Chapter 1 - The Electric Field

Module 1.1 - Definition

The electric field is actually a particular case of the more extended notion of electromagnetic field, considered only from the point of view of its electric properties.

Associated to the electric field are:

• the area in which particles experience electric forces and torques and
• the point function as its characteristic feature.

Module 1.2 - The Electric Field

The electric field in vacuum is studied at a macroscopic level by measuring each point in the field, as well as the magnitude, direction and sense of the force acting upon an electrically charged sample particle.

The test is a particle used to explore the electric field. It has to be as small as possible, so that the force acting upon it may be approximated by the force which would be exerted on it and not by the resultant of forces in the area the particle occupies. For any position in the field, its electric state must be time-invariant.

For the beginning, the study of the electric field is done in vacuum conditions.

Vacuum is a rarefaction limit state meaning a lessening in the particle substance. As a consequence, a point in the vacuum is identified by the vector radii in respect with the origin of a reference system in relative immobility to the particles in the vicinity.

In the electric field, the vacuum force \({\overline F}_{qv}\) exerted upon a particle, depends on the electric charge q and its position in the field spotted by the vectorial radius r.

By bringing two electrically charged particles with different, but positive charges in the same point P(r) in the electric field, we find that the module ratio equals the charge ratio q ₁ and q₂.

By bringing the same particle with charge q in two different points P ₁(r₁) and P₂(r₂) we find that the module ratio equals the ratio of two functions E _v(r₁) and E_v(r₂).

In SI, E_v is marked [V/m] and is the electric field vector in which a force of 1N is exerted upon the particle of charge 1 C.

Since in each point the vector \({\overline E}_{v}\) is uniquely determined, the field lines do not cross.

Denoting by ds the length of the field line oriented in its direction, the differential equation of the field line is the following:

Chapter 2 - The Electric Charge

Module 2.1 - Definition

If in a time-varying physical system the electric charges q1, q2, q3... satisfy the condition that at each moment their sum is nil,

then they make up a complete system of electric charges. If their sum is non-nil they make up an incomplete system.

Module 2.2 - Charge Distribution

We define:

• Volume density \[\rho_v=\lim_{\triangle_v\rightarrow0}\frac{\triangle q}{\triangle V}=\frac{dq}{dV'}\]
• Surface density \[\rho_A=\lim_{\triangle A'\rightarrow0}\frac{\triangle q}{\triangle A'}=\frac{dq}{dA'}\]
• Linear density

Chapter 3 – Coulomb's Law

Module 3.1 - Definition

Charles Coulomb (1736–1806) measured the magnitudes of the electric forces between charged objects using the torsion balance, which he invented (see the video below).

In the image above, given the q1 and q2 electric charges in two particles in vacuum situates at the distance R, forces F₁₂ and F₂₁ exerting over particles 1 and 2 have the following properties:

The resulting formula is: \[F_{12}=F_{21}=\frac1{4{\mathrm{πε}}_0}\cdot\frac{q_1q_2}{R^2}\] where \({\mathrm\varepsilon}_0\) is the vacuum permittivity [F/m].

Module 3.2 - The Coulombian Electric Field

According to the formula \({\overline F}_{qv_T}=q{\overline E}_v\), force F₁₂ equals the product of electric charge q₁ and the vector electric field in vacuum \({\overline E}_{12}\), established by charge q₂.

Thus:

A point charge q, establishes in a certain point P of distance R an electrostatic field whose field vector E_v is radial, in proportion to the charge q and inversely proportional to the square distance.

The strength of the electrostatic field is oriented outward from the particle in case of a positive charge and inward in case of a negative charge.

If charge q is distributed, the strength of the elementary electrostatic field E_v established in vacuum by the elementary charge dq is computed by:

If the charge has a volumetric distribution of density \(\rho_v(r')\), of superficial surface \(\rho_A(r')\) or linear \(\rho_1(r')\) we can consider that:

and vector \({\overline E}_{v}\) we integrate on volume V, surface S and line C.

If there are discrete charges in the field, the strength of the electrostatic field is computed by the following:

Elements of volume, surface, length are given by vector radius \(\overline r'\) and point P by radius \(\overline r\).

The distance between point P and P' is:

and

and \(\nabla=\nabla'\)

If U(R) is a scalar function and \(\overline F(R)\) a vector, then:

• gradU(R) = - grad'U(R)
• dirF(R) = - div'F(R)
• rotF(R) = - rot'F(R)

Since \(\frac{\overline R}{R^3}=-grad\frac1R\)

The defining relation of the electrostatic field strength then becomes: \[{\overline E}_v(r')=-\frac q{4{\mathrm{πε}}_0}grad\frac1R\]

Chapter 4 – The Electric Potential

Module 4.1 - Definition

Under static and stationary regimes, the electric potential does not vary in time U₁₂. If the electric potential is time-varying it is denoted with u₁₂ and is called instantaneous.

The electric potential globally characterises the electric field referring to a given curve \(\Gamma\) between two of its points.

The integration sense, i.e. the sense of the curve arc element \(\Gamma\), is called the reference sense from P₁ to P₂ if point P spotted by the vector radius r, length \(d\overline s\) is assimilated to the differential of vector radius \(d\overline r\).

In an even electric field the electric potential u12 between two points P₁ and P ₂ at distance d is not dependent on the curve \(\Gamma\) shape and has the following relation:

If points P₁ and P₂ are on the same field line and \(\alpha=0\), the electric potential is positive and \(u=E_vd>0\). What will this lead to?

If points P₁ and P₂ are on a line perpendicular on the field lines \(\alpha=\pm\mathrm\pi/2\), then the electric potential is identical nil.

If \({\overrightarrow{\mathrm E}}_\mathrm v\) has in rectangular coordinates the components E_x, E_y, E_z and ds element components dx, dy, dz, electric potential is computed by the relation below:

In a uniform field, the electromotive potential is nil for any closed curve \(\Gamma\).

Module 4.2 - Electrostatic potential; Potential difference

Let us consider a point charge q in the reference origin. The electric potential U₁₂ between points P ₁ and P₂ of curve \(\Gamma\) is computed by:

E_v(r) is the strength of the electrostatic field in a point P on the curve computed by the relation:

Replacing

function V is defined by the relation: \(V=\frac q{4\pi\varepsilon_0}\cdot\frac1r\) and U₁₂=V₁-V₂ where \(V_1=\frac q{4\pi\varepsilon_0}\cdot\frac1{r_1}\); \(V_2=\frac q{4\pi\varepsilon_0}\cdot\frac1{r_2}\)

If point P₂ tends to infinite, the electric potential \(U_{1\infty_{r_2\rightarrow0}}=\lim_{}U_{12}=V_1=\frac q{4\pi\varepsilon_0r_1}\)

The line integral of the electrostatic field on a curve of a certain shape, between point P of distance r from the charge and the point in the infinite is called electrostatic potential V.

On considering P₁ a current point P and P₂ a reference point P₀, potential P is computed by the following relation:

The electric potential U₁₂ equal to the difference between potentials V₁ and V ₂ is called potential difference.

The electrostatic potential V(r) is a continuous point function except for singularities. The choice of reference point P₀ whose reference potential V₀ enters the potential relation V _P is random, provided that the integral of the electric field between points P₀ and P₁ does not have infinite values. In this case, the point P₀ is not specified and V₀ is an additive constant.

The electrostatic potential V is defined according to the definition of the potential energy of a material point:

• Vector E_v is the Coulombian force \({\overline F}_{qv}\) exerted over the point charge unit
• The potential difference is the work L₁₂ from outside needed to move the test particle with unit electric charge from point P ₁ to P₂.

Module 4.3 – Potential Gradient

If we differentiate the relation \(V=V_0-\int_0^P{\overline E}_v\cdot d\overline s\) we get \(dV=gradV\cdot d\overline s\)

By comparing the two relations, we get: \(E_v=-gradV=-\Delta V\)

It results that the strength of the electric field Ev is the gradient with the changed sign of potential V.

Module 4.4 - The Electrostatic Potential of Distributed Charges

According to the superposition principle, the electrostatic potential V established in a certain point, in the vacuum, of n charges q_k is equal to the sum of the electrostatic potentials V _k that each charge would have established in that point.

For a distributed charge, \(dV=\frac{dq}{4\pi\varepsilon_0}\cdot\frac1r\).

Module 4.5 - Equipotential Surface

Since the electrostatic potential is a scalar potential function, we can draw surfaces whose points have the same potential, called equipotential surfaces.

V(r) = V(x,y,z) = const.

Since the strength of the electric field is the gradient of opposed sign of the potential, the field lines of the electric field are perpendicular on equipotential surfaces.

Chapter 5 – The Electric Potential

Module 5.1 – Definition of the Electric Field Flux

Module 5.2 – Gauss’s Law

We consider an outward facing surface \({\overrightarrow S}_\Gamma\) on \(\Gamma\), which has a coverage direction.

The electric inductivity in vacuum \({\overrightarrow D}_v\), in a point P on surface \({\overrightarrow S}_\Gamma\) is computed by the following relation:

And for the electric flux \(\Psi_{S_\Gamma}\) we get:

magnitude \(\int_{S_\Gamma}\frac{dA\;\cos\alpha}{R^2}=\Omega_\Gamma\) is the solid angle under which we may see the curve in the point where the charge is located.

If there are n charges in the field, according to the superposition principle, the electric flux \(\Psi_{S_\Gamma}\) has the following expression:

For closed surfaces, if we consider the charge enclosed, the solid angle in a point inside the closed surface being equal to \(4\mathrm\pi\), we get \(\Psi_\Sigma=q\)

This result may be explained as follows:

• the cone with the vertex in which is the charge whose generators are tangent to the surface \(\Sigma\), determine a curve \(\Gamma\) which separates two open surfaces \(S\Gamma\) and \(S'\Gamma\) \((S\Gamma\cup S'\Gamma=\Sigma)\).

Considering at random a coverage direction for curve \(\Gamma\), the normal versor of one of the open surfaces is identical to the versor of surface \(\Sigma\) and the versor of the other has an opposed direction.

Therefore \(d\Omega_{S_\Gamma}=-d\Omega'_{S'_\Gamma}\) and \(\Omega_\Sigma=\Omega_{S_\Gamma}+\Omega'_{S'_\Gamma}=0\).

Chapter 6 – Application

The field and the electric potential in case of a plane S evenly charged with surface density \(\rho\) of electric charges.

For symmetrical reasons, \({\overrightarrow E}_v\) is \(\perp\) on the plane.

Thus, the flux on the lateral surface is "0".

The flux is reduced to its value by surfaces S and S' of equal areas, A.

From GAUSS's Law: