# Spotlight 1

## Introduction

The many thousands of different chemical reactions carried out simultaneously and successively in living cells are closely coordinated. Electron transport is closely linked to biological activity. Electrons can be carried by diffusible molecules picking up electrons at one location and delivering them to another one as well as being transferred along molecular chains. Moreover, biological polar molecules and polymer structures with energy supply (e.g., microtubules) can get excited and generate an endogenous electromagnetic field with a strong electrical component in their vicinity. The endogenous electrical fields through action on charges, on dipoles and multipoles, and through polarisation exert forces and can drive charges and particles in the cell. It is well known that the majority of proteins are electrically polar and represent electric dipoles.

A protein polymer network e.g., the cytoskeleton is a dynamic organiser of eukaryotic cells. The cytoskeleton exerts forces and generates movements without any major chemical changes. The cytoskeleton reorganises continually as the cell changes its shape, divides, and responds to its environment. Indeed, the fundamental structure of the cytoskeleton formed by microtubules satisfies the basic requirements for excitation of vibrations and generation of an oscillating electromagnetic field.

## A Quick Review of Maxwell's Equations

The general theory of electromagnetic phenomena is based on Maxwell’s equations, which constitute a set of four physical equations relating space and time changes of electric and magnetic fields to their scalar and vector source densities. In a stationary medium, all quantities are evaluated in a reference frame in which the observer and all the surfaces and volumes are assumed to be at rest. For stationary media, Maxwell’s equations in differential form are given by:

$\triangledown&space;\cdot&space;\overrightarrow{D}&space;\:&space;\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:&space;(1.1a)$
$\triangledown&space;\cdot&space;\overrightarrow{B}=0&space;\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:&space;(1.1b)$
$\triangledown&space;\times&space;\overrightarrow{E}=-\frac{\partial&space;\overrightarrow{B}}{\partial&space;t}&space;\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:&space;(1.1c)$
$\triangledown&space;\times&space;\overrightarrow{H}=\overrightarrow{J}+\frac{\partial&space;\overrightarrow{D}}{\partial&space;t}&space;\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:&space;(1.1d)$

In integral form, Maxwell’s equations are written as follows:

$\oint&space;_S&space;\overrightarrow{D}\cdot&space;d\overrightarrow{s}=Q_T(t)&space;\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:&space;(1.2a)$
$\oint&space;_S&space;\overrightarrow{B}\cdot&space;d&space;\overrightarrow{s}=0&space;\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:&space;(1.2b)$
$\oint&space;_\Gamma&space;\overrightarrow{E}\cdot&space;d&space;\overrightarrow{l&space;}=-\int&space;_S&space;\frac{\partial&space;\overrightarrow{B}}{\partial&space;t}&space;\cdot&space;d&space;\overrightarrow{s&space;}&space;\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:&space;(1.2c)$
$\oint&space;_\Gamma&space;\overrightarrow{H}\cdot&space;d&space;\overrightarrow{l&space;}=\left&space;(&space;\overrightarrow{J}+\int&space;_S&space;\frac{\partial&space;\overrightarrow{D}}{\partial&space;t}&space;\right&space;)&space;\cdot&space;d&space;\overrightarrow{s}&space;\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:&space;(1.2d)$

Where:

$\overrightarrow{E}$ = electric field intensity $\left&space;(&space;Vm^{-1}&space;\right&space;)$.
$\overrightarrow{B}$ = magnetic flux density $\left&space;(&space;T&space;\right&space;)$.
$\overrightarrow{D}$ = electric flux density $\left&space;(&space;Cm^{-2}&space;\right&space;)$.
$\overrightarrow{H}$ = magnetic field intensity $\left&space;(&space;Am^{-1}&space;\right&space;)$.
$\rho$ = free electric charge density $\left&space;(&space;Cm^{-3}&space;\right&space;)$.
$Q_T$ = net free charge, in coulombs $\left&space;(&space;C&space;\right&space;)$, inside any closed surface $\left&space;&space;S&space;\right&space;$.
$\overrightarrow{J}$ = free electric current density $\left&space;(&space;Am^{-2}&space;\right&space;)$.

In general, the quantities in (1.1) and (1.2) are arbitrary functions of the position, $\overrightarrow{r}$ and time, $t$. Maxwell’s equations, involve only macroscopic electromagnetic fields and, explicitly, only macroscopic densities of free-charge $\rho$, which are free to move within the medium, giving rise to the free-current densities, $\overrightarrow{J}$. The effect of the macroscopic charges and current densities bound to the medium’s molecules are implicitly included in the auxiliary magnitudes $\overrightarrow{D}$ and $\overrightarrow{H}$ which are related to the electric and magnetic fields, $\overrightarrow{E}$ and $\overrightarrow{B}$ by the so-called constitutive equations that describe the behavior of the medium.

Three of Maxwell’s equations (1.1a), (1.1c), (1.1d), or their alternative integral formulations (1.2a), (1.2c), (1.2d), are normally known by the names of the scientists who derived them. For its similarity with (1.1a), equation (1.1b) is usually termed the Gauss’ law for magnetic fields, for which the integral formulation is given by (1.2b). These four equations as a whole are associated with the name of Maxwell because he was responsible for completing them, adding to Ampère’s original equation, $\bigtriangledown&space;\times&space;\overrightarrow{H}=\overrightarrow{J}$ , the displacement current density term or, in short, the displacement current, $\frac{\partial&space;\overrightarrow{D}}{\partial&space;t}$ , as an additional vector source for the field $\overrightarrow{H}$. This term has the same dimensions as the free- current density but its nature is different because no free charge movement is involved. Its inclusion in Maxwell’s equations is fundamental to predict the existence of electromagnetic waves which can propagate through empty space at the constant velocity of light $c$. The concept of displacement current is also fundamental to deduce from (1.1d) the principle of charge conservation by means of the continuity equation given by:

$\bigtriangledown&space;\cdot&space;\overrightarrow{J}=-\frac{\partial&space;\rho}{\partial&space;t}&space;\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:(1.3a)$

or, in integral form as follows:

$\oint&space;_S&space;\overrightarrow{D}\cdot&space;d\overrightarrow{s}=Q_T(t)&space;\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:&space;(1.3b)$

With these equations, Maxwell validated the concept of field previously introduced by Faraday to explain the remote interactions of charges and currents, and showed not only that the electric and magnetic fields are interrelated but also that they are in fact two aspects of a single concept, the electromagnetic field.

Maxwell’s equations (1.1) can be written without using the artificial fields $\overrightarrow{D}$ and $\overrightarrow{H}$ as follows:

$\bigtriangledown&space;\cdot&space;\overrightarrow{E}=\frac{\rho&space;}{\varepsilon_0}&space;\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:&space;(1.4a)$ $\triangledown&space;\cdot&space;\overrightarrow{B}=0&space;\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:&space;(1.4b)$
$\triangledown&space;\times&space;\overrightarrow{E}=-\frac{\partial&space;\overrightarrow{B}}{\partial&space;t}&space;\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:&space;(1.4c)$
$\triangledown&space;\times&space;\overrightarrow{B}=\mu_0\overrightarrow{J}+\mu_0\epsilon_0\frac{\partial&space;\overrightarrow{E}}{\partial&space;t}&space;\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:&space;(1.4d)$

Where:

$\epsilon_0$ = electric permittivity of free space $\left&space;(&space;8.854&space;\times&space;10^{^{-12}}&space;Fm^{-1}&space;\right&space;)$.
$\mu_0$ = magnetic permeability of free space $\left&space;(&space;4.0\pi&space;\times&space;10^{^{-7}}&space;Hm^{-1}&space;\right&space;)$.

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