The gauss's law establishes that the electric field flux, through an imaginary closed surface, is proportional to the net value of charge of the particles found inside said surface.
Denoting the electric flux through a closed surface as ΦAND and the net charge enclosed by the surface by Qenc, then the following mathematical relationship is established:
ΦAND = c ∙ Qenc
Where c is the constant of proportionality.
To understand the meaning of Gauss's law, it is necessary to explain the concepts involved in its statement: electric charge, electric field and electric field flow through a surface..
Electric charge is one of the fundamental properties of matter. A charged object can have one of two types of charge: positive or negative, although normally the objects are neutral, that is, they have the same amount of negative charge as positive..
Two charged objects of the same type repel each other even when there is no contact between them and they are in a vacuum. On the contrary, when each of the bodies have charges of a different sign, then they attract each other. This type of interaction at a distance is known as electrical interaction..
In the international system of SI units, electric charge is measured in coulombs (C). The negative elemental charge carrier is the electron with load of -1.6 x 10-19C and the positive elemental charge carrier is the proton with a charge value +1.6 x 10-19C. Typically charged bodies have between 10-9C Y 10-3C.
An electrically charged body alters the space in its surroundings, filling it with something invisible called an electric field. To know that this field is present requires a point positive test charge.
If the test charge is placed in a place where there is an electric field, a force appears on it in a certain direction, which is the same as that of the electric field. Field strength is the force on the test charge divided by the amount of charge on the test charge. Then the units of the electric field AND in the International System of units are newton Come in coulomb: [E] = N / C.
Positive point charges produce a radially outward field, while negative charges produce a radially inward directed field. Furthermore, the field produced by a point charge decays with the inverse of the square of the distance to said charge.
Michael Faraday (1791 - 1867) was the first to have a mental image of the electric field, imagining it as lines that follow the direction of the field. In the case of a positive point charge, these lines are radial starting from the center outwards. Where the lines are closer together the field is more intense and less intense where they are further apart.
Positive charges are the sources from which the electric field lines emerge, while negative charges are the sinks of the lines..
Electric field lines do not close in on themselves. In a set of charges the lines leave the positive charges and enter the positive ones, but they can also reach or come from infinity.
They also do not intersect and at each point in space the electric field vector is tangent to the field line and proportional to the density of lines there..
Electric field lines resemble the streamlines of a gently flowing river, hence the concept of electric field flow.
In a region where the electric field is uniform, the flux Φ through a flat surface is the product of the normal component of the field En to this surface, multiplied by the area TO Of the same:
Φ = En ∙ A
Component En is obtained by multiplying the magnitude of the electric field by the cosine of the angle formed between the field and the unit normal vector to the surface area TO. (see figure 4).
Gauss's law can be applied to determine the electric field produced by charge distributions with a high degree of symmetry.
A point charge produces a radial electric field that is outgoing if the charge is positive and incoming otherwise..
Choosing as the Gaussian surface an imaginary sphere of radius R and concentric to the charge Q, at all points on the surface of said sphere the electric field is of equal magnitude and its direction is always normal to the surface. So, in this case the electric field flux is the product of the magnitude of the field and the total area of the spherical surface:
Φ = E ∙ A = E ∙ 4πRtwo
On the other hand, Gauss's law states that: Φ = c ∙ Q, being the constant of proportionality c. When working in units of the international measurement system, the constant c is the inverse of the permittivity of the vacuum, and Gauss's law is formulated like this:
Φ = (1 / εor) ∙ Q
Incorporating the result obtained for the flow to Gauss's law, we have:
E ∙ 4πRtwo = (1 / εor) ∙ Q
And for the magnitude of AND result:
E = (1 / 4πεor) ∙ (Q / Rtwo)
Which fully agrees with Coulomb's law of the electric field of a point charge.
Two point charges lie within a Gaussian surface S arbitrarily. One of them is known to have a value of +3 nC (3 nano-coulomb). If the net electric field flux through the Gaussian surface is 113 (N / C) mtwo, What will be the value of the other load?
Gauss's law states that
ΦAND = (1 / εor) ∙ Qenc
Hence the net enclosed charge is:
Qenc = ΦAND ∙ εor
Substituting the data results:
Qenc = 113 (N / C) mtwo ∙ 8.85 x 10-12 (Ctwo m-two N-1) = 1 x 10-9 C = 1 nC.
But Qenc = + Q - q, where the positive charge has a known value of +3 nC, therefore the charge will necessarily be -2 nC.
In figure 2 there is an arrangement (on the left) of two positive charges, each with a value + q and another arrangement (on the right) with a charge + q and the other -q. Each arrangement is enclosed in an imaginary box with all its 10 cm edges. If | q | = 3 μC, find the net electric field flux through the box for each arrangement.
In the first arrangement the net flow is:
ΦAND = (1 / εor) ∙ (+ q + q) = 678000 (N / C) mtwo
In the arrangement on the right, the net flow through the imaginary box containing the pair of charges is zero.
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