Page 626 - Mechatronics with Experiments

612 MECHATRONICS

B
i
A ps

ds dB i

l
C
dl e r dl
e i r
r
P
ds
B i
A cs
C

(a) (b) (c)
FIGURE 8.6: Magnetic fields due to current (moving charge): (a) magnetic field at any point P
due to current over a general shape conductor, (b) magnetic field around an infinitely long
straight current carrying conductor, (c) magnetic field inside a coil due to current.

which is the area integral of the electric field over a closed surface. The electric flux over a
closed surface is proportional to the net electric charge inside the surface (Figure 8.5d),
q net
Φ = ∮ ⃗ E ⋅ d ⃗ A = (8.31)
E
A o
where d ⃗ A is a differential vector normal to the surface. The line integral of an electric field
between any two points is the electric potential difference between the two points (voltage)
(Figure 8.5e),
B
V AB =− ∫ ⃗ E ⋅ d ⃗ s (8.32)
A
where the d ⃗ s vector is a differential vector that is tangent to the path traveled from A to B.
The magnetic fields are generated by moving charges (Figure 8.6). There are two
sources to generate and sustain a magnetic field:

1. current (moving charge) over a conductor,
2. permanent magnetic materials.
In the case of current carrying conductors, the magnetic field generated by the current
(moving charges) is called an electromagnetic field. In the case of permanent magnet
materials, the magnetic field is generated by orbital rotation of the electrons around the
nucleus and the spin motion of electrons around their own axis of the permanent magnetic
material. The net magnetic field of the material in macro scale is the result of the vector
sum of the magnetic fields of its electrons.
In a non-magnetized material, the net effect of magnetic fields of electrons cancel out
each other. Their alignment in a certain direction, by magnetizing the material, gives the
material non-zero magnetization in a specific orientation. Either way, the magnetic field is
a result of moving charges. The electric field vector starts at charges (i.e., positive charges)
and ends in charges (negative charges). The magnetic field vector encircles the current that
generates it (Figure 8.6). The vector relationship between the current and the magnetic field
it generates follows the right hand rule. If the current is in the direction of the thumb, the