Electric Dipole: Difference between revisions

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An electric dipole is constructed from two point charges, one at position <math>[\frac{d}{2}, 0]</math> and one at position <math>[\frac{-d}{2}, 0]</math>. These point charges are of equal and opposite charge. We then wish to know the electric field due to the dipole at some point <math>p</math> in the plane (see the figure). <math>p</math> can be considered either a distance <math>[x_0, y_0]</math> from the midpoint of the dipole, or a distance <math>r</math> and an angle <math>\theta</math> as in the diagram.
An electric dipole is constructed from two point charges, one at position <math>[\frac{d}{2}, 0]</math> and one at position <math>[\frac{-d}{2}, 0]</math>. These point charges are of equal and opposite charge. We then wish to know the electric field due to the dipole at some point <math>p</math> in the plane (see the figure). <math>p</math> can be considered either a distance <math>[x_0, y_0]</math> from the midpoint of the dipole, or a distance <math>r</math> and an angle <math>\theta</math> as in the diagram.


Then we begin by calculating <math>r_+</math> and <math>r_-</math> the radii from the positive and negative particles to the point <math>p</math>. In this example, we will assume that the positive particle is closer to <math>p</math>, but its simple to modify this derivation for the opposite case. First, we divide <math>r</math> into its x and y components, <math> r_x = r * cos(\theta) </math> and <math> r_y = r * sin(\theta) </math>.
We state that the net electric field at <math>p</math> is <math>E_{net}</math> and has an x and y component, <math>E_{net_x}</math> and <math>E_{net_y}</math>. Then we can individually calculate the x and y components. First we realize that since <math>E_{net} = E_{q_+} + E_{q_-}</math>, <math>E_{net_x} = E_{q_{+x}} + E_{q_{-x}}</math>, similarly for y <math>E_{net_y} = E_{q_{+y}} + E_{q_{-y}}</math>. At this point, its worth noting that <math>E_{q_{+y}} = E_{q_+} * cos(\theta_+)</math>, where <math>\theta_+</math> is the angle from <math>q_{+}</math> to <math>p</math>.


We know that the net electric field at <math>p</math> is equal to the sum of all electric fields in our system, in this case <math>E_{net} = E_{q_-} + E_{q_+}</math>. Then we divide <math>E_{net}</math> into its x and y components and calculate the electric field for each. Let's do y first.
<math>\theta_+</math> and its counterpart <math>\theta_-</math> are not known. However, we can calculate them. We know <math>\theta_+</math> is formed by a triangle with one side length <math>p_y</math> and one side length <math>p_x - \frac{d}{2}</math>. Then <math>sin(\theta_+) = \frac{p_y}{\sqrt{(p_x - \frac{d}{2})+p_y}}</math>, from which you can calculate the angle. This looks disgusting, but a close inspection shows that <math>p_y</math> is the opposite side of the triangle, and the denominator is an expression forming the hypotenuse of the triangle (<math>r_+</math>) from known quantities. A similar method shows that <math>sin(\theta_-) = \frac{p_y}{\sqrt{(p_x + \frac{d}{2})+p_y}}</math>, where once again <math>\sqrt{(p_x + \frac{d}{2})+p_y} = |\vec r_-|</math>.


<math>E_{net_y} = E_{q_{-_y}} + E_{q_{+_y}}</math>. The equation for electric field is <math>\frac{1}{4\pi\epsilon_0} \frac{q}{|\vec r|^2} \hat r</math>. Where <math>\hat r</math> is the unit vector in the direction from the the charge to the point p. We want only the component of this in the y direction, which means
We now have values for <math> d, q, \theta_+, \theta_-, \vec r_+, \vec r_-</math>. This is enough to calculate <math>E_net</math> in both directions. The formula for electric field strength, <math>E = \frac{1}{4\po\epsilon_0} \frac{q}{\|\vec r\|^2} \hat r</math>.


==Examples==
==Examples==

Revision as of 02:08, 4 December 2015

An Electric Dipole is a pair of equal and opposite Point Charges separated by a small distance. Electric dipoles have a number of interesting properties.

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

Mathematical Models

An Exact Model

An Electric Dipole

An electric dipole is constructed from two point charges, one at position [math]\displaystyle{ [\frac{d}{2}, 0] }[/math] and one at position [math]\displaystyle{ [\frac{-d}{2}, 0] }[/math]. These point charges are of equal and opposite charge. We then wish to know the electric field due to the dipole at some point [math]\displaystyle{ p }[/math] in the plane (see the figure). [math]\displaystyle{ p }[/math] can be considered either a distance [math]\displaystyle{ [x_0, y_0] }[/math] from the midpoint of the dipole, or a distance [math]\displaystyle{ r }[/math] and an angle [math]\displaystyle{ \theta }[/math] as in the diagram.

We state that the net electric field at [math]\displaystyle{ p }[/math] is [math]\displaystyle{ E_{net} }[/math] and has an x and y component, [math]\displaystyle{ E_{net_x} }[/math] and [math]\displaystyle{ E_{net_y} }[/math]. Then we can individually calculate the x and y components. First we realize that since [math]\displaystyle{ E_{net} = E_{q_+} + E_{q_-} }[/math], [math]\displaystyle{ E_{net_x} = E_{q_{+x}} + E_{q_{-x}} }[/math], similarly for y [math]\displaystyle{ E_{net_y} = E_{q_{+y}} + E_{q_{-y}} }[/math]. At this point, its worth noting that [math]\displaystyle{ E_{q_{+y}} = E_{q_+} * cos(\theta_+) }[/math], where [math]\displaystyle{ \theta_+ }[/math] is the angle from [math]\displaystyle{ q_{+} }[/math] to [math]\displaystyle{ p }[/math].

[math]\displaystyle{ \theta_+ }[/math] and its counterpart [math]\displaystyle{ \theta_- }[/math] are not known. However, we can calculate them. We know [math]\displaystyle{ \theta_+ }[/math] is formed by a triangle with one side length [math]\displaystyle{ p_y }[/math] and one side length [math]\displaystyle{ p_x - \frac{d}{2} }[/math]. Then [math]\displaystyle{ sin(\theta_+) = \frac{p_y}{\sqrt{(p_x - \frac{d}{2})+p_y}} }[/math], from which you can calculate the angle. This looks disgusting, but a close inspection shows that [math]\displaystyle{ p_y }[/math] is the opposite side of the triangle, and the denominator is an expression forming the hypotenuse of the triangle ([math]\displaystyle{ r_+ }[/math]) from known quantities. A similar method shows that [math]\displaystyle{ sin(\theta_-) = \frac{p_y}{\sqrt{(p_x + \frac{d}{2})+p_y}} }[/math], where once again [math]\displaystyle{ \sqrt{(p_x + \frac{d}{2})+p_y} = |\vec r_-| }[/math].

We now have values for [math]\displaystyle{ d, q, \theta_+, \theta_-, \vec r_+, \vec r_- }[/math]. This is enough to calculate [math]\displaystyle{ E_net }[/math] in both directions. The formula for electric field strength, [math]\displaystyle{ E = \frac{1}{4\po\epsilon_0} \frac{q}{\|\vec r\|^2} \hat r }[/math].

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