Electric field due to a point charge

The concept of electric field was introduced by Faraday during the middle of the 19th century. It is a vector quantity equal to the force experienced by a positive unit charge at any point P of the space.

To get an idea, consider a stationary positive point charge q1 like the one represented in green in the following figure.

The force experienced by a 1 coulomb charge situated at any position P of the space in the vicinity of a charge q1 is given by Coulomb’s law:

Where ur is a unit vector in the radial direction.

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

It can be expressed as the following product of the vacuum permittivity (permittivity of free space or electric constant):

The force experienced by the positive unit charge is repulsive, because q1 is positive too and two charges with the same sign repel each other. At any point of space, the positive unit charge would experience a repulsive force given by Coulomb’s law. It is represented with the green arrows in the previous figure.

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This situation can be interpreted as a disruption due to the charge q1 on its surroundings, so that a test charge situated at a point P of space experiences a force. The electric field E is the vector magnitude that describes this disruption.

The charge q1 creating the electric field E is called a source charge.

Going back to the definition given at the beginning of this page, the electric field due to a point charge is:

The SI units for the electric field strength are N/C or V/m.

The green arrows in the previous figure are called field lines and they are a way to graphically represent the field due to a point charge. As you can see in the figure, the field lines of the electric field start at positive charges, For this reason, a positive charge is called a source of field lines.

The density (number per area) of field lines is proportional to the magnitude of the electric field and they can never intersect because it they did, the electric field would have two different values for the same point in space.

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If the source charge were negative, the positive unit charge (the test charge) would experience an attractive force. Therefore, the field lines would end at the negative charge, which is called a sink of field lines (see next figure).

The electric field satisfies the superposition principle. At any point of space, the net electric field from multiple charges is the sum of the individual electric fields due to each individual charge.

Therefore, the electric field due to a set of N charges is:

The force exerted by an electric field on a test charge q2 at any location of space is given by:

If the electric field is due to a source point charge q1, this force is given by:

This expression is Coulomb’s law. The following figure represents the force experienced by a negative (a) or positive (b) test charge q2 in vicinity of the electric field due to a positive source charge q1:

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