Electrostatic fields

A static field
We call a field static field when it does not change with time explicitly. Electric fields due to static or stationary charge distributions are electrostatic fields. For example, if we have a charge held fixed somewhere in the space, it creates an electrostatic field. And when it is moving, it creates an electrodynamic field (the charge distribution changes with time explicitly). But when I have a static current of charges in a wire, it creates an electrostatic field (and later it will be seen that moving charges create magnetic field too) because although the charges are moving, the charge distribution in the space is not changing with time as we have a fixed current.

The idea of static field must not be confused as it is the same always. Sometimes in electrostatics we encounter problems when the charge distribution do change but we wish to study the final theme when everything is static (as example, when we place a small charge near a conducting surface, it creates induced charges on the conducting surface and thus changes the charge distribution; but in study of electrostatics we are concerned about what the ultimate charge distribution or field will be there and not about what was happening when the charge distribution was changing).

Electric Field
The electric force acting on a charged object can be interpreted as being produced by some property of the space where that object is located. That property of space that leads to forces over the electrostatic charges is called Electric field denoted as E

In mathematical terms, the electric field can be defined precisely in the following way: at any point in space the electric field is a vector $$\vec{E}$$ obtained by dividing the force $$\vec{F}$$ acting on a point charge $$q_0$$ placed in that point, over the value of the charge:

$$\vec{E} = \frac{\vec{F}}{q_0}$$

With
 * $$\vec{E}$$ Vector Electric field
 * $$F$$ Electric force
 * $$q_0$$ Point charge

Electric Potential
The source of the electric field can be either other electrostatic charges, or a varying magnetic field. If the field is produced by static electric charge distribution, it is called electrostatic field. In that case, it can be shown that is is a conservative field; namely, the path integral of the field between two points is independent of the path. As a consequence, it is possible to define a potential function: the potential in any point of space $$(x,y,z)$$ is equal to the path integral of the electrostatic field from an arbitrary reference point $$P_0$$ until the point $$(x,y,z)$$.

$$V(x,y,z) = -\int_{P_0}^{(x,y,z)} \vec{E}\cdot d\vec{l}$$

Thus, the electrostatic potential (also known as voltage) is a scalar field and it can be proved that the gradient of that scalar field is equal to minus the electrostatic field:

$$\vec{E} = -\vec{\nabla} V$$