Page 137 - ITU Journal Future and evolving technologies Volume 2 (2021), Issue 3 – Internet of Bio-Nano Things for health applications

ITU Journal on Future and Evolving Technologies, Volume 2 (2021), Issue 3

Mathematically speaking, one can integrate the 2 − 2
volume in both sides of Eq. 18 , so that one gets: ∫ ( ) = ∫ .

2 − ( ) 2 ( )
∫ ( , ) = 2 ∫ ( , ) + ∫ ( , ) =
2
Where Gauss’s law was employed but in the form of:
− ( − ) ∫ ( , ) ( )

( )
From the side of electricity, ( , ) denotes the ∫ . = ( )
volumetric charge density transported by bacteria, With “ ” a permissivity constant different to " ”
fact that implies that its integration over the volume because equipotential lines might not be inside the
containing the charge turns out to be the net charge: region of electric field in some points of space of
( ) = ∫ ( , ). Under this view the left side of bacteria displacement, so that only a single medium
Eq. 19 is recognized as the instantaneous electric is not expected. Instead up two media or more can
current. Thus, one arrives at: coexist together. A similar procedure can be applied

( ) 2 2 to the second term of the right side of Eq. 21:
= ( ) = ∫ ( , ) + ∫ ( , )
2 ( )
2 ∫ ( , ) = ∫ ( ) ( )
− (1 − 2 ) ∫ ( , ) ( )

and again, one can construct the electric potential
By assuming the cylindrical coordinates system yielding finally that the term of Eq. 27 is linearly
then the variable “x” is now changed to be the radial proportional to the cylindric radius:
coordinate. Then one gets:
( )
∫ ( ) =
( ) = 2 ∫ ( , ) + ∫ ( , )

− − ( )
− ( − ) ( ). ( ) ∫ ( ) = ∫ = . ( )

Now special attention is paid on the first term of the It should be noted again that = . Therefore,
right side. Since ( , ) have been defined as the when previous resulting equations done above are
density of charge transported by bacteria, then this inserted in Eq. 21 then one arrives at:
term is processed as follows: ( ) ( ) ( )
= [− −
2 ( ) 2 ( )
2 ∫ = ∫ ( ) ( )
2 − (1 − )] ( ). ( )

where the volume has been assumed to be = .
The radial derivatives are actually the gradient Thus, one can see that all charges of the right side
operator that acts onto the electric potential [12] have a negative sign. Putting apart the total charge,
given by: then one arrives at:

( ) ( ) ( )
( ) = . ( ) = ( ) [− − − ( − )].

With the balance of units and the incorporation of One can see that the apparition of sign ”-“ on the
permissivity constant one arrives at: right side is because the possible existence of a type
of discharge because the straightforward solution

2
( ) ( )
2 ∫ = ∫ ( ) of Eq. 30 yields the familiar negative exponential.

2
= ∫ ( ) . ( ) Electric interactions between compounds can be

measured through the electric force as dictated by
By which one can identify that in the last equation classical electrodynamics. To explore this, the total
one has the field electric, thus one arrives at the charge of bacteria should be explicitly done, thus
divergence theorem , yielding that this contribution from Eq. 30,
to the BKS equation is proportional to the square of
cylindric radius. With this one gets that: ( ) = [− − − (1 − )] , ( )
( )

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