and tangent to the path, we get:
002
2encl
IB d B d B r I B
r
µπ µ
π⋅ = = = ⇒ =∫ ∫��ℓ ℓ� �
∫�
I
B�
r
Problem:
8. Magnetic field inside and outside of a current carrying conductor: A cylindrical
conductor with radius R carries a current I uniformly distributed over the cross-sectional
area of the conductor. Confirm the relationships given below for the magnetic field in the
interior and the exterior of the conductor.
I
Direction
• The field lines of the solenoid resemble those of a bar
magnet and the field direction is given by the right hand
Sources of Magnetic Field – Solenoids
• If a long straight wire is wound into
a coil of closely spaced loops, the
resulting device is called a solenoid
• It is also known as an electromagnet
since it acts like a magnet only when
it carries a current
S
S
:
:
I
I
B�
solenoid bar magnet
magnet and the field direction is given by the right hand
rule applied to the current through any of the turns
Magnitude
• The magnitude of the field inside a solenoid is constant at
all points far from its ends
where n is the number of turns per unit length
• This expression can be obtained by applying Ampère’s
Law to the solenoid…
0B n Iµ=n #= ℓ
Quiz 2: What is the
direction of the field in the
center of the solenoid
below?
Sources of Magnetic Field – Solenoid field using Ampère’s Law
• Consider a cross-sectional view of a tightly wound solenoid of turn density n,
carrying a current I
• If the solenoid is long compared to its radius, we assume the field inside is uniform
and outside is zero
• Then we can apply Ampère’s Law to a rectangular Amperian a → b → c → d → a:
Comment:
In reality the field is not perfectly
uniform along the axis of a cylinder. Amperian loop
with # turns inside
0 0 0 #
BL #I B I n IL
µ µ µ= ⇒ = =
with # turns inside
Ampère’s law
I