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Engineering LibreTexts

5.12: Electric Potential Field Due to Point Charges

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  • [m0064_V_due_to_Point_Charges]

    The electric field intensity due to a point charge \(q\) at the origin is (see Section [m0102_Coulombs_Law] or [m0014_Gauss_Law_Integral_Form]) \[{\bf E} = \hat{\bf r}\frac{q}{4\pi\epsilon r^2} \label{eEPPCE}\] In Sections [m0061_Force_Energy_and_Potential_Difference] and [m0062_Independence_of_Path], it was determined that the potential difference measured from position \({\bf r}_1\) to position \({\bf r}_2\) is \[V_{21} = - \int_{{\bf r}_1}^{{\bf r}_2} {\bf E} \cdot d{\bf l} \label{m0064_eV12}\]

    This method for calculating potential difference is often a bit awkward. To see why, consider an example from circuit theory, shown in Figure [m0064_fResistor]. In this example, consisting of a single resistor and a ground node, we’ve identified four quantities:

    • The resistance \(R\)

    • The current \(I\) through the resistor

    • The node voltage \(V_1\), which is the potential difference measured from ground to the left side of the resistor

    • The node voltage \(V_2\), which is the potential difference measured from ground to the right side of the resistor

    Let’s say we wish to calculate the potential difference \(V_{21}\) across the resistor. There are two ways this can be done:

    • \(V_{21}=-IR\)

    • \(V_{21}=V_2-V_1\)

    The advantage of the second method is that it is not necessary to know \(I\), \(R\), or indeed anything about what is happening between the nodes; it is only necessary to know the node voltages. The point is that it is often convenient to have a common datum – in this example, ground – with respect to which the potential differences at all other locations of interest can be defined. When we have this, calculating potential differences reduced to simply subtracting predetermined node potentials.

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    So, can we establish a datum in general electrostatic problems that works the same way? The answer is yes. The datum is arbitrarily chosen to be a sphere that encompasses the universe; i.e., a sphere with radius \(\to\infty\). Employing this choice of datum, we can use Equation [m0064_eV12] to define \(V({\bf r})\), the potential at point \({\bf r}\), as follows: \[\boxed{ V({\bf r}) \triangleq - \int_{\infty}^

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    {\bf E} \cdot d{\bf l} } \label{m0064_eVP}\]

    The electrical potential at a point, given by Equation [m0064_eVP], is defined as the potential difference measured beginning at a sphere of infinite radius and ending at the point \({\bf r}\). The potential obtained in this manner is with respect to the potential infinitely far away.

    In the particular case where \({\bf E}\) is due to the point charge at the origin: \[V({\bf r}) = - \int_{\infty}^

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    \left[ \hat{\bf r}\frac{q}{4\pi\epsilon r^2} \right] \cdot d{\bf l}\] The principle of independence of path (Section [m0062_Independence_of_Path]) asserts that the path of integration doesn’t matter as long as the path begins at the datum at infinity and ends at \({\bf r}\). So, we should choose the easiest such path. The radial symmetry of the problem indicates that the easiest path will be a line of constant \(\theta\) and \(\phi\), so we choose \(d{\bf l}=\hat{\bf r}dr\). Continuing: so \[\boxed{ V({\bf r}) = + \frac{q}{4\pi\epsilon r} } \label{m0064_eV}\] (Suggestion: Confirm that Equation [m0064_eV] is dimensionally correct.) In the context of the circuit theory example above, this is the “node voltage” at \({\bf r}\) when the datum is defined to be the surface of a sphere at infinity. Subsequently, we may calculate the potential difference from any point \({\bf r}_1\) to any other point \({\bf r}_2\) as \[V_{21} = V({\bf r}_2)-V({\bf r}_1)\] and that will typically be a lot easier than using Equation [m0064_eV12].

    It is not often that one deals with systems consisting of a single charged particle. So, for the above technique to be truly useful, we need a straightforward way to determine the potential field \(V({\bf r})\) for arbitrary distributions of charge. The first step in developing a more general expression is to determine the result for a particle located at a point \({\bf r}'\) somewhere other than the origin. Since Equation [m0064_eV] depends only on charge and the distance between the field point \({\bf r}\) and \({\bf r}'\), we have \[V({\bf r};{\bf r}') \triangleq + \frac{q'}{4\pi\epsilon \left|{\bf r}-{\bf r}'\right|} \label{m0064_eVd}\] where, for notational consistency, we use the symbol \(q'\) to indicate the charge. Now applying superposition, the potential field due to \(N\) charges is \[V({\bf r}) = \sum_{n=1}^N { V({\bf r};{\bf r}_n) }\] Substituting Equation [m0064_eVd] we obtain: \[\boxed{ V({\bf r}) = \frac{1}{4\pi\epsilon} \sum_{n=1}^N { \frac{q_n}{\left|{\bf r}-{\bf r}_n\right|} } } \label{m0064_eVN}\]

    Equation [m0064_eVN] gives the electric potential at a specified location due to a finite number of charged particles.

    The potential field due to continuous distributions of charge is addressed in Section [m0065_V_due_to_a_Continuous_Distribution_of_Charge].