Magnetic Field

 

MAGNETIC FIELD & MAGNETIC FORCES
(Chapter 27)

SOURCES OF MAGNETIC FIELD
(Chapter 28) appears below Ch 27

 

KEY TERMS: permanent magnet, magnetic monopole,
Tesla, Gauss, magnetic field line, magnetic flux, Weber,
mass spectrometer, magnetic dipole moment or
magnetic moment, solenoid, Hall effect.

 

 

There are no individual magnetic poles
(or magnetic charges). Electric charges can
be separated, but magnetic poles always come in pairs
- one north and one south.

Opposite poles (N and S)
attract and like
poles (N and N,
or S and S) repel.

These bar magnets will remain "permanent"
until something
happens to eliminate
the alignment of
atomic magnets
in the bar of
iron, nickel,
or cobalt.
Fig. 27.1 Bar magnets (permanent magnets)
  

 

  

The Earth's magnetic
field appears
to come from
a giant bar
magnet, but
with its south pole located
up near the
Earth's north pole
(near Canada).

The magnetic field lines come out of the Earth
near Antarctica and enter near Canada.
Fig. 27.3 Earth's magnetic field
  

 

 

 

 

 

The connection between electric current and
magnetic field was first observed when the
presence of a current in a wire near a magnetic compass affected the direction of the compass needle. We now know
that current gives rise
to magnetic fields, just
as electric charge gave
rise to electric fields.
Fig. 27.5 Compass near a current-carrying wire
  

 

 

 

The direction of the cross product can be obtained
by using a
right-hand rule:
FINGERS of the right
hand point in the
direction of the
FIRST vector (v) in
the cross product, then
adjust your wrist so that
you can bend your
fingers(at the knuckles!) toward the direction of
the second vector (B);
extend the thumb to
get the direction
of the force.

Fig. 27.6 Magnetic force acting on a moving charge
  

 

 

 

 
Fig. 27.11 Magnetic field lines of a permanent magnet, cylindrical coil, iron-core electromagnet, straight
current-carrying wire, and a circular
current-carrying loop.
  

 

 

 
Fig. 27.13 Magnetic flux through an area element dA is defined to be dFB = B dA cos f
 

 

 

A charged particle
moving in a plane perpendicular to a magnetic field will
move in a circular
orbit with the magnetic force playing the role
of centripetal force.
The direction of the
force is given by the right-hand rule.

Equating the centripetal force with the magnetic force and solving for R the radius of the circular path we get

mv^2 / R = q v B and

R = m v / q B

 

We will observe
electrons travelling in circular motion
in the lab.

Fig. 27.15 Orbit of charged particle
in a magnetic field

  

 

 

 

The Earth's surface and its inhabitants are protected from dangerous
cosmic radiation (energetic
protons) from
the Sun by the Earth's
magnetic field.


Using the right-hand rule one can see that positively charged particles coming
from the Sun
will be deflected in an easterly direction.

Fig. 27.18 Van Allen radiation
belts around the Earth
  

 

 

 
Fig. 27.21Velocity selector for charged particles
  

 

An electric field and a magnetic field placed at right
angles to each other can function as a "velocity selector." When the force up = force down in (b)
above, the charge will travel in a straight (horizontal)
line. The speed can be obtained from the equation

q v B = q E, or v= E / B

 
   

 

Charged particles leaving a velocity selector (with a known velocity) can be inserted into a chamber with
a magnetic field as shown.

In the circular orbit equation above
R = mv / q B

we can substitute

v = E / B to get

R = m E / q B^2

from which we can solve
for m / v,
the mass-to-charge ratio. Knowing
the charge (ionized state)
and the measured radius
we can find the mass of
the particle.
Fig. 27.22 Mass spectrometer
  

 

 

 

 

 

Similar to the force
on a moving charge
in a B field, we have
for a conductor of length L carrying a current of I in a B
field the force experienced by the conductor:

F = I L x B
Fig. 27.23 Force on a moving charge
in a current-carrying conductor
  

 

 

 
Fig. 27.24 Magnetic force on straight wire segment
  

 

 

 
Fig. 27.25 Magnetic forces - reversing the direction of
the B field or the direction of the current I reverses
the direction of the force.
  

 

 

 

 

 

 

 
Fig. 27.26 Loudspeaker - the signal (or voltage) from an amplifier causes a current to flow in the voice coil which
is in a B field as shown. The current experiences a force along the axis of the coil. If the signal is an AC signal of
a certain frequency the coil will vibrate back and forth at that frequency causing the speaker cone to vibrate
and put out a sound wave.
 

 

 

 

 

 
Fig. 27.29 Magnetic forces on a current-carrying loop
  

 

A current-carrying loop in a B field is the basis of an electric motor.
Using the right-hand rule one can see that the forces actingon the wire will cause the loop to rotate. Changing the current direction at the right time will cause the loop to continue rotating on the motor shaft.
 
   

 

 

 

 
Fig. 27.30 The right-hand rule determines the direction
of the magnetic moment of a current-carrying loop. This
is also the direction of the loop's area vector A;
the loop's magnetic moment is m = I A.
  

 

 

TORQUE

t = m x B

 

From the right-
hand rule we see that the torque vector t is directed into the page or screen. The torque tends to rotate the solenoid in a clockwise direction.
Fig. 27.32 Torque on a solenoid
in a magnetic field
  

 

 

 

 

 

 
Fig. 27.34 Current loops in a non-uniform magnetic field. The axis of the bar magnet is perpendicular to the plane
of the loop and passes through the center of the loop. In
(a) the net force on the loop is toward the right and
in (b) the net force on the loop is toward the left.
(Note: When m and B are in the same direction the
loop is attracted toward the magnet.)
  

 

 

 

 

(a) Randomly oriented atomic magnetic moments in an unmagnetized iron bar.

 

(b) In a magnetized piece of iron
the atomic magnetic moments are aligned. The net magnetic moment of the bar magnet points
from its south pole to its north
pole (inside the magnet).

 

(c) In a magnetic field the torque
on a bar magnet tends to align
the magnet's dipole moment with
the direction of the B field.

t = m x B
Fig. 27.35 Atomic
magnetic moments
  

 

 

 

 

 

 

The B field of the bar magnet causes a net magnetic moment
in the object.

 

(Note: When m and B are in the same direction the loop
is attracted toward
the magnet. See Fig. 27.34 above.)
Fig. 27.36 Bar magnetic attracts
iron object
  

 

 

 

Fig. 27.37 DC motor.
Use torque (t = m x B) or force (F = I L x B) to see how
the current loop rotates. The direction of the current
must be reversed at the right time to keep the loop
rotating continuously in the same direction.

  

 

 

 
Fig. 27.39 Hall effect.
The forces on the charge carriers in a conductor in a magnetic field give rise to a voltage (Vab) across the width
of the conductor. The polarity (sign) of Vab is evidence of the fact that the electrons move and carry the charge in conductors. (The protons are massive and fixed in the
solid metal lattice.)
  

 

 

 
Fig. 27.54 Linear motor.
The force (F = I L x B) on the moving charges in the
purple bar cause the bar to move to the right - a linear motor! Use the right hand rule to obtain direction of
force on the moving charges (current) in the purple bar.
  

 

 

 

Fig. 27.67 Electromagnetic pump.
Current flowing vertically through the liquid metal experiences a force (F = I L x B) along the tube as shown. The charges (in the current)
are the electrons on the atoms of the (liquid) metal,
thus the metal experiences a force F and the
liquid metal moves.

  

For an intesting website (HyperPhysics) with a pictorial
descriptionof magnetic fields and their properties have
a look at:

http://230nsc1.phy-astr.gsu.edu/hbase/hframe.html

and click on Electricity and Magnetism,

then Magnetic Fields, etc., etc.

 

 

 

SOURCES OF MAGNETIC FIELD (Chapter 28)

 

KEY TERMS: source point, field point, principle of superposition of magnetic fields, law of Biot and
Savart, Ampere, Ampere's law, displacement current.

 

 

 
Fig. 28.1Magnewtic field lines B due to a moving
charge. The positive charge is moving with a
velocity v as shown. Use right hand rule to
determine the direction of the B field.
  

 

 

 
Fig. 28.2 Electric and magnetic forces on two moving protons. Coulomb's law indicates the repulsive electric
force between the two positive charges, and the
magnetic force (F = q v x B) is caused by the upper
charge moving in the B field of the lower moving charge.
  

 

 

 
Fig. 28.3 Magnetic field due to a current element. The current is moving in a direction as shown. Use the Law of
Biot and Savart (B = I dl x r mo / 4pr^3)
and the right hand rule to determine the magnitude and
direction of the B field.
  

 

 

 
Fig. 28.5 Magnetic field B due to long straight current-carrying wire. Use a right hand rule to get the direction
of the infinitesimal dB.
  

 

 

 
Fig. 28.6 Magnetic field B lines around long straight conductor. Note the use of the right hand rule to
get the direction of the current.
  

 

 

 

Fig. 28.9 Parallel conductors carrying currents in same direction. The current I' is moving in the B field caused
by the current I, so it experiences a force (F = I' L x B).

  

 

 

 
Fig. 28.12 Magnetic field B caused by circular current loop.
After adding (integrating) all the vector components
caused by all the infinitesimal I dl the total B field will be
in a direction along the x-axis. From symmetry we can see that the vectors components in the yz-plane will all cancel leaving only the component of B in the x-direction.
  

 

 

 
Fig. 28.15 Some integration paths for line integral of B
  

 

 

 
Fig. 28.18 Ampere's law can be used to find B both
inside and outside a solid wire carrying a current I as shown. The right hand rule gives the direction of B.
 

 

 

 
Fig. 28.21The magnetic field B caused by long current-carrying solenoid. Ampere's law reduces to four
straight-line paths of integration.
The integral over three of the paths will be zero.
  

 

 

 
Fig. 28.24 Magnetic dipole moment (m = I A) of an orbiting electron.
The right-hand rule determines the direction of the magnetic moment of a
current-carrying loop. The direction of the electron's angular momentum vector L can be obtained using the right hand rule for angular momentum.
 

 

 

 
Fig. 29.22 The displacement current, as the capacitor
is charged by Ic, can be regarded as the source of
the B field. (See chapter 29)
  

 

 

 
Fig. 28.42 Magnetic fields are associated with a signal-carrying coaxial cable.
If the current is the same magnitude in each direction, the magnetic field outside the coaxial cable is zero.
The absence of B fields around a coaxial
cable results in no interference in nearby electrical equipment
and wires - an important result.
  

 

 

 

 

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© 2008 J. F. Becker