The principle of superposition of electric fields

The main problem from the section of electrostatics is formulated in this way: from the given distribution in space and the magnitude of electric charges (field sources), determine the value of the intensity vector E at all points of the field. The solution of this problem is possible on the basis of such a concept as the principle of superposition of electric fields (the principle of the independence of the action of electric fields): the intensity of any electric field of the system of charges will be equal to the geometric sum of the fields that are created by each of the charges.

Charges that create an electrostatic field can be distributed in space either diskertno, or continuously. In the first case , the field strength :

N

E = Σ Ei₃

I = t,

Where Ei is the strength at a certain point in the field space created by one i-th charge of the system, and n is the total number of discrete charges that are part of the system.

An example of a solution to a problem based on the principle of superposition of electric fields. So to determine the intensity of the electrostatic field, which is created in a vacuum by stationary point charges q₁, q₂, ..., qn, we use the formula:

N

E = (1 / 4πε₀) Σ (qi / rφi) ri

I = t,

Where ri is the radius vector drawn from the point charge qi to the considered point of the field.

Let us give one more example. Determination of the intensity of the electrostatic field, which is created in a vacuum by an electric dipole.

An electric dipole is a system of two charges of the same magnitude and, at the same time, opposite in sign, q> 0 and -q, the distance I between which is relatively small in comparison with the distance of the points under consideration. The shoulder of the dipole will be the vector l, which is directed along the dipole axis to a positive charge from the negative and is numerically equal to the distance I between them. The vector pₑ = ql is the electric moment of the dipole (dipole electric moment).

The strength E of the dipole field at any point:

E = E₊ + E₋,

Where E₊ and E₋ are the electric charge fields q and -q.

Thus, at point A, which is located on the axis of the dipole, the dipole field strength in vacuum will be equal to

E = (1 / 4πε₀) (2pₑ / r³)

At point B, which is located on the perpendicular, restored to the axis of the dipole from its center:

E = (1 / 4πε₀) (pₑ / r³)

At an arbitrary point M, sufficiently far from the dipole (r≥l), the modulus of its field strength is equal to

E = (1 / 4πε₀) (pₑ / r³) √3cosθ + 1

In addition, the principle of superposition of electric fields consists of two statements:

1. The Coulomb force of the interaction of two charges does not depend on the presence of other charged bodies.
2. Suppose that the charge q interacts with a system of charges q1, q2,. . . , Qn. If each of the charges of the system acts on the charge q with the force F₁, F₂, ..., Fn, respectively, the resultant force F applied to the charge q from the side of the given system is equal to the vector sum of the individual forces:
   F = F₁ + F₂ + ... + Fn.

Thus, the principle of superposition of electric fields allows us to arrive at one important statement.

As is known, the law of universal gravitation is valid not only for point masses, but also for balls with a spherically symmetric mass distribution (in particular, for a sphere and point mass); Then r is the distance between the centers of the balls (from the point mass to the center of the sphere). This fact follows from the mathematical form of the law of universal gravitation and the principle of superposition.

Since the formula of the Coulomb law has the same structure as the law of universal gravitation, and the principle of superposition of fields is also satisfied for the Coulomb force, one can draw a similar conclusion: according to Coulomb's law, two charged balls (a point charge with a sphere) will interact, provided that the balls have Spherically symmetric charge distribution; The value of r in this case will be the distance between the centers of the balls (from the point charge to the sphere).

That is why the intensity of the field of a charged sphere is outside the sphere the same as for a point charge.

But in electrostatics, unlike gravity, with such a notion as superposition of fields, one must be careful. For example, when the positively charged metal spheres converge, the spherical symmetry breaks down: positive charges, mutually repulsing, will tend to the most distant parts of the balls (the centers of positive charges will be farther apart than the centers of the balls). Therefore, the repulsive force of the balls in this case will be less than the value obtained from the Coulomb law when the distance between centers is substituted for r.