Electric Dipole

An Electric Dipole is a pair of equal and opposite Point Charges separated by a small distance.

claimed by Jmorton32 (talk) 02:52, 19 October 2015 (EDT)

Mathematical Models

An Exact Model

An Electric Dipole

Since an electric dipole is made up of 2 electric point charges, the electric field of the dipole can be calculated by summing the electric fields contributed by each point charge. In the example, the field at point P, [math]\displaystyle{ E_{P} }[/math] is equivalent to the sum of the electric field from the positive charge [math]\displaystyle{ q+ }[/math] and the negative charge [math]\displaystyle{ q- }[/math]. In other words, [math]\displaystyle{ E_{P} = E_{P_{q+}} + E_{P_{q-}} }[/math].

Substituting in the equation for the electric field from a point charge we get [math]\displaystyle{ \vec E_{P} = \frac{1}{4\pi\epsilon_{0}} \times \frac{q_{+}}{|r_{+}|^{2}} \times \hat r_{+} + \frac{1}{4\pi\epsilon_{0}} \times \frac{q_{-}}{|r_{-}|^{2}} \times \hat r_{-} }[/math].

A simple refactoring gives [math]\displaystyle{ \vec E_{P} = \frac{1}{4\pi\epsilon_{0}} \times (\frac{q_{+}}{|r_{+}|^{2}}\hat r_{+} + \frac{q_{-}}{|r_{-}|^{2}} \hat r_{-}) }[/math].

Since [math]\displaystyle{ q_+ = -1 * q_- }[/math], this is equivalent to [math]\displaystyle{ \vec E_{P} = \frac{q_+}{4\pi\epsilon_{0}} \times (\frac{\hat r_{+}}{|r_{+}|^{2}} - \frac{\hat r_{-}}{|r_{-}|^{2}}) }[/math]

[math]\displaystyle{ |r_+| \text{ and } |r_-| }[/math] can then be calculated as [math]\displaystyle{ |r_+| = \sqrt{(|r_x| + \frac{d}{2})^2 + |r_y|^2} }[/math], by decomposing [math]\displaystyle{ \vec r }[/math] into its components and factoring.

By a similar method, [math]\displaystyle{ |r_-| = \sqrt{(|r_x| - \frac{d}{2})^2 + |r_y|^2} }[/math].

By substituting [math]\displaystyle{ |\vec r| \cos(\theta) \text{ and } |\vec r| \sin(\theta) }[/math] for [math]\displaystyle{ |r_x| \text{ and } |r_y| }[/math] respectively, we get

[math]\displaystyle{ |r_+| = \sqrt{(|\vec r| \cos(\theta) + \frac{d}{2})^2 + (|\vec r| \sin(\theta))^2} }[/math] and [math]\displaystyle{ |r_-| = \sqrt{(|\vec r| \cos(\theta) - \frac{d}{2})^2 + (|\vec r| \sin(\theta))^2} }[/math]

We next expand the squared terms inside the radical: [math]\displaystyle{ \sqrt{(|\vec r| \cos(\theta))^2 + \frac{d |\vec r| \cos(\theta)}{2} + \frac{d^2}{4} +(|\vec r| \sin(\theta))^2} }[/math]

Rearranging, [math]\displaystyle{ \sqrt{|\vec r|^2 * (\cos(\theta)^2 + \sin(\theta)^2) + \frac{d |\vec r| \cos(\theta)}{2} + \frac{d^2}{4}} = \sqrt{|\vec r|^2 + \frac{d |\vec r| \cos(\theta)}{2} + \frac{d^2}{4}} = |r_+| }[/math]

By a similar method, [math]\displaystyle{ |r_-| = \sqrt{|\vec r|^2 - \frac{d |\vec r| \cos(\theta)}{2} + \frac{d^2}{4}} }[/math]

Plugging this into the earlier equation, we get the monstrosity [math]\displaystyle{ E_{P} = \frac{q_+}{4\pi\epsilon_{0}} \times (\frac{\hat r_{+}}{|\vec r|^2 + \frac{d |\vec r| \cos(\theta)}{2} + \frac{d^2}{4}} - \frac{\hat r_{-}}{|\vec r|^2 - \frac{d |\vec r| \cos(\theta)}{2} + \frac{d^2}{4}}) }[/math]

To subtract the fractions, we first put them in like terms:

[math]\displaystyle{ \frac{\hat r_{+}}{|\vec r|^2 + \frac{d |\vec r| \cos(\theta)}{2} + \frac{d^2}{4}} \times \frac{|\vec r|^2 - \frac{d |\vec r| \cos(\theta)}{2} + \frac{d^2}{4}}{|\vec r|^2 - \frac{d |\vec r| \cos(\theta)}{2} + \frac{d^2}{4}}) = \frac{\hat r_{+}\times (|\vec r|^2 - \frac{d |\vec r| \cos(\theta)}{2} + \frac{d^2}{4})}{|\vec r|^4 - \frac{|\vec r|^2 d |\vec r| \cos(\theta)}{2} + \frac{|\vec r|^2 d^2}{4} + \frac{|\vec r|^2 d |\vec r| \cos(\theta)}{2} - \frac{d^2 |\vec r|^2 \cos(\theta)^2}{4} + \frac{d^3|\vec r|cos(\theta)}{8} + \frac{|\vec r|^2 d^2}{4} - \frac{d^3|\vec r|cos(\theta)}{8} + \frac{d^4}{16}} }[/math]

Cancelling like terms simplifies this to [math]\displaystyle{ \frac{\hat r_{+}\times (|\vec r|^2 - \frac{d |\vec r| \cos(\theta)}{2} + \frac{d^2}{4})}{|\vec r|^4 + \frac{|\vec r|^2 d^2}{2} -\frac{d^2 |\vec r|^2 \cos(\theta)^2}{4} + \frac{d^4}{16}} }[/math]

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