Biomedical Engineering Reference
In-Depth Information
WORKED EXAMPLE
Consider a 0.5 m long solenoid with a diameter of 30 mm consisting of five layers of windings
of 600 turns each and carrying a current of 3 A. The magnetic flux at the centre of the solenoid is
=
μ
o
NI
L
B
10
−
7
600
×
5
×
3
=
4
π
×
0
.
5
m
2
=
0
.
0226 weber
/
Note that the solenoid diameter and the number of layers of turns do not enter the equation
so long as the length to diameter ratio remains large.
If the air core within the solenoid is replaced with a metal (usually iron), then the
magnetic field is increased in proportion to the relative permeability,
μ
r
, of the material
=
μ
r
μ
o
NI
L
B
(3.4)
NI
L
=
μ
where
is the permeability of the material.
The relative permeability is dependent on the magnetic field and is also a function
of frequency and saturation. Typical values of
μ
r
are 200 for magnetic iron and 100 for
nickel at a magnetic flux density of 0.002 weber/m
2
. Special alloys can have much higher
permeability with permalloy (78.5% Ni and 21.5% Fe) reaching 8000 and mumetal (75%
Ni, 2% Cr, 5% Cu, and 18% Fe) reaching 20,000.
If the air core of the solenoid in the previous example is replaced with an iron core
with
μ
μ
r
=
200, the magnetic flux at its center will increase to 0
.
0226
×
200
=
4
.
52
weber/m
2
When a conductor of length
l
(m) cuts through a magnetic field with strength
B
(weber/m
2
)
at a velocity
v
(m/s), a voltage
ε
(V) is developed across the conductor if the
respective directions of the various components are orthogonal.
ε
=
Bl
v
(3.5)
Once again, the direction of the induced current is determined by the right-hand rule,
where the direction of the magnetic field is defined by the thumb, the direction of motion
by the middle finger, and the direction of current flow (if the circuit is closed) by the index
finger.
One final important relationship is Faraday's law, which states that the voltage
(V)
induced in a coil is equal to product of the number of turns,
N
, and the rate of change of
the magnetic flux,
d
ϕ/
dt
, where
ϕ
is measured in webers.
ε
=
N
d
ε
dt
(3.6)
It is not important whether the change in flux is induced by moving a permanent
magnet within the coil or by changing the current through the primary winding of a
transformer to induce a changing flux in the secondary.
