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4.1 The Electric Flux due to an Electric Field
We have already shown how electric field can be described by lines of force. A line of force is an imaginary line drawn in such a way that its direction at any point is the same as the direction of the field at that point. Field lines never intersect, since only one line can pass through a single point.
The Electric flux (F) is a measure of the number of electric field lines penetrating some surface of area A.

Case one:
The electric flux for a plan surface perpendicular to a uniform electric field
To calculate the electric flux we recall that the number of lines per unit area is proportional to the magnitude of the electric field. Therefore, the number of lines penetrating the surface of area A is proportional to the product EA. The product of the electric filed E and the surface area A perpendicular to the field is called the electric flux F.

The electric flux F has a unit of N.m2/C.

Case Two

Where q is the angle between the electric field E and the normal to the surface

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إذاً يكون الفيض ذا قيمة عظمى عندما يكون السطح عمودياً على المجال أي q = 0 ويكون ذا قيمة صغرى عندما يكون السطح موازياً للمجال أي عندما q = 90. لاحظ هنا أن المتجه هو متجه المساحة وهو عمودي دائما على المساحة وطوله يعبر عن مقدار المساحة.


Case Three

The net flux through the surface is proportional to the net number of lines penetrating the surface

والمقصود بـ net number of linesأي عدد الخطوط الخارجة من السطح (إذا كانت الشحنة موجبة) - عدد الخطوط الداخلة إلى السطح (إذا كانت الشحنة سالبة). Example 4.1
What is electric flux F for closed cylinder of radius R immersed in a uniform electric field as shown in figure?


Solution





</SPAN>Example 4.2
In figure 4.8 two equal and opposite charges of 2Q and -2Q what is the flux F for the surfaces S1, S2, S3 and S4.
Solution
For S1 the flux F = zero
For S2 the flux F = zero
For S3 the flux F = +2Q/ eo
For S4 the flux F = -2Q/ eo


4.4 Gauss’s Law
Gauss law is a very powerful theorem, which relates any charge distribution to the resulting electric field at any point in the vicinity of the charge. As we saw the electric field lines means that each charge q must have q/eo flux lines coming from it. This is the basis for an important equation referred to as Gauss’s law. Note the following facts:
1.If there are charges q1, q2, q3, ......qn inside a closed (gaussian) surface, the total number of flux lines coming from these charges will be

(q1 + q2 + q3 + ....... +qn)/eo (4.4)
2.The number of flux lines coming out of a closed surface is the integral of over the surface,
We can equate both equations to get Gauss law which state that the net electric flux through a closed gaussian surface is equal to the net charge inside the surface divided by eo
Gauss’s law (4.5)
where qin is the total charge inside the gaussian surface.

Gauss’s law states that the net electric flux through any closed gaussian surface is equal to the net electric charge inside the surface divided by the permittivity.

4.5 Gauss’s law and Coulomb’s law

We can deduce Coulomb’s law from Gauss’s law by assuming a point charge q, to find the electric field at point or points a distance r from the charge we imagine a spherical gaussian surface of radius r and the charge q at its center as shown in figure 4.10.

Because E is constant for all points on the sphere, it can be factored from the inside of the integral sign, then

&THORN;&THORN;
(4.6)

Now put a second point charge qo at the point, which E is calculated. The magnitude of the electric force that acts on it F = Eqo

Electric Flux التدفق الكهربي

4.6 Conductors in electrostatic equilibrium
A good electrical conductor, such as copper, contains charges (electrons) that are free to move within the material. When there is no net motion of charges within the conductor, the conductor is in electrostatic equilibrium.
Conductor in electrostatic equilibrium has the following properties:
1.Any excess charge on an isolated conductor must reside entirely on its surface. (Explain why?) The answer is when an excess charge is placed on a conductor, it will set-up electric field inside the conductor. These fields act on the charge carriers of the conductor (electrons) and cause them to move i.e. current flow inside the conductor. These currents redistribute the excess charge on the surface in such away that the internal electric fields reduced to become zero and the currents stop, and the electrostatic conditions restore.

2.The electric field is zero everywhere inside the conductor. (Explain why?) Same reason as above
In figure 4.11 it shows a conducting slab in an external electric field E. The charges induced on the surface of the slab produce an electric field, which opposes the external field, giving a resultant field of zero in the conductor.





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Figure 4.14

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