The Coulomb Law and superposition principle can lead to divergence theorem which is valid for bilateral, axial and spherical charged objects. Let’s divide the surface Σ, by the parts dS. Module of dS is equal to the square of dS (Figure below), and is directed along the normal n.

 Electric field flow through the surface Σ
Electric field flow through the surface Σ

Elementary flow dФ of the electric field E through the piece of surface dS is a  dΦE=EndS.

En is a E projection on the normal n to the surface element dS. To simplify this case, the electric field flow is a scalar multiplication of the vectors E and dS(E¯,dS¯)=E*dS*cosα=En*dS

So the electric field E that flows through the surfaceΣ:    ΦE=ΣEndS

Electric field flow has the following properties:

  • Electric field flow through the surface is proportional to the number of electric field lines through this surface. As flow is a vector term and vector direction has to be taken into account.
  • Superposition principle is valid for electric field flow, and states that if surface Σ is the amount of N charges q1 then total electric field flow ΦE is an algebraic sum of electric field flow ΦEi, created by every single dot charge of the surface ΦE=ΣEn(r¯)dS,  En(r¯)=Ni=1 Eni(r¯) (Figure below).
ΦE=Σ [Ni=1 Eni(r¯) dS=Ni=1 [Σ Eni(r¯) dS]=Ni=1 ΦEi
 Electric field from every charge in a considered field of a surface Σ
Electric field from every charge in a considered field of a surface Σ

Based on the conclusions above, Gauss theorem can be introduced. Gauss theorem states the following: electric field flow ΦE in a vacuum through the closed surface ∑ is proportional to the total charge of this surface:

ΦE=1ε0Ni=1 qi, orΣ EndS=1ε0Ni=1 qi

Let’s consider several cases of charged surfaces.

  1. Spherical surface surrounds point charge. (Figure below). Electric field structure for this case is radial, and it is reverse proportional to the square of distance from point charge to some point in space. In this case, for every small piece of spherical surface vector E coincides to the dS  Then we can switch from the sign of sum Σ to the integral.
    ΦE=Σ EndS=Σ E(r) dS=Σ (14πε0qr2)dS=14πε0qr24πr2=qε0
Point charge surrounded by spherical surface
Point charge surrounded by spherical surface

2. Let’s consider a single charge shifted from the centre of the sphere (Figure below). In this case angles between vectors E and dS  for every point on the sphere are different, and electric field has different values. Electric field flow through the sphere is equal to the quantity of electric field lines, crossing its surface. This means that the result should be the same as for case 1. We can also use integration to find the electric field flow through the surface. According to the Gauss theorem

ΦEn=EndS=(E¯,dS¯)=E*dS*cosα.   EndS=14πε0qr2dS*cosα . Here dS*cosαr2 is a measure of solid angle for dΩ. So ΦE=Σ q4πε0dΩ=qε0.

3. System of N point charges q1...qi is placed inside and outside of the closed random form surface. Every single charge inside the surface creates electric field flow  ΦEi=1ε0qi.

Every single charge outside the surface does not create an electric field. So for N charges inside the surface  ΦE=Ni=1 ΦEi=Ni=1 1ε0qi.

In the case of extended one-, two- or three-dimensional charge, integration must be used in Gauss theorem instead of sum sign. So curved, surface or volume integral will be used:

Σ EndS=1ε0L λdl;   Σ EndS=1ε0Σ σdS;   Σ EndS=1ε0Ω ρdV

where  λ, σ, ρ –are linear, surface and volume charge density; L, Σ, Ω – space areas have distributed charge.

The most comfortable way to use Gauss theorem is when the system is characterised with flat, axis or spherical symmetry – where you can switch from the surface integral to the multiplication of scalar quantities E and dS.

Task 3: Prove that the electric field outside the uniformly charged sphere (dielectric , ε) on the distance r bigger than R, sphere radius, is the same as the electric field of a single point charge q.

Task 4: Determine electric field E(r) of the uniformly charged infinite cylinder with permittivity ε, and radius R :  a. inside the cylinder; b. outside the cylinder. ρ is a charge volume density, r is a distance from the cylinder axis.

There is a potential difference in electric field id scalar value. It is also an electric field characteristic, and it is very useful, as it is easier to use scalar quantities. We consider all the charges in an electrostatic field, when electric forces in electrostatic fields are conservative. Forces are considered to be conservative if their work does not depend on the object of the force movement trajectory. This work is determined only by initial and final coordinates of a moving object.

Stationary point electric charge is the source of central field force. According to the superposition principle, work wasted on the movement of a single charge in the system of stationary charges is equal to the algebraic sum of works from all the separate charges. This field is also a field of conservative forces. So work of electrostatic forces characterises the electric field, and depends on the considered charge.

The potential difference between two points in electric field 1 and 2 is the ration of work (of electric field) needed to move considered charge q from point 1 in point 2, to the value of considered charge φ1φ2=A12fieldq.

The measurement unit is Volts (V). So work needs to be done to move charge q from point 1 to point 2:  A12field=q(φ1φ2).

A potential difference is the energy characteristics of an electric field.

Let’s consider the point in space x0 where the electric field is absent and  φ(x0)=0.

So, the electric field energy needed to move the charge from point x0 to a random point of the electric field is φ01=A01q.

Let’s express the field work  A01  as characteristics of the field EA01=01Fdl=q01Edl  and  φ0φ1=01Edl.

Let’s consider a potential of point charge. First, assuming  φ()=0.  The point in space we are calculating the electric field potential is distanced from the charge to the distance r. To simplify the task, let’s assume the electric field lines are in a radial straight line from the charge. Then φ(x)=xx0 Erdr.

In this case φ(r)=rq4πε01r2dr=q4πε01r.  Potential energy of a point charge in the electric field is U(x)=q*φ(x).

For two point charges q1 and q2 interaction energy is U(r)=14πε0q1*q2εr.

Equipotential surfaces are energy characteristics of an electric field that characterises the moving charge in an electric field. Work along these surfaces is 0. It’s at its maximum along the directions, where there is maximum density of equipotential surfaces. Electric fields here is also maximum. Equipotential surfaces and electric field lines are perpendicular. (Figure below). Work of any charge movement along the equipotential surface is 0, so it’s possible only if the electric field is directed normally to the surfaces. There is a very easy mathematical explanation

0dφ=0,  dA=q*El*dl=0,  El=0 then  Eldl.  Here are equipotential surfaces for cases we discussed above. (Figure below).

 Equipotential surfaces and electric field lines
Equipotential surfaces and electric field lines
Equipotential surfaces for different charge systems
Equipotential surfaces for different charge systems

Potential of system point charges

Field potential of a system of point charges is characterised by the superposition principle φ=Ni=1 φi.

The proof of this statement is based on the facts that the potential of a field is work that needs to move a charge to the point of space where the field is equal to 0. And in the fact that work is an additive term. For system of charges in the field φ=14πε0Ni=1 qiri (Figure below).

System of point charges
System of point charges

Let’s use the potential superposition principle for calculating field potential on the axis of a charged circle, with radius R and charge q, where x is the distance from the circle’s centre. (Figure below).

The circle can be divided to the point charges q1. And according to the superposition principle φx=i 14πε0qir=14πε01R2+x2i qi=14πε0 qR2+x2.

Electrostatic field and potential difference


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