The main task from the section of electrostatics is formulated as follows: 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 to this problem is possible on the basis of such a concept as the principle of superposition of electric fields (the principle of independence of the action of electric fields): the intensity of an electric field of a system of charges will be equal to the geometric sum of the field strengths that are created by each of the charges.
The charges that create the electrostatic field can be distributed in space either discretely or continuously. In the first case , the field strength :
n
E = Σ Ei₃
i = t
where Ei is the tension at a certain point in the field space created by one i-th charge of the system, and n is the total number of discount charges that are part of the system.
An example of solving a problem, which is based on the principle of superposition of electric fields. So, to determine the strength of the electrostatic field, which is created in a vacuum by fixed point charges q₁, q₂, ..., qn, we use the formula:
n
E = (1 / 4πε₀) Σ (qi / r³i) ri
i = t
where ri is the radius vector drawn from the point charge qi to the considered point of the field.
We 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 identical in absolute value and, at the same time, of opposite charges in sign q> 0 and –q, the distance I between which is relatively small in comparison with the distance of the points under consideration. Shoulder of a dipole will be called the vector l, which is directed along the axis of the dipole to a positive charge from a negative one 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 intensity E of the dipole field at any point:
E = E₊ + E₋,
where ₊ and ₋ are the electric field strengths 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 a perpendicular restored to the axis of the dipole from its middle:
E = (1 / 4πε₀) (pₑ / r³)
At an arbitrary point M, sufficiently remote from the dipole (r≥l), the modulus of its field strength is
E = (1 / 4πε₀) (pₑ / r³) √3cosϑ + 1
In addition, the principle of superposition of electric fields consists of two statements:
- The Coulomb force of interaction of two charges does not depend on the presence of other charged bodies.
- Suppose that the charge q interacts with the 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, then the resulting force F applied to the charge q from the side of this 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 come to one important statement.
As you know, the law of gravity is valid not only for point masses, but also for balls with a spherically-symmetric distribution of mass (in particular, for a ball and point mass); then r is the distance between the centers of the balls (from the point mass to the center of the ball). 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, we can draw a similar conclusion: according to Coulomb's law, two charged balls (point charge with a ball) 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 ball).
That is why the field strength of a charged ball will turn out to be the same outside the ball as that of a point charge.
But in electrostatics, unlike gravity, with such a concept as superposition of fields, one must be careful. For example, when positively charged metal balls approach each other, spherical symmetry will be broken: positive charges, mutually repelling, will tend to the most distant parts of the balls (the centers of positive charges will be farther from each other 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 substituting the distance between the centers instead of r.