5.2: Electric Field Due to Point Charges

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The electric field intensity associated with a single particle bearing charge \(q_1\), located at the origin, is (Section 5.1)

\[{\bf E}({\bf r}) = \hat{\bf r}\frac{q_1}{4\pi\epsilon r^2} \nonumber \]

If this particle is instead located at some position \({\bf r}_1\), then the above expression may be written as follows:

\[{\bf E}({\bf r};{\bf r}_1) = \frac{{\bf r}-{\bf r}_1}{\left|{\bf r}-{\bf r}_1\right|}~\frac{q_1}{4\pi\epsilon \left|{\bf r}-{\bf r}_1\right|^2} \nonumber \]

or, combining like terms in the denominator:

\[{\bf E}({\bf r};{\bf r}_1) = \frac{{\bf r}-{\bf r}_1}{\left|{\bf r}-{\bf r}_1\right|^3}~\frac{q_1}{4\pi\epsilon} \nonumber \]

Now let us consider the field due to multiple such particles. Under the usual assumptions about the permittivity of the medium (Section 2.8), the property of superposition applies. Using this principle, we conclude:

The electric field resulting from a set of charged particles is equal to the sum of the fields associated with the individual particles.

Stated mathematically:

\[{\bf E}({\bf r}) = \sum_{n=1}^{N}{\bf E}({\bf r};{\bf r}_n) \nonumber \] where \(N\) is the number of particles. Thus, we have

\[{\bf E}({\bf r}) = \frac{1}{4\pi\epsilon} \sum_{n=1}^{N} { \frac{{\bf r}-{\bf r}_n}{\left|{\bf r}-{\bf r}_n\right|^3}~q_n} \nonumber \]

Contributors and Attributions

Ellingson, Steven W. (2018) Electromagnetics, Vol. 1. Blacksburg, VA: VT Publishing. https://doi.org/10.21061/electromagnetics-vol-1 Licensed with CC BY-SA 4.0 https://creativecommons.org/licenses/by-sa/4.0. Report adoption of this book here. If you are a professor reviewing, adopting, or adapting this textbook please help us understand a little more about your use by filling out this form.