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Engineering LibreTexts

5.2: Electric Field Due to Point Charges

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  • [m0103_E_due_to_Point_Charges]

    The electric field intensity associated with a single particle bearing charge \(q_1\), located at the origin, is (Section [m0102_Coulombs_Law]) \[{\bf E}({\bf r}) = \hat{\bf r}\frac{q_1}{4\pi\epsilon r^2}\] If this particle is instead located at some position \({\bf r}_1\), then the above expression may be written as follows: \[{\bf E}({\bf r};{\bf r}_1) = \frac{{\bf r}-{\bf r}_1}{\left|{\bf r}-{\bf r}_1\right|}~\frac{q_1}{4\pi\epsilon \left|{\bf r}-{\bf r}_1\right|^2}\] or, combining like terms in the denominator: \[{\bf E}({\bf r};{\bf r}_1) = \frac{{\bf r}-{\bf r}_1}{\left|{\bf r}-{\bf r}_1\right|^3}~\frac{q_1}{4\pi\epsilon}\]

    Now let us consider the field due to multiple such particles. Under the usual assumptions about the permittivity of the medium (reminder: Section [m0007_Properties_of_Materials]), 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: \[{\bf E}({\bf r}) = \sum_{n=1}^{N}{\bf E}({\bf r};{\bf r}_n)\] where \(N\) is the number of particles. Thus, we have \[{\bf E}({\bf r}) = \frac{1}{4\pi\epsilon} \sum_{n=1}^{N} { \frac{{\bf r}-{\bf r}_n}{\left|{\bf r}-{\bf r}_n\right|^3}~q_n}\]