Biomedical Engineering Reference
In-Depth Information
theorem, there is a bandwidth, or equivalently an averaging time,
over which thermal noise and the non-random E or B fields are
compared(BarnesandSeyed-Madani,1987).Thevolume(orspatial
dimension) of comparative measurements must also be taken into
account since for small volumes, statistical fluctuations due to
space charge density become relatively larger. For example, at the
molecular level, noise fields are of the order of 10
8
Vm
−
1
.Inthe
following, the effects of both random noise fields and non-random
appliedfieldsaresimplycompared,ratherthancomparingthefields
themselves.
The
x
componentofthetranslationalLorentzequationofmotion
is written as
x
=
v
x
0
t
col
+
1
2
qE
x
m
t
col
, where the
x
component of the
thermal velocity is
v
x
0
,and
t
col
is the time since the last collision.
Hencethethermalstepis
v
x
0
t
col
,whichisrandom,whiletheoffsetis
1
2
qE
x
m
t
col
,which isnon-random.
At this point we need to decide upon a suitable averaging time
and volume size for our comparison of the effects due to noise
and due to the E fields of interest. The averaging time in the
general case relates to the method of measurement, whether man-
made experimental equipment, or perhaps some biological means
of sensing E fields. Our measuring volume will also depend on the
measurement technique, whether experimental or biological. We
can estimate that the averaging time (
t
ave
) might range from some
milliseconds upwards, and our volume will have a length associated
with the volume (
l
vol
) that might range from atomic dimensions,
0.1 nm upwards. When considering E field effects upon a calcium
ion, the ion's mobility,
μ
Ca
, links the averaging time
t
ave
, with the
measurement length
l
vol
; if we choose either
t
ave
or
l
vol
, we specify
the other. To determine the volume, a measurement area must be
specified; if we are investigating E field effects, this area will form
a current probe, sensing a flow of charged particles per area, a
currentdensity.Like
l
vol
,thisareamightrangefrom0.1nmupwards,
depending on current density.
Taking
t
ave
=
0.1 seconds, we shall assume a very dilute solution
ofcalciumions;ion-ioninteractionsareignoredinoursimplestudy.
Examiningthesizeofthethermalstep,
v
x
0
t
col
,thisisestimatedtobe
2.6
×
10
−
12
mandtheoffset
2
qE
m
t
col
estimatedtobe4.0
×
10
−
22
m,
where we take
E
x
to be 1 V m
−
1
and the mean thermal velocity is
