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Chapter 9
Magnetic Forces, Materials and Inductance
The magnetic field B is defined from the Lorentz Force Law, and specifically from the
magnetic force on a moving charge:
F = qv x B
1. The force is perpendicular to both the velocity v of the charge q and the magnetic field B.
2. The magnitude of the force is F = qvB sin where is the angle < 180 degrees between the
velocity and the magnetic field. This implies that the magnetic force on a stationary charge or
a charge moving parallel to the magnetic field is zero.
3. The direction of the force is given by the right hand rule. The force relationship above is in
the form of a vector product.
From the force relationship above it can be deduced that the units of magnetic field are
Newton seconds /(Coulomb meter) or Newton per Ampere meter. This unit is named the Tesla.
It is a large unit, and the smaller unit Gauss is used for small fields like the Earth's magnetic
field. A Tesla is 10,000 Gauss. The Earth's magnetic field is on the order of half a Gauss.
Force On A Moving Charge
Lorentz Force Law
Both the electric field and magnetic field can be defined from the
Lorentz force law:
The electric force is straightforward, being in the direction of the electric
field if the charge q is positive, but the direction of the magnetic part of the
force is given by the right hand rule.
Force On A Moving Charge
Force on a Differential Current
dF = dQv x B
J
 v v
dF
J  Bdv
dF
 v dv v  B
dF
J  Bdv
Jdv
dQ
 v dv
F

 J  B dv
 vol
F

 K  B dS
S
KdS IdL
dF
K  BdS
F


 I dL  B I  B_x_dL


dF
IdL  B
F
IL  B
Example 9.1
 3

3
 6  1 
Fy1  2 10  3 10 
dx
x 


 1

H
I
 az
2  x
az  ax
Fy1  6.592 10
 2

3
 6  1 
Fx1 2 10  3 10 
dy  1
3 


 0

az  ay
Fx1  4  10
 1

3
 6  1 
Fy2  2 10  3 10 
dx
x 


 3

az  ax
Fy2  6.592 10
 0 
Fx2 2 10  3 10   1 dy  1
 2



az  ay
Fx2  1.2  10
3
B o H
 6 
F  Fx1  Fx2  Fy1  Fy2
F

I  BxdL

9
9
9
8
9
F  8  10
The net force on the loop is in the -ax direction
Force Between Differential Current Elements
Example 9.2
D9.4
Force And Torque On A Closed Circuit
F

I  B_x_dL

F

IB   1 dL

T
R F
Force And Torque On A Closed Circuit
dT
IdS  B
Magnetic Dipole Moment dm
dm IdS
dT
T
dm  B
IS  B m  B
Force And Torque On A Closed Circuit
Build a DC Motor – Hands On
Explain Operation
Force And Torque On A Closed Circuit
DC Motor - Illustration
Example 9.3 and 9.4
The Nature of Magnetic Materials
Magnetic Materials
Magnetic Materials may be classified as diamagnetic, paramagnetic, or ferromagnetic on
the basis of their susceptibilities. Diamagnetic materials, such as bismuth, when placed in an external
magnetic field, partly expel the external field from within themselves and, if shaped like a rod, line up
at right angles to a non-uniform magnetic field. Diamagnetic materials are characterized by constant,
small negative susceptibilities, only slightly affected by changes in temperature.
Paramagnetic materials, such as platinum, increase a magnetic field in which they are
placed because their atoms have small magnetic dipole moments that partly line up with the external
field. Paramagnetic materials have constant, small positive susceptibilities, less than 1/1,000 at room
temperature, which means that the enhancement of the magnetic field caused by the alignment of
magnetic dipoles is relatively small compared with the applied field. Paramagnetic susceptibility is
inversely proportional to the value of the absolute temperature. Temperature increases cause greater
thermal vibration of atoms, which interferes with alignment of magnetic dipoles.
Ferromagnetic materials, such as iron and cobalt, do not have constant susceptibilities; the
magnetization is not usually proportional to the applied field strength. Measured ferromagnetic
susceptibilities have relatively large positive values, sometimes in excess of 1,000. Thus, within
ferromagnetic materials, the magnetization may be more than 1,000 times larger than the external
magnetizing field, because such materials are composed of highly magnetized clusters of atomic
magnets (ferromagnetic domains) that are more easily lined up by the external field.
Magnetization and Permeability
Magnetization and Permeability
Example 9.5
Magnetic Boundary Conditions
The Magnetic Circuit
The Magnetic Circuit
The Magnetic Circuit
Inductance and Mutual Inductance
A self-induced electromotive force opposes the change that brings it about.
Consequently, when a current begins to flow through a coil of wire, it undergoes an
opposition to its flow in addition to the resistance of the metal wire. On the other hand,
when an electric circuit carrying a steady current and containing a coil is suddenly
opened, the collapsing, and hence diminishing, magnetic field causes an induced
electromotive force that tends to maintain the current and the magnetic field and may
cause a spark between the contacts of the switch. The self-inductance of a coil, or
simply its inductance, may thus be thought of as electromagnetic inertia, a property that
opposes changes both in currents and in magnetic fields.
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