# Electric field due to a continuous distribution of charge

A continuous distribution of charge is a conceptual model used to mathematically describe the electric charge of a macroscopic object. Although the electric charge is quantized, that is, all electric charges are multiples of a fundamental unit of charge, it is convenient to consider it as continuous to calculate the electric field due to a charged object. Depending on the shape and dimensions of the object that creates the electric field, we can identify three different types of charge density:

• Linear charge density λ: is the charge density per unit length. It is used to describe the charge of an object with a length far greater than the two other dimensions; for instance a wire.
• Surface charge density σ: is the charge density per unit surface area. It is the charge density of a planar object; for instance a disc or a plate.
• Volume charge density ρ: is the charge density per unit volume. It is used when the three dimensions of an object are relevant; for instance to calculate the electric field due to a charged sphere.

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Usually, the charge density depends on the coordinates. But, in simple situations, the charge density is homogeneous (it is the same for all points of the object) and it can be considered to be constant.

The electric field due to a continuous distribution of charge is given by calculating the electric field due to a charge element and later by integrating it over the whole object.

An object with a total electric charge q is represented in the following figure. To calculate the field at any point P of space, we choose a point charge element dq. We calculate the electric field due to this charge element and then we integrate it over the whole object.

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The electric field due to a charge element dq is given by:

Where k is the Coulomb constant and its value in SI units in vacuum is:

Or, if we express it as a product of the vacuum permittivity (permittivity of free space or electric constant) ε0:, it is equal to:

The total electric field is the integral over the whole charged object:

In general, this integral is difficult to solve, except if the charged object has a high level of symmetry. You can see how to calculate the electric field due to simple charge distributions in the pages below.

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