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The mathematical definition of the electric field is developed as follows. Coulomb's law gives the force between two point charges (infinitesimally small charged objects) as
where
In the SI system of units, force is given in newtons, charge in coulombs, and distance in metres. Thus, has units of C²/Nm².
This was known empirically. Suppose one of the charges is taken to be fixed, and the other one to be a moveable "test charge". Note that according to this equation, the force on the test object is proportional to its charge. The electric field is defined as the proportionality constant between charge and force:
However, note that this equation is only true in the case of electrostatics, that is to say, when there is nothing moving. The more general case of moving charges causes this equation to become the Lorentz equation.
The above paragraph is perhaps misleading that the equation could refer to electrodynamics by referring to a "moveable test charge"; however, it is important to understand that by "moveable", this means that the charge to be moved to, and held at, any position.
In the case of electrostatics, where velocity of both particles is zero, the above equation holds.
According to Equation (1) above, electric field is dependent on position. The electric field due to any single charge falls off as the square of the distance from that charge.
Electric fields follow the superposition principle. If more than one charge is present, the total electric field at any point is equal to the vector sum of the respective electric fields that each object would create in the absence of the others.
If this principle is extended to an infinite number of infinitesimally small elements of charge, the following formula results:
where ρ is the charge density , or the amount of charge per unit volumeVolume (also called capacity is a quantification of how much space an object occupies. The SI unit for volume is the cubic metre (American spelling meter). The volume of a solid object is a numerical value given to describe the three-dimensional concept o.
The electric field is equal to minus the gradientMultivariate calculus In vector calculus, gradient is a vector-valued operator that acts on a scalar field. The gradient of a scalar field is a vector field called gradient field which shows its rate and direction of change. For example, consider a room. of the electric potential. If several spatially distributed charges generate such an electric potential, e.g. in a solidA solid is a state of matter, characterized by a definite volume and a definite shape (i. it resists deformation). Within a solid, atoms/ molecules are relatively close together, or "rigid"; however, this does not prevent the solid from becoming deformed, an electric field gradientMathematically the electric field gradient (EFG) is the second derivative of the electrical potential V : It is an important structural property of a crystalline solid, where it is defined at the location of a nucleus. The EFG is non-zero only if the char may be defined.
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