Electric Field Due To Point Electric Charges

So in a simple way we can define the electrostatic field considering the force exerted by a point charge on a unit charge. In other words we can define the electric field as the force per unit charge.

To detect an electric field of a charge q, we can introduce a test charge q₀ and measure the force acting on it.

\(\overrightarrow {F}\) = \( \frac{1}{4~\pi~ε_0} \frac {q q_0}{r^2} \hat r \)

Thus the force exerted per unit charge is:

\( \overrightarrow{E} \) = \( \frac {\overrightarrow F}{q_0}\) = \( \frac {1}{4~\pi~ε_0 } \frac{q}{r^2} \hat r \)

Note that the electric field is a vector quantity that is defined at every pint in space, the value of which is dependent only upon the radial distance from q.

The test charge q₀ itself has the ability to exert an electric field around it. Hence, to prevent the influence of the test charge, we must ideally make it as small as possible.

Thus,

\( \overrightarrow{E}\)
=\(  \lim_{{q_0}\to 0} \frac {\overrightarrow{F}}{q_0} \) = \( \lim_{{q_0}\to 0} \frac {1}{4~\pi~ε_0 } \frac{qq_0}{r^2} \hat r × \frac {1}{q_0} \) = \( \frac {1}{4~\pi~ε_0 } \frac{q}{r^2} \hat r \)

This is the electric field of a point charge. Also, observe that it exhibits spherical symmetry since the electric field has the same magnitude on every point of an imaginary sphere centered around the charge q.

Electric field due to positive and negative charges

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