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# Electric Field in Vacuum   short version

## Electromagnetic interaction

Interaction between electric charges is called electromagnetic interaction, and is one of four fundamental interactions of matter. Nevertheless, the electromagnetic interaction takes the leading role among these interactions, since overwhelming majority of phenomena surrounding us have the electromagnetic nature.

Formally the theory of electricity is similar to the theory of gravity, since the forces of interaction between the charges q1, q2 and the masses m1, m2 separated by distance, r, have similar form:

electric force

gravitational force

where ke and G are fundamental physics constants

Nevertheless, electricity and gravity differ greatly.  If we examine the ratio of electric repulsion to gravitational attraction between two electrons, we find the ratio . This is the largest number in the universe, which can be measured by modern science and so has the physical meaning. It is interesting to note that the same value has the ratio of the largest distance (the radius of the universe) to the smallest distance (the radius of the nucleus) that can be measured. Also, the same value has the ratio of the largest time interval (the age of the universe) to the smallest time interval (the time light takes to cross the nucleus). This shows that electricity and gravity principally differ one from another, and represent two diametrically opposite sides of matter at the present level of our knowledge.

## Electric Charges

The atom consists of elementary particles: protons, neutrons and electrons. In addition to these particles there are many other elementary particles in the universe. These particles constitute matter in the universe, and arrive on Earth in the form of cosmic rays. Elementary particles are also produced on Earth inside physics laboratories by using powerful accelerators. There are over 400 elementary particles known at present, and this number increases every year as the power of the accelerators increases. The investigation of elementary particles is on the foremost edge of the modern physics.

Any elementary particle has a set of strictly defined properties, which are the same for the given type of particles and cannot be changed without destroying the particle. In electricity the most important property of the particle, is its electric charge. The electric charge is just the number assigned to the particle, which is called charge number, or simply, the charge. The charges of all famous elementary particles have one of five possible values: , , , where  is a fundamental physics constant, called elementary charge, measured in coulombs. The charge of the proton is , the charge of the neutron is , and the charge of the electron is

The charge is a scalar value, so the total electric charge of a object consisting of Np protons and Ne electrons is given by the simple formula

When the number of protons exceeds the number of electrons the object is positively charged, and on the contrary, when the number of electrons exceeds the number of protons the object is negatively charged. The neutrons have zero charge and so cannot change the charge of a substance.

The proton is heavy particle. The mass of proton is nearly 1800 times the mass of electron, so the number of protons in the substance usually remains constant, and its charge depends upon the total number of its electrons.

## Charge Conservation Law

One of the main laws on nature is the charge conservation law, which states that the net electric charge in any isolated system remains constant

This law, along with energy conservation law, rules over conversions in matter. All nuclear reactions obey this law. For example, a neutron can decay into proton and electron, so that the net charge remains zero.

## Electric force

The fundamental characteristic of an electric charge is its ability to exert a force on another charge. Unlike charges are attracted to each other and like charges are repelled from each other.

Coulomb’s law states that the magnitude of an electric force between charges Q and q, separated by distance r, is given by

where is a fundamental physics constant called permittivity of vacuum.

Coulomb's law defines the unit of charge. According to this law two charges of 1 coulomb each, separated by a distance of 1 meter, experience a force given by

This is a very large value, so the charge 1 coulomb is extremely large. The charges of usual objects, as a rule, are measured in nanocoulombs (), or microcoulombs ().

Electric f orce is vector defined by its magnitude and direction. Let us direct the distance vector, , from Q to q.

Now the electric force can be expressed in vector form, and the Coulomb's law in vector form becomes

where the force is produced on q by Q

Here the positive sign of corresponds to the repulsive force directed as shown in the above diagram. The negative sign of corresponds to attraction when the direction of force is opposite to that shown in the diagram.

It is important to note that the above Coulomb's law is valid for point charges, when the dimensions of charges are much less than the distance between them. We will show below that the Coulomb's law is also valid for uniformly charged spheres.

## The electric field

In accordance with Coulomb's law, any charge Q produces a force field around itself, which is called the electric field. If this charge is immovable, the electric field is called electrostatic field. This field can be measured by a small test charge q fixed at any point at distance  from the charge Q. According to Coulomb's law the force on the test charge is directly proportional to its charge, so the ratio of this force to the value of the test charge does not depend upon the test charge q and is the unique characteristics of charge Q. This ratio is called the electric field intensity, , or just electric field, defined as the following vector

Thus the electric field is equal to the electric force per unit charge placed in this field. The unit of the electric field is newton per coulomb

The other unit of the electric field, frequently used, is volt per meter. We will show further that these units are the same.

Using Coulomb's law we get the vector of the electric field produced by a point charge Q

with magnitude

Now we can see that this field does not depend upon the test charge q and depends only on the charge producing this field and the distance where it is measured.

The vector of this electric field is directed from the charge Q for positive charge and toward the charge for negative charge. This is shown in the diagram below at an arbitrary point P

Any electric field can be defined graphically by means of the electric field lines, as shown below

The electric field lines are drawn as curves so that the tangent line to the curve at arbitrary point P is directed along the vector of the electric field at this point, and the density of lines is directly proportional to the magnitude of the electric field

where N is the number of lines crossing a small area A oriented normally to the electric field with the center at the point P, and s is an insignificant arbitrary scale parameter the same for all points.

Taking s = 1 we can rewrite the above formula in form

where the sign "" means numerical equality without taking units into account

The electric field with  constant everywhere in both the magnitude and the direction is called a uniform electric field. The electric field lines of uniform field are shown below

According to above formula the uniform electric field has a constant density of the electric field lines.

The electric field from a point charge is not uniform. Here the electric field lines are directed radially as shown below for positive (Q>0) and negative (Q<0) charges respectively

Applying formulas for magnitude of electric field and lines density , we get the density of field lines

Thus the electric field of a point charge has radial symmetry. Using , we get the total number of electric field lines for the electric field of a point charge

We got very important result for the point charge, that the total number of electric field lines is defined only by the value of the charge producing this electric field.

## Superposition Principle

Now consider the more complex case, when the electric field is produced by two point charges Q1 and Q2, located at points defined by vectors  and   respectively, where we want to find the net electric field . We can readily get the electric fields  and  produced by charges Q1 and Q2 taken separately by using Coulomb's law:

The experiment shows that the net electric field is equal to the vector sum of individual fields

Putting the above formulas gives

The vector  can be readily determined graphically by parallelogram rule, which states that the vector  is defined by the diagonal of the parallelogram with sides  and . This is shown in the diagram below

The above equation is a mathematical notation of Superposition Principle for two charges. In general the Superposition Principle states that the net electric field produced at any point by a system on n charges  is equal to the vector sum of all individual fields produced by each charge at this point

or

where  is position vector of point P where the electric field is defined with respect to charge

The Superposition Principle seems to be evident. Nevertheless it cannot be derived from any fundamentals of Physics. Like the Coulomb's law, it is an experimental fact.

As an example, find the electric field, produced by a ring of radius R uniformly charged by charge Q, on the axis of the ring at a distance  from its center

Subdivide the ring into n pairs of diametrically opposite small portions each of charge , so that these portions can be considered as point charges. Then the electric fields produced by the two different portions of the pair at a point P are given respectively by:

From Superposition Principle we find that the resultant field produced by one portion is given by

Thus  is directed along the axis of the ring.

Any other pair of opposite portions produces an electric field equal in magnitude and direction to . So, according to the Superposition Principle, the net electric field is given by

The above example gives a powerful algorithm for the calculation of an electric field of any charged object with arbitrary form and charge distribution. All we should do for this purpose is subdivide the object into n small charged portions and apply Superposition Principle using numerical integration over the volume of the object by a computer. The higher the number n the more accurate is the value of the electric field.