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Movement of charged particles in electric fields
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When particles have an electrical charge, in addition to thermal agitation they can experience forces produced by electric fields. These forces generate a directed displacement known as electrical drift, in which the particles acquire an average speed proportional to the intensity of the applied field. The resulting movement depends on both the electrical force that drives the particles and the viscous resistance of the medium that opposes their movement. This mechanism constitutes the physical basis of ionic transport, electrical conduction and numerous electrochemical and biological processes.
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Electric Field Strength
Description
Conceptually, the electric field can be understood as a way of storing energy and direction in space. The electric charge senses that field and responds by moving under the action of the resulting force.
The electric field strength corresponds to the force that a charged particle experiences when it is within an electric field. This force can be expressed as:
 $F_x = q \cdot E_x$
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where Force ($F_x$), Electric charge of the particle ($q$) and Electric field ($E_x$).
The equation shows that the greater the electric field, the greater the force exerted on the particle. Likewise, particles with larger charges experience stronger forces. The sign of the charge also determines the direction of movement: positive charges are accelerated in the direction of the electric field, while negative charges are propelled in the opposite direction.
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Distance between two Points
Description
Two points and a volume associated with each of them are defined, within which the concentration of particles is determined.
Distance between two points.
Subsequently, Distancia de Posiciones ($\Delta x$) is calculated as the difference between Position 2 ($x_2$) and Position 1 ($x_1$):
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$\Delta x = x_2 - x_1$
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Electric Potential Difference
Description
The electric potential is measured at each point and the difference between both potentials is subsequently determined:
Electric potential difference between two points.
Electric Potential Difference ($\Delta V$) is obtained by subtracting Electric potential in 1 ($V_1$) from Electric potential in 2 ($V_2$):
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$\Delta V = V_2 - V_1$
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Electric Field
Description
Electric potential, on the other hand, represents electrical potential energy per unit charge. While the electric field describes local forces, the potential describes how electrical energy is distributed in space.
The electric field appears precisely when the potential changes from one point to another. If there is a spatial potential difference, charges will tend to move toward regions of lower potential energy. Mathematically, the field corresponds to the speed with which the potential changes in space, that is, to its spatial derivative:
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$E_x = - \displaystyle\frac{ dV }{ dx }$
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The negative sign indicates that the field points toward where the potential decreases most rapidly. In other words, positive charges are spontaneously pushed downhill in the electric potential landscape.
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Electric mobility
Description
Electrical mobility describes how easily a charged particle can move through a medium when an electric field acts. It represents the ability of the ion or particle to respond to electrical thrust while simultaneously considering the viscous resistance of the environment.
Conceptually, a charge immersed in a fluid experiences two opposite effects. The electric field tries to accelerate it, while the viscosity of the medium generates a friction force that opposes the movement. After a short interval, both forces balance and the particle reaches a constant average velocity called the drift velocity.
Mobility indicates precisely how much speed the particle acquires for each unit of applied electric field.
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$v_x = \mu_q \cdot E_x$
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where Relative velocity between the particle and the medium ($v_x$), Electric mobility ($\mu_q$) and Electric field ($E_x$). If the field increases, the force on the charge also increases and the particle moves faster. Likewise, particles with greater mobility respond more efficiently to the field and reach higher speeds.
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Particle Flow
Description
The particle flux corresponds to the number of particles that cross a surface perpendicular to the direction of movement per unit of time and per unit of area. To calculate it, it is considered that Concentration ($C$) represents the number of particles per unit of volume. If the particles move with a Relative velocity between the particle and the medium ($v_x$), then during a time interval $\Delta t$ they will travel a distance $v \Delta t$.
If a surface $S$ is considered perpendicular to the movement, the swept volume during that interval will be $S \cdot v \cdot \Delta t$. Since each unit volume contains $C$ particles, the total number of particles that will cross the surface will be:
$\Delta N = C \cdot S \cdot v \cdot \Delta t$
Therefore, the particle flux can be expressed as:
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$J_x = C \cdot v_x$
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with Particle Flow by Diffusion ($J_x$), Concentration ($C$) and Relative velocity between the particle and the medium ($v_x$).
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Particle Flow by Electric Field
Description
If the particles have a Electric charge of the particle ($q$) and are in the presence of a Electric field ($E_x$) they experience a Force ($F_x$).
This force accelerates the particles in the direction of the field if the charge is positive, or in the opposite direction if it is negative. However, due to collisions and the effective viscosity of the medium, the particles do not continue to accelerate indefinitely, but instead reach Relative velocity between the particle and the medium ($v_x$).
If Concentration ($C$), then Particle Flow by Electric Field ($\vec{J}$) corresponds to the number of particles that cross a surface perpendicular to the movement per unit of time and per unit of area. This flow can be expressed as:
The drift speed depends on how easily the particles respond to Electric field ($E_x$). This property is described by Electric mobility ($\mu_q$), defined by:
The equation shows that particles with greater mobility reach higher speeds under the same electric field.
On the other hand, the electric field can be interpreted as the spatial variation of the electric potential. In one dimension:
where the negative sign indicates that the field points towards where the potential decreases most rapidly.
Substituting these relations we finally obtain:
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Movement of charged particles in electric fields
Description
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