Electric Field

The term "field" used in physics means the space in which some quantity is defined for each point of this space. If this is a vector quantity, we have a vector field. If the quantity is a scalar, we have a scalar field.
If we measure the temperature of water at different points of a swimming pool, we get the temperature field for this pool. This will be a scalar field as temperature is a scalar quantity, it does not have direction only magnitude. If, in turn, we measure the weight of a given object at different points above the Earth's surface, we will get the gravitational field, which is a vector field as the weight of an object is a vector quantity, it has magnitude as well as direction.
In the chapter concerning electricity we will be defining the electric field. Such a field exists in the vicinity of any electric charge or charges. If we consider electric charges which do not move, we can call it more precisely – an electrostatic field.
The electric field E is a vector field and its value at any point is defined as the ratio of the force F acting on a positive test point charge q₀, to the magnitude of this charge
E = F / q₀     (1) letters in bold, according to the convention used in scientific literature, represent a vector quantity. We use this "bold" notation alternately with "arrows" notation which use a small arrow above the symbol representing the vector quantity.
The test charge q₀ should be very small so we can assume that it does not alter the electric field we are measuring. By "point charge" we understand a charge located on an object so small, that it can be treated as a mathematical point.
The electric field can be visualized by drawing lines representing the direction of this field. They are called electric field lines.

They are constructed in such a way that:
a) the direction of the straight field line gives the direction of the electric field E,
b) if the field line is curved, the tangent to it at any point gives the direction of E at this point,
c) the number of lines passing trough a unit area perpendicular to these lines represent the magnitude of the electric field.

On the figure below there are examples of electric field lines for two cases of charge distribution.

Figure.1 Electric field lines for two cases of charge distribution. Part A represents electric field lines arising from a positive point charge, part B field lines between to linearly distributed positive and negative charges.

If you want to examine electric field lines for different configurations of charges creating it, see for example the following pictures on this page -   Electric field lines
You can imagine electric field lines as lines drawn by a small positive charge inserted into this field and allowed to move under influence of the force acting on it. From Eq. 1 it follows that the force F exerted on an electric charge Q in a field E is

F = Q E     (2) One of the main tasks in teaching electrostatics is to calculate the electric field produced by different distributions of charges. The general recipe for such calculation is:
a) insert a small positive test charge q0 into a given point of the electric field,
b) calculate the magnitude of an electrostatic force exerted on this charge,
c) calculate the magnitude of electric field from Eq.1
d) assign the direction to this field as the direction of the force acting on a test charge

The simplest case is an electric field at point P at distance r from a point charge Q - see Fig.2. We put a small positive charge q₀ at distance r from this charge Q and calculate the electrostatic force exerted on it. Fig._2

Fig.2 Drawing for electric field calculation at distance r from a charge Q.
The electrostatic force acting on charge q₀ is given by Coulomb's law F = Q q₀ / (4π ε₀ r²)      (3) Notice, that we omit the signs of charges and we do not use vector notation as at this point of the electric field calculation we calculate only the magnitude of the electrostatic force between charges Q and q₀. This equation is true for vacuum as we know from the Coulomb law Chapter of this tutorial.
On the basis of Eq.1 we find the magnitude of the electric field as E = F/q₀ = Q / (4π ε₀ r²)     (4) The direction of the field produced by negative charge –Q is from point P to the charge Q along the line r, as this is the direction of the electrostatic force. The calculation of the electric field for different distributions of electric charges will be given as a problem to this paragraph.

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