Physics Book - User contributions [en]

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2022-04-24T00:14:42Z

Aramirez81: ckck

== Summary ==
ckck

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2022-04-24T00:08:29Z

Aramirez81: 3 cssss

== Summary ==
3 cssss

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2022-04-24T00:05:10Z

Aramirez81: Adiabatic demag

== Summary ==
Adiabatic demag

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2022-04-24T00:04:26Z

Aramirez81:

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2022-04-23T23:30:45Z

Aramirez81:

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2022-04-23T23:10:20Z

Aramirez81: Isochor (Slope of constant volume) reaching the same value of entropy as temperature is decreased towards absolute zero.

== Summary ==
Isochor (Slope of constant volume) reaching the same value of entropy as temperature is decreased towards absolute zero.

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2022-04-23T23:09:33Z

Aramirez81: Isochor (Slope of constant volume) reaching the same value of entropy as temperature is decreased towards absolute zero.

== Summary ==
Isochor (Slope of constant volume) reaching the same value of entropy as temperature is decreased towards absolute zero.

Electric Dipole

2022-04-23T20:04:15Z

Aramirez81:

==Summary==

An electric dipole is made up of two point charges that have equal but opposite electric charges (q) and are separated by a short distance (d).

[[File:dipo.jpg|An Electric Dipole]]

The electric field is proportional to the cube of the distance from the dipole, and is dependent on whether you’re moving along the line separating the two charges or perpendicular to it.

A dipole can be created, for example, when you place a neutral atom in an electric field, because of the movement of electrons, the atom polarizes (negative charge on one side, positive charge on the other) and yields a separation of charge.

A prime example of this is the molecule for water, which forms a 105 degree angle between the two hydrogens connected to the oxygen. Since the oxygen has a greater electronegativity, it pulls more strongly on the electrons shared by the oxygen and hydrogen atoms and that end of the molecule becomes more negatively charged compared to the hydrogen end. Therefore, the net electric dipole points towards the oxygen atom.

[[File:Water.png|300px|thumb|Dipole moment of water]]

==Mathematical Models==

===An Exact Model===
[[File:Phys2212 dipole image.PNG|300px|thumb|Polarization by an electric field]]
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.

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>.

<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})^2+p_y^2}}</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})^2+p_y^2}}</math>, where once again <math>\sqrt{(p_x + \frac{d}{2})^2+p_y^2} = |\vec r_-|</math>.

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 general formula for electric field strength from a [[Point Charge]] is <math>E = \frac{1}{4\pi\epsilon_0} \frac{q}{|\vec r|^2} \hat r</math>. Then <math>|E_+| = \frac{1}{4\pi\epsilon_0} \frac{q_+}{|\vec r_+|^2}</math> and <math>|E_-| = \frac{1}{4\pi\epsilon_0} \frac{q_-}{|\vec r_-|^2}</math>. We want solely the magnitude in this case because we can calculate direction and component forces using sin and cosine. Its worth noting that we can expand <math>r_+, r_-</math> to the form in the denominator of the sine and cosine. We will use this later.

First we calculate <math>E_{net_y}</math>. <math>E_{net_y} = E_{+_y} + E_{-_y} = \frac{1}{4\pi\epsilon_0} \frac{q_+}{|\vec r_+|^2} sin(\theta_+) + \frac{1}{4\pi\epsilon_0} \frac{q_-}{|\vec r_-|^2} sin(\theta_-)</math>.

Then we combine some terms, noting that <math> q_+ = -q_-</math>. <math>E_{net_y} = \frac{q_+}{4\pi\epsilon_0} * \Bigg(\frac{1}{|\vec r_+|^2}sin(\theta_+) + \frac{-1}{|\vec r_-|^2}sin(\theta_-)\Bigg)</math>

Now it gets ugly, we expand our radii and sines. To recap, <math>sin(\theta_+) = \frac{p_y}{\sqrt{(p_x - \frac{d}{2})^2+p_y^2}}</math>, <math>sin(\theta_-) = \frac{p_y}{\sqrt{(p_x + \frac{d}{2})^2+p_y^2}}</math>, <math>|r_+| = \sqrt{(p_x - \frac{d}{2})^2 +p_y^2}</math> and <math>|r_-| = \sqrt{(p_x + \frac{d}{2})^2 +p_y^2}</math>, giving us

<math>E_{net_y} =
\frac{q_+}{4\pi\epsilon_0} *
\Bigg(
\frac{1}{
(p_x - \frac{d}{2})^2 +p_y^2
}
\frac{p_y}{\sqrt{(p_x - \frac{d}{2})^2+p_y^2}} +
\frac{-1}{
(p_x + \frac{d}{2})^2 +p_y^2
}
\frac{p_y}{\sqrt{(p_x + \frac{d}{2})^2+p_y^2}}
\Bigg)</math>

Finally we can combine more terms, the denominators of the expanded sines are the square roots of the radii. We can also pull out the negative sign.

<math>E_{net_y} =
\frac{q_+}{4\pi\epsilon_0}
\Bigg(
\frac{p_y}{
\Big((p_x - \frac{d}{2})^2 +p_y^2 \Big)^\frac{3}{2}
}
-
\frac{p_y}{
\Big((p_x + \frac{d}{2})^2 +p_y^2 \Big)^\frac{3}{2}
}

\Bigg)</math> That's as simplified as possible.

Much of the derivation for the x direction is similar. The major difference is that instead of calculating the sine, opposite over hypotenuse, we want cosine, adjacent over hypotenuse. That is, where <math>sin(\theta_+) = \frac{p_y}{\sqrt{(p_x - \frac{d}{2})^2+p_y^2}}</math>, <math>cos(\theta_+) = \frac{p_x - \frac{d}{2}}{\sqrt{(p_x - \frac{d}{2})^2+p_y^2}}</math>. By using this and its counterpart for <math>\theta_-</math>, the result is that

<math>E_{net_x} =
\frac{q_+}{4\pi\epsilon_0}
\Bigg(
\frac{p_x - \frac{d}{2}}{
\Big((p_x - \frac{d}{2})^2 +p_y^2 \Big)^\frac{3}{2}
}
-
\frac{p_x + \frac{d}{2}}{
\Big((p_x + \frac{d}{2})^2 +p_y^2 \Big)^\frac{3}{2}
}

\Bigg)</math>. These provide exact formulae for the electric field due to an electric dipole anywhere on the two-dimensional plane, and they translate easily into 3-dimensions.

==Special Cases==
We can simplify the solution for many cases

===On the Parallel Axis===
On the parallel axis, we begin with the now known formula <math>E_{net_x} =
\frac{q_+}{4\pi\epsilon_0}
\Bigg(
\frac{p_x - \frac{d}{2}}{
\Big((p_x - \frac{d}{2})^2 +p_y^2 \Big)^\frac{3}{2}
}
-
\frac{p_x + \frac{d}{2}}{
\Big((p_x + \frac{d}{2})^2 +p_y^2 \Big)^\frac{3}{2}
}

\Bigg)</math>. Since we are on the parallel axis, we know that <math>E_{net_y} = 0</math>, and <math>p_y = 0</math>.

Simplifies to

<math>E_{net_x} =
\frac{q_+}{4\pi\epsilon_0}
\Bigg(
\frac{p_x - \frac{d}{2}}{
\Big((p_x - \frac{d}{2})^2 \Big)^\frac{3}{2}
}
-
\frac{p_x + \frac{d}{2}}{
\Big((p_x + \frac{d}{2})^2 \Big)^\frac{3}{2}
}

\Bigg)</math>.

Then, combining exponents and reducing the fraction:
<math>E_{net_x} =
\frac{q_+}{4\pi\epsilon_0}
\Bigg(
\frac{1}{
(p_x - \frac{d}{2})^2
}
-
\frac{1}{
(p_x + \frac{d}{2})^2
}

\Bigg)</math>.

Then, we can combine these fractions. to simplify the calculations, replace <math>\frac{d}{2}</math> with <math>a</math>.

<math>E_{net_x} =
\frac{q_+}{4\pi\epsilon_0}
\Bigg(
\frac{1}{
(p_x - a)^2
}
-
\frac{1}{
(p_x + a)^2
}

\Bigg) =

\frac{q_+}{4\pi\epsilon_0}
\Bigg(\frac{4p_x a}{(p_x^2 + a^2)^2}
\Bigg)

=

\frac{q_+ 4 a}{4\pi\epsilon_0}
\Bigg(\frac{p_x}{(p_x^2 + a^2)^2}
\Bigg)
</math>.

This is the formula. When <math>p_x >> a</math>, we can assume that <math>p_x^2 + a^2</math> is very close to <math>p_x^2</math>. Then

<math>E_{net_x} \approx
\frac{q_+ 4 a}{4\pi\epsilon_0}
\Bigg(\frac{p_x}{(p_x^2)^2}
\Bigg) =

\frac{q_+ 4 a}{4\pi\epsilon_0}
\Bigg(\frac{p_x}{p_x^4}
\Bigg)
=
\frac{1}{4\pi\epsilon_0}
\Bigg(\frac{4 a q_+}{p_x^3}
\Bigg)</math>

===On the Perpendicular Axis===
We can do a similar simplification for the perpendicular axis. We know that <math>E_{net_y} = 0</math> because the vertical forces from both point charges cancel, leaving only horizontal forces.

<math>
E_{net_x} =
\frac{q_+}{4\pi\epsilon_0}
\Bigg(
\frac{p_x - \frac{d}{2}}{
\Big((p_x - \frac{d}{2})^2 +p_y^2 \Big)^\frac{3}{2}
}
-
\frac{p_x + \frac{d}{2}}{
\Big((p_x + \frac{d}{2})^2 +p_y^2 \Big)^\frac{3}{2}
}

\Bigg)</math>

In this case though, <math>p_x = 0</math>

<math>
E_{net_x} =
\frac{q_+}{4\pi\epsilon_0}
\Bigg(
\frac{- \frac{d}{2}}{
\Big(( - \frac{d}{2})^2 +p_y^2 \Big)^\frac{3}{2}
}
-
\frac{\frac{d}{2}}{
\Big((\frac{d}{2})^2 +p_y^2 \Big)^\frac{3}{2}
}

\Bigg)</math>

Once again, we say <math>a = \frac{d}{2}</math>.

<math>
E_{net_x} =
\frac{q_+}{4\pi\epsilon_0}
\Bigg(
\frac{-a}{
\Big(( - a)^2 +p_y^2 \Big)^\frac{3}{2}
}
-
\frac{a}{
\Big(a^2 +p_y^2 \Big)^\frac{3}{2}
}

\Bigg)
=
\frac{q_+}{4\pi\epsilon_0}
\Bigg(
\frac{-a}{
\Big(a^2 +p_y^2 \Big)^\frac{3}{2}
}
-
\frac{a}{
\Big(a^2 +p_y^2 \Big)^\frac{3}{2}
}

\Bigg)
=\frac{q_+}{4\pi\epsilon_0}
\Bigg(
\frac{-2a}{
\Big(a^2 +p_y^2 \Big)^\frac{3}{2}
}
\Bigg)
</math>

And this is our result.

Once again, when <math>d</math> is much smaller than <math> p_y</math>, <math>a</math> is also small, so we can assume that the denominator is just <math>p_y</math>. This allows us to simplify the resulting equation to

<math>E_{net_x} \approx \frac{q_+}{4\pi\epsilon_0} \frac{-2a}{p_y^3} </math>

==Examples==

===Simple===

A dipole is located at the origin, and is composed of charged particles with charge <math>+e</math> and <math>-e</math>, separated by a distance <math>9 \times10^{-10}</math> along the <math>y</math> axis. The <math>+e</math> charge is on the <math>+y</math> axis. Calculate the force on a proton due to this dipole at a location <math>< 0, 0, 3 \times 10^{-8} ></math> meters.

<div class="toccolours mw-collapsible mw-collapsed">
===Click for Solution===
<div class="mw-collapsible-content">
The center of the dipole is at the origin and there is a proton along the z axis. In this case, we apply the perpendicular from of the electric field equation. In this case, since <math>r >> d</math>, we can also use an approximate solution. Therefore, we apply the formula <math>E_{net} = \frac{q}{4\pi\epsilon_0} \frac{-2a}{r^3}</math>. Since <math>a = \frac{d}{2}</math>, and r is the distance to the proton, we can plug in the values and solve for the net electric field.

[[File:Phys2212 sample simple.PNG | 300px]]

<math>1.6\times 10^{-19} \times 9 \times 10^9
\frac{-9 \times 10^{-10}}
{3 \times 10^{-8^3}} = -48000 \frac{N}{C}</math> on the y axis, as a vector: <math><0, -48000, 0></math>.

However, we aren't done since we want to know the force. We know that <math>F = qE</math> and in this case, both <math>q</math>, the charge on the proton and <math>E</math>, the electric field, are known. Thus the solution is <math>-48000 \times 1.6 \times 10^{-19} = -7.68 \times 10^{-15}</math> on the y axis, or <math><0, -7.68 \times 10^{-15}, 0></math>.
</div>
</div>

===Middling===
A ball of mass <math>M</math> and radius <math>R</math> is given an unknown negative charge spread uniformly over its surface. The ball is hanging from a thread and can move freely. A distance <math>L</math> directly below the center of the ball, a small permanent dipole is oriented such that the dipole axis is parallel with the center of the ball. The dipole has a dipole moment <math>p = qs</math>, with a distance <math>s</math> between the positive and negative charges of the dipole, and a mass <math>m</math>. The positive charge of the dipole is oriented closer to the center of the ball.

a) calculate the charge on the ball to levitate the dipole

b) the dipole is turned 90 degrees clockwise, without changing its position relative to the ball, what effect does this have on the ball?

<div class="toccolours mw-collapsible mw-collapsed">

===Click for Solutions===
<div class="mw-collapsible-content">
a) Because the dipole is small, we can assume that <math> s << L </math>. We wish to find the force on the dipole such that it can equal the force due to gravity. Once again, <math>F = qE</math> since by newton's third law, for a force exerted on the ball by the dipole, there is an equal and opposite for exerted on the dipole by the ball. That is <math>F_G = F_E</math>, so <math>qE = mg</math> (where <math>g</math> is the acceleration due to gravity). Therefore, in this case we wish to find the force on the ball, meaning the electric field from the dipole and the charge on the ball, <math>Q</math>. The field from the dipole is, since we are on the parallel axis, <math>E = \frac{1}{4\pi\epsilon_0} \frac{2p}{L^3}</math>. Putting this together, we get <math>mg = |Q| \frac{1}{4\pi\epsilon_0} \frac{2p}{L^3}</math>

Solving for <math>|Q|</math>: <math>|Q| = \Bigg(\frac{1}{4\pi\epsilon_0}\Bigg)^{-1} \frac{mgL^3}{2p}</math>.

However, we know that since the positive charge of the dipole is closer to the ball, the charge on the ball must be negative to create an attractive force. <math>|Q| > 0</math>, so our final answer is <math>Q = -\Bigg(\frac{1}{4\pi\epsilon_0}\Bigg)^{-1} \frac{mgL^3}{2p}</math>

b) By rotating the dipole clockwise the direction of the electric field at the location of the ball changes. Since the positive end of the dipole is to the right, and the negative end to the left of the dipole, the electric field from the dipole acting on the ball is oriented toward the left. However, since the ball has negative charge, this results in a force on the ball to the right.
</div>
</div>

===Concept Question===
Is it possible for a permanent electric dipole to have a net (total) charge of zero?

<div class="toccolours mw-collapsible mw-collapsed">

===Click for Solution===
<div class="mw-collapsible-content">
Permanent dipoles occur when two atoms in a molecule have a great difference in their electronegativity; one atom attracts electrons more than the other, becoming more negative, while the other atom becomes more positive. A permanent magnet, such as a bar magnet, owes its magnetism to the magnetic dipole moment of the electron. A molecule with a permanent dipole moment is called a polar molecule. A molecule is polarized when it carries an induced dipole. A non-degenerate (S-state) atom can have only a zero permanent dipole.

</div>
</div>

===Practice Test Problem===

[[File:Exampleprac.jpg|400px]]

==Electric Field of an Electric Dipole==
The electric field of an electric dipole can be constructed as a vector sum of the point charge fields of the two charges. As can be seen in the graphics, the electric field always points towards the negative particle and points away from the positive particle. This is an important characteristic which can be used to determine which end is positive and which is negative in a dipole.

Direction of electric dipole:

[[File:dipd.gif]]

Electric Field:

[[File:edip2.gif]]

In this class, most of the questions with either be on axis or perpendicular to the dipole, so learning the direction of the electric field at these locations is important. This is done by decomposing the two vectors from each charged particle of the dipole, as shown in the positive x-axis location in the picture below. Where the positive particle has an electric field pointing away, and the negative particle electric field points inward.

[[File:Phys2212 dipole electric field.PNG]]

==Torque==
===Derivation===
Consider a dipole with the following arbitrary orientation in a uniform electric field:

[[File:dipole_torque_2.gif]]

Note how the electric field will exert an an electric force on each of the point charges. Since the electric field is uniform and charge of each point charge is equal and opposite, the electric force exerted on each point charge will be equal and opposite. The magnitude of each force is simply the force on a point charge, or <math> F = qE </math>. The component of this force perpendicular to the dipole axis can be written as <math> F_{\bot}= qE\sin \theta</math>, where theta is the angle between the electric field and the dipole. It is this perpendicular force which causes rotational motion, and thus is the force component of the applied torque. Since the forces are separated by the distance of the dipole, it can be generalized that an electric field produces the following torque on an electric dipole:

<math> \tau\ = p \times E = Eqd\sin\theta</math>

===Direction===
The direction of the torque can be found using the right hand rule, as it will always be perpendicular to the dipole axis and applied force. The cross product relationship also indicates that when the dipole is parallel to the electric field, no torque will be acting on it.
[[File:dipole_t.gif]]

It is important to note that since the forces on each of the point charges are equal and opposite, the net force on the dipole in a uniform field is 0, which also explains why there is no linear motion by the dipole. Also, the torque from the electric field will align the orientation of the dipole to be parallel with the electric field. This is because at this point the force applied to each of the point charges is 0 because <math> F = Eqd \sin 0 = 0 </math>.

===Energy and Work===

The torque that rotates the dipole moves the orientation of the dipole from a higher energy configuration to a lower energy configuration as shown below.

[[File:dipole_torque.gif]]

This also means that any rotation done against this energy gradient requires work. The amount of work required to rotate a dipole from its low energy configuration to an angle <math>\theta_0</math> can be derived as:

<math>\int_{0}^{\theta_0}\tau d\theta = \int_{0}^{\theta_0}Eqd\sin\theta d\theta = \int_{0}^{\theta_0}Ep\sin\theta d\theta = Ep(1-\cos\theta_0)</math>

Calculating potential energy is a matter of convention. The standard convention it the potential energy is zero when the dipole is perpendicular to the electric field. With this in mind, the potential energy of a dipole can be calculated as:

<math> U = Ep \cos\theta = -p \dot\ E </math>

Again, this would indicate the potential energy is maximized when <math> \cos\theta = 1 </math>, or when the dipole is in the low energy configuration.

===Nonuniform Electric Field===
When the electric field is not uniform, the two point charges will not feel equal and opposite forces at all times, meaning that the net force on the dipole will not be zero. The force on the dipole will be in the direction of where the electric field has the steepest increase. The derivation for the net force on a dipole in a non-uniform electric field is extremely complex [http://bolvan.ph.utexas.edu/~vadim/classes/17f/dipole.pdf], but can be explained in simple terms. Consider an non-uniform electric field E with a gradient <math>\nabla</math>. For any dipole placed in this system, the difference between the electric field at small intervals between the two point charges can be expanded as a power series to calculate the total difference in electric field. It can be proven that in an ideal dipole, all the subleading terms vanish, and all that is left is the leading term of this series. Thus, the general formula for net force on a dipole in a non-uniform electric field can be written as:

<math> F = ( p \dot\ \nabla ) E(r) </math>

where <math> E(r) </math> is the electric field at the center of the dipole.

==Electric Dipole Concept Map==
This concept map illustrates the other fields and forces caused by the electric dipole.

[[File:dipolecon.gif]]

==Connectedness==
Dipoles are incredibly common in physics, chemistry, and other natural sciences. While not specific to electric dipoles, much of the mathematics taught in advanced algorithms is relevant to the study of dipoles in nature, specifically certain randomized algorithms useful in computer science can be used to effectively simulate and predict natural phenomena having to do with dipole forces and the arrangement of many dipoles.

Dipoles are useful in determining the behavior of certain molecules with each other. Polar molecules can act as electric dipoles, such as the water molecule mentioned earlier. This gives polar molecules certain properties when in a solution. Dipoles are the basis for polarity in molecules, which leads to other important properties such as hydrophilicity, which is very important in industry as well as in your body. The cells in the body are surrounded by a selectively permeable membrane. The outer and inner ends of this membrane are polar, while the middle part is non polar. This polarity is very important in determining which molecules enter and exit the cells in our body and therefore how the cells maintain homeostasis. Dipoles are also very common in regards to magnets, which have several applications including Maglev trains and even metal detectors.

To specifically Biomedical Engineering, dipole interactions can help break down molecules as a result of ion polarization and separation.

==History==

Electric dipoles have been understood since the mid to late 1800s. However, atomic dipoles could only be understood after the discovery of the correct model of the atom by Bohr in 1913. Based on this knowledge, atomic dipoles were used in a lot of technology. Even though electric dipoles are a newer concept, the human understanding of magnetic dipoles goes way back to the ancient Greeks who discovered magnetite, which had magnetic properties.

== See also ==

[http://www.physicsbook.gatech.edu/Magnetic_Dipole Magnetic Dipole]

===External links===

[https://en.wikipedia.org/wiki/Electric_charge Electric Charge]

[https://en.wikibooks.org/wiki/Physics_Exercises/Electrostatics Additional Dipole Derivations]

[https://en.wikipedia.org/wiki/Electric_dipole_moment Electric Dipole Moment]

[https://en.wikipedia.org/wiki/Dipole Dipole]

[http://hyperphysics.phy-astr.gsu.edu/hbase/electric/diptor.html Electric Dipole Torque]

==References==

[http://education.jlab.org/qa/historymag_01.html Magnet History]

[https://en.wikipedia.org/wiki/Bohr_model Bohr Model]

[http://hyperphysics.phy-astr.gsu.edu/hbase/electric/diph2o.html Electric Dipole]

[[Category:Fields]]

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2022-04-23T20:03:43Z

Aramirez81: /* Thermal Energy, Dissipation, and Transfer of Energy */

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* Interactive physics simulations [https://phet.colorado.edu/en/simulations/category/physics PhET]
* OpenStax intro physics textbooks: [https://openstax.org/details/books/university-physics-volume-1 Vol1], [https://openstax.org/details/books/university-physics-volume-2 Vol2], [https://openstax.org/details/books/university-physics-volume-3 Vol3]
* The Open Source Physics project is a collection of online physics resources [http://www.opensourcephysics.org/ OSP]
* A resource guide compiled by the [http://www.aapt.org/ AAPT] for educators [http://www.compadre.org/ ComPADRE]
* The Feynman lectures on physics are free to read [http://www.feynmanlectures.caltech.edu/ Feynman]
* Final Study Guide for Modern Physics II created by a lab TA [https://docs.google.com/document/d/1_6GktDPq5tiNFFYs_ZjgjxBAWVQYaXp_2Imha4_nSyc/edit?usp=sharing Modern Physics II Final Study Guide]

== Resources ==
* Commonly used wiki commands [https://en.wikipedia.org/wiki/Help:Cheatsheet Wiki Cheatsheet]
* A guide to representing equations in math mode [https://en.wikipedia.org/wiki/Help:Displaying_a_formula Wiki Math Mode]
* A page to keep track of all the physics [[Constants]]
* A listing of [[Notable Scientist]] with links to their individual pages

<div style="float:left; width:30%; padding:1%;">

==Physics 1==
===Week 1===
<div class="toccolours mw-collapsible mw-collapsed">
====GlowScript 101====
<div class="mw-collapsible-content">
*[[Python Syntax]]
*[[GlowScript]]
</div>
</div>

<div class="toccolours mw-collapsible mw-collapsed">

====VPython====

<div class="mw-collapsible-content">
*[[VPython]]
*[[VPython basics]]
*[[VPython Common Errors and Troubleshooting]]
*[[VPython Functions]]
*[[VPython Lists]]
*[[VPython Loops]]
*[[VPython Multithreading]]
*[[VPython Animation]]
*[[VPython Objects]]
*[[VPython 3D Objects]]
*[[VPython Reference]]
*[[VPython MapReduceFilter]]
*[[VPython GUIs]]
</div>
</div>

<div class="toccolours mw-collapsible mw-collapsed">

====Vectors and Units====
<div class="mw-collapsible-content">
*[[Vectors]]
*[[SI Units]]
</div>
</div>

<div class="toccolours mw-collapsible mw-collapsed">

====Interactions====
<div class="mw-collapsible-content">
*[[Types of Interactions and How to Detect Them]]
</div>
</div>

<div class="toccolours mw-collapsible mw-collapsed">
====Velocity and Momentum====
<div class="mw-collapsible-content">
*[[Newton's First Law of Motion]]
*[[Mass]]
*[[Velocity]]
*[[Speed]]
*[[Speed vs Velocity]]
*[[Relative Velocity]]
*[[Derivation of Average Velocity]]
*[[2-Dimensional Motion]]
*[[3-Dimensional Position and Motion]]
</div>
</div>

===Week 2===
<div class="toccolours mw-collapsible mw-collapsed">
====Momentum and the Momentum Principle====
<div class="mw-collapsible-content">
*[[Linear Momentum]]
*[[Newton's Second Law: the Momentum Principle]]
*[[Impulse and Momentum]]
*[[Net Force]]
*[[Inertia]]
*[[Acceleration]]
*[[Relativistic Momentum]]

</div>
</div>

<div class="toccolours mw-collapsible mw-collapsed">

====Iterative Prediction with a Constant Force====
<div class="mw-collapsible-content">
*[[Iterative Prediction]]
</div>
</div>

===Week 3===
<div class="toccolours mw-collapsible mw-collapsed">

====Analytic Prediction with a Constant Force====
<div class="mw-collapsible-content">

*[[Kinematics]]
*[[Projectile Motion]]
</div>
</div>

<div class="toccolours mw-collapsible mw-collapsed">

====Iterative Prediction with a Varying Force====
<div class="mw-collapsible-content">
*[[Fundamentals of Iterative Prediction with Varying Force]]
*[[Spring_Force]]
*[[Simple Harmonic Motion]]


*[[Iterative Prediction of Spring-Mass System]]
*[[Terminal Speed]]
*[[Predicting Change in multiple dimensions]]
*[[Two Dimensional Harmonic Motion]]
*[[Determinism]]
</div>
</div>

===Week 4===
<div class="toccolours mw-collapsible mw-collapsed">
====Fundamental Interactions====
<div class="mw-collapsible-content">
*[[Gravitational Force]]
*[[Gravitational Force Near Earth]]
*[[Gravitational Force in Space and Other Applications]]
*[[3 or More Body Interactions]]

*[[Electric Force]]
*[[Introduction to Magnetic Force]]
*[[Strong and Weak Force]]
*[[Reciprocity]]
*[[Conservation of Momentum]]
</div>
</div>

===Week 5===
<div class="toccolours mw-collapsible mw-collapsed">
====Properties of Matter====
<div class="mw-collapsible-content">
*[[Kinds of Matter]]
*[[Ball and Spring Model of Matter]]
*[[Density]]
*[[Length and Stiffness of an Interatomic Bond]]
*[[Young's Modulus]]
*[[Speed of Sound in Solids]]
*[[Malleability]]
*[[Ductility]]
*[[Weight]]
*[[Hardness]]
*[[Boiling Point]]
*[[Melting Point]]
*[[Change of State]]
</div>
</div>

===Week 6===
<div class="toccolours mw-collapsible mw-collapsed">
====Identifying Forces====
<div class="mw-collapsible-content">
*[[Free Body Diagram]]
*[[Inclined Plane]]
*[[Compression or Normal Force]]
*[[Tension]]
</div>
</div>
<div class="toccolours mw-collapsible mw-collapsed">

====Curving Motion====
<div class="mw-collapsible-content">
*[[Curving Motion]]
*[[Centripetal Force and Curving Motion]]
*[[Perpetual Freefall (Orbit)]]
</div>
</div>

===Week 7===
<div class="toccolours mw-collapsible mw-collapsed">
====Energy Principle====
<div class="mw-collapsible-content">
*[[Energy of a Single Particle]]
*[[Kinetic Energy]]
*[[Work/Energy]]
*[[The Energy Principle]]
*[[Conservation of Energy]]
</div>
</div>

===Week 8===
<div class="toccolours mw-collapsible mw-collapsed">
====Work by Non-Constant Forces====
<div class="mw-collapsible-content">
*[[Work Done By A Nonconstant Force]]
</div>
</div>
<div class="toccolours mw-collapsible mw-collapsed">
====Potential Energy====
<div class="mw-collapsible-content">
*[[Potential Energy]]
*[[Potential Energy of Macroscopic Springs]]
*[[Spring Potential Energy]]
*[[Ball and Spring Model]]
*[[Gravitational Potential Energy]]
*[[Energy Graphs]]
*[[Escape Velocity]]
</div>
</div>

===Week 9===
<div class="toccolours mw-collapsible mw-collapsed">
====Multiparticle Systems====
<div class="mw-collapsible-content">
*[[Center of Mass]]
*[[Multi-particle analysis of Momentum]]
*[[Potential Energy of a Multiparticle System]]
*[[Work and Energy for an Extended System]]
*[[Internal Energy]]
**[[Potential Energy of a Pair of Neutral Atoms]]
</div>
</div>

===Week 10===
<div class="toccolours mw-collapsible mw-collapsed">
====Choice of System====
<div class="mw-collapsible-content">
*[[System & Surroundings]]
</div>
</div>
<div class="toccolours mw-collapsible mw-collapsed">
====Thermal Energy, Dissipation, and Transfer of Energy====
<div class="mw-collapsible-content">
*[[Thermal Energy]]
*[[Specific Heat]]
*[[Calorific Value(Heat of combustion)]]
*[[First Law of Thermodynamics]]
*[[Second Law of Thermodynamics and Entropy]]
*[[Temperature]]
*[[Transformation of Energy]]
*[[The Maxwell-Boltzmann Distribution]]
*[[Air Resistance]]
*[[The Third Law of Thermodynamics]]
</div>
</div>
<div class="toccolours mw-collapsible mw-collapsed">

====Rotational and Vibrational Energy====
<div class="mw-collapsible-content">
*[[Translational, Rotational and Vibrational Energy]]
*[[Rolling Motion]]
</div>
</div>

===Week 11===
<div class="toccolours mw-collapsible mw-collapsed">
====Different Models of a System====
<div class="mw-collapsible-content">
*[[Point Particle Systems]]
*[[Real Systems]]
</div>
</div>
<div class="toccolours mw-collapsible mw-collapsed">
====Friction====
<div class="mw-collapsible-content">
*[[Friction]]
*[[Static Friction]]
*[[Kinetic Friction]]
</div>
</div>

===Week 12===
<div class="toccolours mw-collapsible mw-collapsed">
====Conservation of Momentum====
<div class="mw-collapsible-content">
*[[Conservation of Momentum]]
</div>
</div>
<div class="toccolours mw-collapsible mw-collapsed">
====Collisions====
<div class="mw-collapsible-content">
*[[Newton's Third Law of Motion]]
*[[Collisions]]
*[[Elastic Collisions]]
*[[Inelastic Collisions]]
*[[Maximally Inelastic Collision]]
*[[Head-on Collision of Equal Masses]]
*[[Head-on Collision of Unequal Masses]]
*[[Scattering: Collisions in 2D and 3D]]
*[[Rutherford Experiment and Atomic Collisions]]
*[[Coefficient of Restitution]]
</div>
</div>

===Week 13===
<div class="toccolours mw-collapsible mw-collapsed">
====Rotations====
<div class="mw-collapsible-content">
*[[Rotational Kinematics]]
*[[Eulerian Angles]]
</div>
</div>
<div class="toccolours mw-collapsible mw-collapsed">

====Angular Momentum====
<div class="mw-collapsible-content">
*[[Total Angular Momentum]]
*[[Translational Angular Momentum]]
*[[Rotational Angular Momentum]]
*[[The Angular Momentum Principle]]
*[[Angular Impulse]]
*[[Predicting the Position of a Rotating System]]
*[[The Moments of Inertia]]
*[[Right Hand Rule]]
</div>
</div>

===Week 14===
<div class="toccolours mw-collapsible mw-collapsed">
====Analyzing Motion with and without Torque====
<div class="mw-collapsible-content">
*[[Torque]]
*[[Torque 2]]
*[[Systems with Zero Torque]]
*[[Systems with Nonzero Torque]]
*[[Torque vs Work]]
*[[Gyroscopes]]
</div>
</div>

===Week 15===
<div class="toccolours mw-collapsible mw-collapsed">
====Introduction to Quantum Concepts====
<div class="mw-collapsible-content">
*[[Bohr Model]]
*[[Energy graphs and the Bohr model]]
*[[Quantized energy levels]]
*[[Electron transitions]]
*[[Entropy]]
</div>
</div>
</div>

<div style="float:left; width:30%; padding:1%;">

==Physics 2==
===Week 1===
<div class="toccolours mw-collapsible mw-collapsed">
====3D Vectors====

<div class="mw-collapsible-content">
*[[Vectors]]
*[[Right-Hand Rule]]
*[[Right Hand Rule]]
</div>
</div>

<div class="toccolours mw-collapsible mw-collapsed">

====Electric field====
<div class="mw-collapsible-content">
*[[Electric Field]]
*[[Electric Field and Electric Potential]]
</div>
</div>

<div class="toccolours mw-collapsible mw-collapsed">

====Electric force====
<div class="mw-collapsible-content">
*[[Electric Force]]
*[[Lorentz Force]]
</div>
</div>

<div class="toccolours mw-collapsible mw-collapsed">

====Electric field of a point particle====
<div class="mw-collapsible-content">
*[[Point Charge]]
</div>
</div>

<div class="toccolours mw-collapsible mw-collapsed">

====Superposition====
<div class="mw-collapsible-content">
*[[Superposition Principle]]
*[[Superposition principle]]
</div>
</div>

<div class="toccolours mw-collapsible mw-collapsed">
====Dipoles====
<div class="mw-collapsible-content">
*[[Electric Dipole]]
*[[Magnetic Dipole]]
</div>
</div>

===Week 2===
<div class="toccolours mw-collapsible mw-collapsed">
====Interactions of charged objects====
<div class="mw-collapsible-content">
*[[Electric Field]]
*[[Electric Potential]]
*[[Electric Force]]
*[[Lorentz Force]]
</div>
</div>

<div class="toccolours mw-collapsible mw-collapsed">
====Tape experiments====
<div class="mw-collapsible-content">
*[[Polarization]]
*[[Electric Polarization]]
</div>
</div>

<div class="toccolours mw-collapsible mw-collapsed">
====Polarization====
<div class="mw-collapsible-content">
*[[Polarization]]
*[[Electric Polarization]]
*[[Polarization of an Atom]]
</div>
</div>

===Week 3===
<div class="toccolours mw-collapsible mw-collapsed">
====Conductors and Insulators====
<div class="mw-collapsible-content">
*[[Conductivity and Resistivity]]
*[[Insulators]]
*[[Potential Difference in an Insulator]]
*[[Conductors]]
*[[Polarization of a conductor]]
</div>
</div>

<div class="toccolours mw-collapsible mw-collapsed">

====Charging and Discharging====
<div class="mw-collapsible-content">
*[[Charge Transfer]]
*[[Electrostatic Discharge]]
*[[Charged Conductor and Charged Insulator]]
</div>
</div>

===Week 4===
<div class="toccolours mw-collapsible mw-collapsed">
====Field of a charged rod====
<div class="mw-collapsible-content">
*[[Field of a Charged Rod|Charged Rod]]
</div>
</div>

<div class="toccolours mw-collapsible mw-collapsed">

====Field of a charged ring/disk/capacitor====
<div class="mw-collapsible-content">
*[[Charged Ring]]
*[[Charged Disk]]
*[[Charged Capacitor]]
</div>
</div>

<div class="toccolours mw-collapsible mw-collapsed">
====Field of a charged sphere====
<div class="mw-collapsible-content">
*[[Charged Spherical Shell]]
*[[Field of a Charged Ball]]
</div>
</div>

===Week 5===
<div class="toccolours mw-collapsible mw-collapsed">
====Potential energy====
<div class="mw-collapsible-content">
*[[Potential Energy]]
</div>
</div>

<div class="toccolours mw-collapsible mw-collapsed">

====Electric potential====
<div class="mw-collapsible-content">
*[[Electric Potential]]
*[[Path Independence of Electric Potential]]
*[[Potential Difference Path Independence, claimed by Aditya Mohile]]
*[[Potential Difference in a Uniform Field]]
*[[Potential Difference of Point Charge in a Non-Uniform Field]]
</div>
</div>
<div class="toccolours mw-collapsible mw-collapsed">
====Sign of a potential difference====
<div class="mw-collapsible-content">
*[[Sign of a Potential Difference]]
</div>
</div>
<div class="toccolours mw-collapsible mw-collapsed">
====Potential at a single location====
<div class="mw-collapsible-content">
*[[Electric Potential]]
*[[Potential Difference at One Location]]
</div>
</div>

<div class="toccolours mw-collapsible mw-collapsed">

====Path independence and round trip potential====
<div class="mw-collapsible-content">
*[[Path Independence of Electric Potential]]
*[[Potential Difference Path Independence, claimed by Aditya Mohile]]
</div>
</div>

===Week 6===
<div class="toccolours mw-collapsible mw-collapsed">
====Electric field and potential in an insulator====
<div class="mw-collapsible-content">
*[[Potential Difference in an Insulator]]
*[[Electric Field in an Insulator]]
</div>
</div>
<div class="toccolours mw-collapsible mw-collapsed">
====Moving charges in a magnetic field====
<div class="mw-collapsible-content">
*[[Magnetic Field]]
*[[Magnetic Force]]
*[[Lorentz Force]]
</div>
</div>
<div class="toccolours mw-collapsible mw-collapsed">
====Biot-Savart Law====
<div class="mw-collapsible-content">
*[[Biot-Savart Law]]
*[[Biot-Savart Law for Currents]]
</div>
</div>
<div class="toccolours mw-collapsible mw-collapsed">

====Moving charges, electron current, and conventional current====
<div class="mw-collapsible-content">
*[[Moving Point Charge]]
*[[Current]]
</div>
</div>

===Week 7===
<div class="toccolours mw-collapsible mw-collapsed">
====Magnetic field of a wire====
<div class="mw-collapsible-content">
*[[Magnetic Field of a Long Straight Wire]]
</div>
</div>

<div class="toccolours mw-collapsible mw-collapsed">

====Magnetic field of a current-carrying loop====
<div class="mw-collapsible-content">
*[[Magnetic Field of a Loop]]
</div>
</div>

<div class="toccolours mw-collapsible mw-collapsed">
====Magnetic field of a Charged Disk====
<div class="mw-collapsible-content">
*[[Magnetic Field of a Disk]]
</div>
</div>

<div class="toccolours mw-collapsible mw-collapsed">
====Magnetic dipoles====
<div class="mw-collapsible-content">
*[[Magnetic Dipole Moment]]
*[[Bar Magnet]]
</div>
</div>

<div class="toccolours mw-collapsible mw-collapsed">
====Atomic structure of magnets====
<div class="mw-collapsible-content">
*[[Atomic Structure of Magnets]]
</div>
</div>

===Week 8===
<div class="toccolours mw-collapsible mw-collapsed">
====Steady state current====
<div class="mw-collapsible-content">
*[[Steady State]]
*[[Non Steady State]]
</div>
</div>

<div class="toccolours mw-collapsible mw-collapsed">
====Kirchoff's Laws====
<div class="mw-collapsible-content">
*[[Kirchoff's Laws]]
</div>
</div>

<div class="toccolours mw-collapsible mw-collapsed">
====Electric fields and energy in circuits====
<div class="mw-collapsible-content">
*[[Electric Potential Difference]]
</div>
</div>

<div class="toccolours mw-collapsible mw-collapsed">

====Macroscopic analysis of circuits====
<div class="mw-collapsible-content">
*[[Series Circuits]]
*[[Parallel Circuits]]
*[[Parallel Circuits vs. Series Circuits*]]
*[[Loop Rule]]
*[[Node Rule]]
*[[Fundamentals of Resistance]]
*[[Problem Solving]]
</div>
</div>

===Week 9===
<div class="toccolours mw-collapsible mw-collapsed">
====Electric field and potential in circuits with capacitors====
<div class="mw-collapsible-content">
*[[Charging and Discharging a Capacitor]]
*[[RC Circuit]]
*[[R Circuit]]
*[[AC and DC]]
</div>
</div>

<div class="toccolours mw-collapsible mw-collapsed">
====Magnetic forces on charges and currents====
<div class="mw-collapsible-content">
*[[Magnetic Force]]
*[[Lorentz Force]]
*[[Motors and Generators]]
*[[Applying Magnetic Force to Currents]]
*[[Magnetic Force in a Moving Reference Frame]]
*[[Right-Hand Rule]]
*[[Analysis of Railgun vs Coil gun technologies]]
</div>
</div>

<div class="toccolours mw-collapsible mw-collapsed">

====Electric and magnetic forces====
<div class="mw-collapsible-content">
*[[Electric Force]]
*[[Magnetic Force]]
*[[Lorentz Force]]
*[[VPython Modelling of Electric and Magnetic Forces]]
</div>
</div>

<div class="toccolours mw-collapsible mw-collapsed">

====Velocity selector====
<div class="mw-collapsible-content">
*[[Lorentz Force]]
*[[Combining Electric and Magnetic Forces]]
</div>
</div>

===Week 10===
<div class="toccolours mw-collapsible mw-collapsed">

====Hall Effect====
<div class="mw-collapsible-content">
*[[Hall Effect]]
<h1><strong>Alayna Baker Spring 2020</strong></h1>
[[File:Hall Effect 1.jpg]]
[[File:Hall Effect 2.jpg]]

*[[Right-Hand Rule]]
*[[Motional Emf]]
*[[Magnetic Force]]
*[[Magnetic Torque]]
</div>
</div>

<div class="toccolours mw-collapsible mw-collapsed">

]]]====Motional EMF====

<div class="mw-collapsible-content">
*[[Motional Emf]]
<h1><strong>Adeline Boswell Fall 2019</strong></h1>
[[File:Motional EMF Example.jpg]]

*[[Motional Emf using Faraday's Law]]
</div>
</div>http://www.physicsbook.gatech.edu/Special:RecentChangesLinked/Main_Page

<div class="toccolours mw-collapsible mw-collapsed">

If you have a bar attached to two rails, and the rails are connected by a resistor, you have effectively created a circuit. As the bar moves, it creates an "electromotive force"

[[File:MotEMFCR.jpg]]

====Magnetic force====
<div class="mw-collapsible-content">
*[[Magnetic Force]]
*[[Lorentz Force]]
</div>
</div>

<div class="toccolours mw-collapsible mw-collapsed">
====Magnetic torque====
<div class="mw-collapsible-content">
*[[Magnetic Torque]]
*[[Right-Hand Rule]]
</div>
</div>

===Week 12===

<div class="toccolours mw-collapsible mw-collapsed">
====Gauss's Law====
<div class="mw-collapsible-content">
*[[Gauss's Flux Theorem]]
*[[Gauss's Law]]
*[[Magnetic Flux]]
</div>
</div>

<div class="toccolours mw-collapsible mw-collapsed">
====Ampere's Law====
<div class="mw-collapsible-content">
*[[Ampere's Law]]
*[[Ampere-Maxwell Law]]
*[[Magnetic Field of Coaxial Cable Using Ampere's Law]]
*[[Magnetic Field of a Long Thick Wire Using Ampere's Law]]
*[[Magnetic Field of a Toroid Using Ampere's Law]]
*[[Magnetic Field of a Solenoid Using Ampere's Law]]
*[[The Differential Form of Ampere's Law]]
</div>
</div>

===Week 13===
<div class="toccolours mw-collapsible mw-collapsed">
====Semiconductors====
<div class="mw-collapsible-content">
*[[Semiconductor Devices]]
</div>
</div>

<div class="toccolours mw-collapsible mw-collapsed">
====Faraday's Law====
<div class="mw-collapsible-content">
*[[Faraday's Law]]
*[[Motional Emf using Faraday's Law]]
*[[Lenz's Law]]

</div>
</div>

<div class="toccolours mw-collapsible mw-collapsed">

====Maxwell's equations====
<div class="mw-collapsible-content">
*[[Gauss's Law]]
*[[Magnetic Flux]]
*[[Ampere's Law]]
*[[Faraday's Law]]
*[[Maxwell's Electromagnetic Theory]]
</div>
</div>

===Week 14===
<div class="toccolours mw-collapsible mw-collapsed">
====Circuits revisited====
<div class="mw-collapsible-content">

</div>
</div>

<div class="toccolours mw-collapsible mw-collapsed">

====Inductors====
<div class="mw-collapsible-content">
*[[Inductors]]
*[[Current in an LC Circuit]]
*[[Current in an RL Circuit]]
</div>
</div>

===Week 15===
<div class="toccolours mw-collapsible mw-collapsed">
==== Electromagnetic Radiation ====
<div class="mw-collapsible-content">
*[[Electromagnetic Radiation]]
</div>
</div>

<div class="toccolours mw-collapsible mw-collapsed">
====Sparks in the air====
<div class="mw-collapsible-content">
*[[Sparks in Air]]
*[[Spark Plugs]]
</div>
</div>

<div class="toccolours mw-collapsible mw-collapsed">
====Superconductors====
<div class="mw-collapsible-content">
*[[Superconducters]]
*[[Superconductors]]
*[[Meissner effect]]
</div>
</div>
</div>

<div style="float:left; width:30%; padding:1%;">

==Physics 3==

===Week 1===
<div class="toccolours mw-collapsible mw-collapsed">
====Classical Physics====
<div class="mw-collapsible-content">
</div>
</div>

===Week 2===
<div class="toccolours mw-collapsible mw-collapsed">
====Special Relativity====
<div class="mw-collapsible-content">
*[[Frame of Reference]]
*[[Einstein's Theory of Special Relativity]]
*[[Time Dilation]]
*[[Einstein's Theory of General Relativity]]
*[[Albert A. Micheleson & Edward W. Morley]]
*[[Magnetic Force in a Moving Reference Frame]]
</div>
</div>

===Week 3===
<div class="toccolours mw-collapsible mw-collapsed">
====Photons====
<div class="mw-collapsible-content">
*[[Spontaneous Photon Emission]]
*[[Light Scattering: Why is the Sky Blue]]
*[[Lasers]]
*[[Electronic Energy Levels and Photons]]
*[[Quantum Properties of Light]]
*[[The Photoelectric Effect]]
</div>
</div>

===Week 4===
<div class="toccolours mw-collapsible mw-collapsed">
====Matter Waves====
<div class="mw-collapsible-content">
*[[Wave-Particle Duality]]
*[[Particle in a 1-Dimensional box]]
*[[Heisenberg Uncertainty Principle]]
</div>
</div>

<div class="toccolours mw-collapsible mw-collapsed">

====Schrödinger Equation====
<div class="mw-collapsible-content">
*[[Solution for a Single Free Particle]]
<div class="mw-collapsible-content">
*[[Solution for a Single Particle in an Infinite Quantum Well - Darin]]
<div class="mw-collapsible-content"></div>
*[[Solution for a Single Particle in a Semi-Infinite Quantum Well]]

===Week 5===
<div class="toccolours mw-collapsible mw-collapsed">
====Wave Mechanics====
<div class="mw-collapsible-content">
*[[Standing Waves]]
*[[Wavelength]]
*[[Wavelength and Frequency]]
*[[Mechanical Waves]]
*[[Transverse and Longitudinal Waves]]
</div>
</div>

<div class="toccolours mw-collapsible mw-collapsed">
====Quantum Mechanics====
<div class="mw-collapsible-content">
*[[Quantum Tunneling through Potential Barriers]]
</div>
</div>

===Week 6===
<div class="toccolours mw-collapsible mw-collapsed">
====Rutherford-Bohr Model====
<div class="mw-collapsible-content">
*[[Rutherford Experiment and Atomic Collisions]]
*[[Bohr Model]]
*[[Quantized energy levels]]
*[[Energy graphs and the Bohr model]]
</div>
</div>

===Week 7===
<div class="toccolours mw-collapsible mw-collapsed">
====The Hydrogen Atom====
<div class="mw-collapsible-content">
*[[Quantum Theory]]
*[[Atomic Theory]]
</div>
</div>

===Week 8===
<div class="toccolours mw-collapsible mw-collapsed">
====Many-Electron Atoms====
<div class="mw-collapsible-content">
*[[Quantum Theory]]
*[[Atomic Theory]]
*[[Pauli exclusion principle]]
</div>
</div>

===Week 9===
<div class="toccolours mw-collapsible mw-collapsed">
====Molecules====
<div class="mw-collapsible-content">
</div>
*[[Molecules]]
</div>

===Week 10===
<div class="toccolours mw-collapsible mw-collapsed">
====Statistical Physics====
<div class="mw-collapsible-content">
</div>
*[[Application of Statistics in Physics]]
*[[Temperature & Entropy]]
</div>
<div class="mw-collapsible-content">

===Week 11===
<div class="toccolours mw-collapsible mw-collapsed">
====Condensed Matter Physics====
<div class="mw-collapsible-content">
</div>
</div>

===Week 12===
<div class="toccolours mw-collapsible mw-collapsed">
====The Nucleus====
<div class="mw-collapsible-content">
*[[Nucleus]]
</div>
</div>

===Week 13===
<div class="toccolours mw-collapsible mw-collapsed">
====Nuclear Physics====
<div class="mw-collapsible-content">
*[[Nuclear Fission]]
*[[Nuclear Energy from Fission and Fusion]]
</div>
</div>

===Week 14===
<div class="toccolours mw-collapsible mw-collapsed">
====Particle Physics====
<div class="mw-collapsible-content">
*[[Elementary Particles and Particle Physics Theory]]
*[[String Theory]]
</div>
</div>
</div>

Electric Dipole

2022-04-15T03:21:03Z

Aramirez81:

CLAIMED BY ANTHONY RAMIREZ (Spring 2022)
==Summary==

An electric dipole is made up of two point charges that have equal but opposite electric charges (q) and are separated by a short distance (d).

[[File:dipo.jpg|An Electric Dipole]]

The electric field is proportional to the cube of the distance from the dipole, and is dependent on whether you’re moving along the line separating the two charges or perpendicular to it.

A dipole can be created, for example, when you place a neutral atom in an electric field, because of the movement of electrons, the atom polarizes (negative charge on one side, positive charge on the other) and yields a separation of charge.

A prime example of this is the molecule for water, which forms a 105 degree angle between the two hydrogens connected to the oxygen. Since the oxygen has a greater electronegativity, it pulls more strongly on the electrons shared by the oxygen and hydrogen atoms and that end of the molecule becomes more negatively charged compared to the hydrogen end. Therefore, the net electric dipole points towards the oxygen atom.

[[File:Water.png|300px|thumb|Dipole moment of water]]

==Mathematical Models==

===An Exact Model===
[[File:Phys2212 dipole image.PNG|300px|thumb|Polarization by an electric field]]
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.

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>.

<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})^2+p_y^2}}</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})^2+p_y^2}}</math>, where once again <math>\sqrt{(p_x + \frac{d}{2})^2+p_y^2} = |\vec r_-|</math>.

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 general formula for electric field strength from a [[Point Charge]] is <math>E = \frac{1}{4\pi\epsilon_0} \frac{q}{|\vec r|^2} \hat r</math>. Then <math>|E_+| = \frac{1}{4\pi\epsilon_0} \frac{q_+}{|\vec r_+|^2}</math> and <math>|E_-| = \frac{1}{4\pi\epsilon_0} \frac{q_-}{|\vec r_-|^2}</math>. We want solely the magnitude in this case because we can calculate direction and component forces using sin and cosine. Its worth noting that we can expand <math>r_+, r_-</math> to the form in the denominator of the sine and cosine. We will use this later.

First we calculate <math>E_{net_y}</math>. <math>E_{net_y} = E_{+_y} + E_{-_y} = \frac{1}{4\pi\epsilon_0} \frac{q_+}{|\vec r_+|^2} sin(\theta_+) + \frac{1}{4\pi\epsilon_0} \frac{q_-}{|\vec r_-|^2} sin(\theta_-)</math>.

Then we combine some terms, noting that <math> q_+ = -q_-</math>. <math>E_{net_y} = \frac{q_+}{4\pi\epsilon_0} * \Bigg(\frac{1}{|\vec r_+|^2}sin(\theta_+) + \frac{-1}{|\vec r_-|^2}sin(\theta_-)\Bigg)</math>

Now it gets ugly, we expand our radii and sines. To recap, <math>sin(\theta_+) = \frac{p_y}{\sqrt{(p_x - \frac{d}{2})^2+p_y^2}}</math>, <math>sin(\theta_-) = \frac{p_y}{\sqrt{(p_x + \frac{d}{2})^2+p_y^2}}</math>, <math>|r_+| = \sqrt{(p_x - \frac{d}{2})^2 +p_y^2}</math> and <math>|r_-| = \sqrt{(p_x + \frac{d}{2})^2 +p_y^2}</math>, giving us

<math>E_{net_y} =
\frac{q_+}{4\pi\epsilon_0} *
\Bigg(
\frac{1}{
(p_x - \frac{d}{2})^2 +p_y^2
}
\frac{p_y}{\sqrt{(p_x - \frac{d}{2})^2+p_y^2}} +
\frac{-1}{
(p_x + \frac{d}{2})^2 +p_y^2
}
\frac{p_y}{\sqrt{(p_x + \frac{d}{2})^2+p_y^2}}
\Bigg)</math>

Finally we can combine more terms, the denominators of the expanded sines are the square roots of the radii. We can also pull out the negative sign.

<math>E_{net_y} =
\frac{q_+}{4\pi\epsilon_0}
\Bigg(
\frac{p_y}{
\Big((p_x - \frac{d}{2})^2 +p_y^2 \Big)^\frac{3}{2}
}
-
\frac{p_y}{
\Big((p_x + \frac{d}{2})^2 +p_y^2 \Big)^\frac{3}{2}
}

\Bigg)</math> That's as simplified as possible.

Much of the derivation for the x direction is similar. The major difference is that instead of calculating the sine, opposite over hypotenuse, we want cosine, adjacent over hypotenuse. That is, where <math>sin(\theta_+) = \frac{p_y}{\sqrt{(p_x - \frac{d}{2})^2+p_y^2}}</math>, <math>cos(\theta_+) = \frac{p_x - \frac{d}{2}}{\sqrt{(p_x - \frac{d}{2})^2+p_y^2}}</math>. By using this and its counterpart for <math>\theta_-</math>, the result is that

<math>E_{net_x} =
\frac{q_+}{4\pi\epsilon_0}
\Bigg(
\frac{p_x - \frac{d}{2}}{
\Big((p_x - \frac{d}{2})^2 +p_y^2 \Big)^\frac{3}{2}
}
-
\frac{p_x + \frac{d}{2}}{
\Big((p_x + \frac{d}{2})^2 +p_y^2 \Big)^\frac{3}{2}
}

\Bigg)</math>. These provide exact formulae for the electric field due to an electric dipole anywhere on the two-dimensional plane, and they translate easily into 3-dimensions.

==Special Cases==
We can simplify the solution for many cases

===On the Parallel Axis===
On the parallel axis, we begin with the now known formula <math>E_{net_x} =
\frac{q_+}{4\pi\epsilon_0}
\Bigg(
\frac{p_x - \frac{d}{2}}{
\Big((p_x - \frac{d}{2})^2 +p_y^2 \Big)^\frac{3}{2}
}
-
\frac{p_x + \frac{d}{2}}{
\Big((p_x + \frac{d}{2})^2 +p_y^2 \Big)^\frac{3}{2}
}

\Bigg)</math>. Since we are on the parallel axis, we know that <math>E_{net_y} = 0</math>, and <math>p_y = 0</math>.

Simplifies to

<math>E_{net_x} =
\frac{q_+}{4\pi\epsilon_0}
\Bigg(
\frac{p_x - \frac{d}{2}}{
\Big((p_x - \frac{d}{2})^2 \Big)^\frac{3}{2}
}
-
\frac{p_x + \frac{d}{2}}{
\Big((p_x + \frac{d}{2})^2 \Big)^\frac{3}{2}
}

\Bigg)</math>.

Then, combining exponents and reducing the fraction:
<math>E_{net_x} =
\frac{q_+}{4\pi\epsilon_0}
\Bigg(
\frac{1}{
(p_x - \frac{d}{2})^2
}
-
\frac{1}{
(p_x + \frac{d}{2})^2
}

\Bigg)</math>.

Then, we can combine these fractions. to simplify the calculations, replace <math>\frac{d}{2}</math> with <math>a</math>.

<math>E_{net_x} =
\frac{q_+}{4\pi\epsilon_0}
\Bigg(
\frac{1}{
(p_x - a)^2
}
-
\frac{1}{
(p_x + a)^2
}

\Bigg) =

\frac{q_+}{4\pi\epsilon_0}
\Bigg(\frac{4p_x a}{(p_x^2 + a^2)^2}
\Bigg)

=

\frac{q_+ 4 a}{4\pi\epsilon_0}
\Bigg(\frac{p_x}{(p_x^2 + a^2)^2}
\Bigg)
</math>.

This is the formula. When <math>p_x >> a</math>, we can assume that <math>p_x^2 + a^2</math> is very close to <math>p_x^2</math>. Then

<math>E_{net_x} \approx
\frac{q_+ 4 a}{4\pi\epsilon_0}
\Bigg(\frac{p_x}{(p_x^2)^2}
\Bigg) =

\frac{q_+ 4 a}{4\pi\epsilon_0}
\Bigg(\frac{p_x}{p_x^4}
\Bigg)
=
\frac{1}{4\pi\epsilon_0}
\Bigg(\frac{4 a q_+}{p_x^3}
\Bigg)</math>

===On the Perpendicular Axis===
We can do a similar simplification for the perpendicular axis. We know that <math>E_{net_y} = 0</math> because the vertical forces from both point charges cancel, leaving only horizontal forces.

<math>
E_{net_x} =
\frac{q_+}{4\pi\epsilon_0}
\Bigg(
\frac{p_x - \frac{d}{2}}{
\Big((p_x - \frac{d}{2})^2 +p_y^2 \Big)^\frac{3}{2}
}
-
\frac{p_x + \frac{d}{2}}{
\Big((p_x + \frac{d}{2})^2 +p_y^2 \Big)^\frac{3}{2}
}

\Bigg)</math>

In this case though, <math>p_x = 0</math>

<math>
E_{net_x} =
\frac{q_+}{4\pi\epsilon_0}
\Bigg(
\frac{- \frac{d}{2}}{
\Big(( - \frac{d}{2})^2 +p_y^2 \Big)^\frac{3}{2}
}
-
\frac{\frac{d}{2}}{
\Big((\frac{d}{2})^2 +p_y^2 \Big)^\frac{3}{2}
}

\Bigg)</math>

Once again, we say <math>a = \frac{d}{2}</math>.

<math>
E_{net_x} =
\frac{q_+}{4\pi\epsilon_0}
\Bigg(
\frac{-a}{
\Big(( - a)^2 +p_y^2 \Big)^\frac{3}{2}
}
-
\frac{a}{
\Big(a^2 +p_y^2 \Big)^\frac{3}{2}
}

\Bigg)
=
\frac{q_+}{4\pi\epsilon_0}
\Bigg(
\frac{-a}{
\Big(a^2 +p_y^2 \Big)^\frac{3}{2}
}
-
\frac{a}{
\Big(a^2 +p_y^2 \Big)^\frac{3}{2}
}

\Bigg)
=\frac{q_+}{4\pi\epsilon_0}
\Bigg(
\frac{-2a}{
\Big(a^2 +p_y^2 \Big)^\frac{3}{2}
}
\Bigg)
</math>

And this is our result.

Once again, when <math>d</math> is much smaller than <math> p_y</math>, <math>a</math> is also small, so we can assume that the denominator is just <math>p_y</math>. This allows us to simplify the resulting equation to

<math>E_{net_x} \approx \frac{q_+}{4\pi\epsilon_0} \frac{-2a}{p_y^3} </math>

==Examples==

===Simple===

A dipole is located at the origin, and is composed of charged particles with charge <math>+e</math> and <math>-e</math>, separated by a distance <math>9 \times10^{-10}</math> along the <math>y</math> axis. The <math>+e</math> charge is on the <math>+y</math> axis. Calculate the force on a proton due to this dipole at a location <math>< 0, 0, 3 \times 10^{-8} ></math> meters.

<div class="toccolours mw-collapsible mw-collapsed">
===Click for Solution===
<div class="mw-collapsible-content">
The center of the dipole is at the origin and there is a proton along the z axis. In this case, we apply the perpendicular from of the electric field equation. In this case, since <math>r >> d</math>, we can also use an approximate solution. Therefore, we apply the formula <math>E_{net} = \frac{q}{4\pi\epsilon_0} \frac{-2a}{r^3}</math>. Since <math>a = \frac{d}{2}</math>, and r is the distance to the proton, we can plug in the values and solve for the net electric field.

[[File:Phys2212 sample simple.PNG | 300px]]

<math>1.6\times 10^{-19} \times 9 \times 10^9
\frac{-9 \times 10^{-10}}
{3 \times 10^{-8^3}} = -48000 \frac{N}{C}</math> on the y axis, as a vector: <math><0, -48000, 0></math>.

However, we aren't done since we want to know the force. We know that <math>F = qE</math> and in this case, both <math>q</math>, the charge on the proton and <math>E</math>, the electric field, are known. Thus the solution is <math>-48000 \times 1.6 \times 10^{-19} = -7.68 \times 10^{-15}</math> on the y axis, or <math><0, -7.68 \times 10^{-15}, 0></math>.
</div>
</div>

===Middling===
A ball of mass <math>M</math> and radius <math>R</math> is given an unknown negative charge spread uniformly over its surface. The ball is hanging from a thread and can move freely. A distance <math>L</math> directly below the center of the ball, a small permanent dipole is oriented such that the dipole axis is parallel with the center of the ball. The dipole has a dipole moment <math>p = qs</math>, with a distance <math>s</math> between the positive and negative charges of the dipole, and a mass <math>m</math>. The positive charge of the dipole is oriented closer to the center of the ball.

a) calculate the charge on the ball to levitate the dipole

b) the dipole is turned 90 degrees clockwise, without changing its position relative to the ball, what effect does this have on the ball?

<div class="toccolours mw-collapsible mw-collapsed">

===Click for Solutions===
<div class="mw-collapsible-content">
a) Because the dipole is small, we can assume that <math> s << L </math>. We wish to find the force on the dipole such that it can equal the force due to gravity. Once again, <math>F = qE</math> since by newton's third law, for a force exerted on the ball by the dipole, there is an equal and opposite for exerted on the dipole by the ball. That is <math>F_G = F_E</math>, so <math>qE = mg</math> (where <math>g</math> is the acceleration due to gravity). Therefore, in this case we wish to find the force on the ball, meaning the electric field from the dipole and the charge on the ball, <math>Q</math>. The field from the dipole is, since we are on the parallel axis, <math>E = \frac{1}{4\pi\epsilon_0} \frac{2p}{L^3}</math>. Putting this together, we get <math>mg = |Q| \frac{1}{4\pi\epsilon_0} \frac{2p}{L^3}</math>

Solving for <math>|Q|</math>: <math>|Q| = \Bigg(\frac{1}{4\pi\epsilon_0}\Bigg)^{-1} \frac{mgL^3}{2p}</math>.

However, we know that since the positive charge of the dipole is closer to the ball, the charge on the ball must be negative to create an attractive force. <math>|Q| > 0</math>, so our final answer is <math>Q = -\Bigg(\frac{1}{4\pi\epsilon_0}\Bigg)^{-1} \frac{mgL^3}{2p}</math>

b) By rotating the dipole clockwise the direction of the electric field at the location of the ball changes. Since the positive end of the dipole is to the right, and the negative end to the left of the dipole, the electric field from the dipole acting on the ball is oriented toward the left. However, since the ball has negative charge, this results in a force on the ball to the right.
</div>
</div>

===Concept Question===
Is it possible for a permanent electric dipole to have a net (total) charge of zero?

<div class="toccolours mw-collapsible mw-collapsed">

===Click for Solution===
<div class="mw-collapsible-content">
Permanent dipoles occur when two atoms in a molecule have a great difference in their electronegativity; one atom attracts electrons more than the other, becoming more negative, while the other atom becomes more positive. A permanent magnet, such as a bar magnet, owes its magnetism to the magnetic dipole moment of the electron. A molecule with a permanent dipole moment is called a polar molecule. A molecule is polarized when it carries an induced dipole. A non-degenerate (S-state) atom can have only a zero permanent dipole.

</div>
</div>

===Practice Test Problem===

[[File:Exampleprac.jpg|400px]]

==Electric Field of an Electric Dipole==
The electric field of an electric dipole can be constructed as a vector sum of the point charge fields of the two charges. As can be seen in the graphics, the electric field always points towards the negative particle and points away from the positive particle. This is an important characteristic which can be used to determine which end is positive and which is negative in a dipole.

Direction of electric dipole:

[[File:dipd.gif]]

Electric Field:

[[File:edip2.gif]]

In this class, most of the questions with either be on axis or perpendicular to the dipole, so learning the direction of the electric field at these locations is important. This is done by decomposing the two vectors from each charged particle of the dipole, as shown in the positive x-axis location in the picture below. Where the positive particle has an electric field pointing away, and the negative particle electric field points inward.

[[File:Phys2212 dipole electric field.PNG]]

==Torque==
===Derivation===
Consider a dipole with the following arbitrary orientation in a uniform electric field:

[[File:dipole_torque_2.gif]]

Note how the electric field will exert an an electric force on each of the point charges. Since the electric field is uniform and charge of each point charge is equal and opposite, the electric force exerted on each point charge will be equal and opposite. The magnitude of each force is simply the force on a point charge, or <math> F = qE </math>. The component of this force perpendicular to the dipole axis can be written as <math> F_{\bot}= qE\sin \theta</math>, where theta is the angle between the electric field and the dipole. It is this perpendicular force which causes rotational motion, and thus is the force component of the applied torque. Since the forces are separated by the distance of the dipole, it can be generalized that an electric field produces the following torque on an electric dipole:

<math> \tau\ = p \times E = Eqd\sin\theta</math>

===Direction===
The direction of the torque can be found using the right hand rule, as it will always be perpendicular to the dipole axis and applied force. The cross product relationship also indicates that when the dipole is parallel to the electric field, no torque will be acting on it.
[[File:dipole_t.gif]]

It is important to note that since the forces on each of the point charges are equal and opposite, the net force on the dipole in a uniform field is 0, which also explains why there is no linear motion by the dipole. Also, the torque from the electric field will align the orientation of the dipole to be parallel with the electric field. This is because at this point the force applied to each of the point charges is 0 because <math> F = Eqd \sin 0 = 0 </math>.

===Energy and Work===

The torque that rotates the dipole moves the orientation of the dipole from a higher energy configuration to a lower energy configuration as shown below.

[[File:dipole_torque.gif]]

This also means that any rotation done against this energy gradient requires work. The amount of work required to rotate a dipole from its low energy configuration to an angle <math>\theta_0</math> can be derived as:

<math>\int_{0}^{\theta_0}\tau d\theta = \int_{0}^{\theta_0}Eqd\sin\theta d\theta = \int_{0}^{\theta_0}Ep\sin\theta d\theta = Ep(1-\cos\theta_0)</math>

Calculating potential energy is a matter of convention. The standard convention it the potential energy is zero when the dipole is perpendicular to the electric field. With this in mind, the potential energy of a dipole can be calculated as:

<math> U = Ep \cos\theta = -p \dot\ E </math>

Again, this would indicate the potential energy is maximized when <math> \cos\theta = 1 </math>, or when the dipole is in the low energy configuration.

===Nonuniform Electric Field===
When the electric field is not uniform, the two point charges will not feel equal and opposite forces at all times, meaning that the net force on the dipole will not be zero. The force on the dipole will be in the direction of where the electric field has the steepest increase. The derivation for the net force on a dipole in a non-uniform electric field is extremely complex [http://bolvan.ph.utexas.edu/~vadim/classes/17f/dipole.pdf], but can be explained in simple terms. Consider an non-uniform electric field E with a gradient <math>\nabla</math>. For any dipole placed in this system, the difference between the electric field at small intervals between the two point charges can be expanded as a power series to calculate the total difference in electric field. It can be proven that in an ideal dipole, all the subleading terms vanish, and all that is left is the leading term of this series. Thus, the general formula for net force on a dipole in a non-uniform electric field can be written as:

<math> F = ( p \dot\ \nabla ) E(r) </math>

where <math> E(r) </math> is the electric field at the center of the dipole.

==Electric Dipole Concept Map==
This concept map illustrates the other fields and forces caused by the electric dipole.

[[File:dipolecon.gif]]

==Connectedness==
Dipoles are incredibly common in physics, chemistry, and other natural sciences. While not specific to electric dipoles, much of the mathematics taught in advanced algorithms is relevant to the study of dipoles in nature, specifically certain randomized algorithms useful in computer science can be used to effectively simulate and predict natural phenomena having to do with dipole forces and the arrangement of many dipoles.

Dipoles are useful in determining the behavior of certain molecules with each other. Polar molecules can act as electric dipoles, such as the water molecule mentioned earlier. This gives polar molecules certain properties when in a solution. Dipoles are the basis for polarity in molecules, which leads to other important properties such as hydrophilicity, which is very important in industry as well as in your body. The cells in the body are surrounded by a selectively permeable membrane. The outer and inner ends of this membrane are polar, while the middle part is non polar. This polarity is very important in determining which molecules enter and exit the cells in our body and therefore how the cells maintain homeostasis. Dipoles are also very common in regards to magnets, which have several applications including Maglev trains and even metal detectors.

To specifically Biomedical Engineering, dipole interactions can help break down molecules as a result of ion polarization and separation.

==History==

Electric dipoles have been understood since the mid to late 1800s. However, atomic dipoles could only be understood after the discovery of the correct model of the atom by Bohr in 1913. Based on this knowledge, atomic dipoles were used in a lot of technology. Even though electric dipoles are a newer concept, the human understanding of magnetic dipoles goes way back to the ancient Greeks who discovered magnetite, which had magnetic properties.

== See also ==

[http://www.physicsbook.gatech.edu/Magnetic_Dipole Magnetic Dipole]

===External links===

[https://en.wikipedia.org/wiki/Electric_charge Electric Charge]

[https://en.wikibooks.org/wiki/Physics_Exercises/Electrostatics Additional Dipole Derivations]

[https://en.wikipedia.org/wiki/Electric_dipole_moment Electric Dipole Moment]

[https://en.wikipedia.org/wiki/Dipole Dipole]

[http://hyperphysics.phy-astr.gsu.edu/hbase/electric/diptor.html Electric Dipole Torque]

==References==

[http://education.jlab.org/qa/historymag_01.html Magnet History]

[https://en.wikipedia.org/wiki/Bohr_model Bohr Model]

[http://hyperphysics.phy-astr.gsu.edu/hbase/electric/diph2o.html Electric Dipole]

[[Category:Fields]]

Electric Dipole

2022-04-15T03:16:55Z

Aramirez81:

Claimed by jmorton32 (2015) and edited by Shivani (Spring 2016)
and claimed by Hyder Hasnain (Fall 2016)
Claimed by Joshua Jacob (Fall 2017)
Edited Further by Shivani Mehrotra (Fall 2017)
Claimed by Daniel Gamero (Fall 2018)
Claimed by Anthony Ramirez (Spring 2022)
==Summary==

An electric dipole is made up of two point charges that have equal but opposite electric charges (q) and are separated by a short distance (d).

[[File:dipo.jpg|An Electric Dipole]]

The electric field is proportional to the cube of the distance from the dipole, and is dependent on whether you’re moving along the line separating the two charges or perpendicular to it.

A dipole can be created, for example, when you place a neutral atom in an electric field, because of the movement of electrons, the atom polarizes (negative charge on one side, positive charge on the other) and yields a separation of charge.

A prime example of this is the molecule for water, which forms a 105 degree angle between the two hydrogens connected to the oxygen. Since the oxygen has a greater electronegativity, it pulls more strongly on the electrons shared by the oxygen and hydrogen atoms and that end of the molecule becomes more negatively charged compared to the hydrogen end. Therefore, the net electric dipole points towards the oxygen atom.

[[File:Water.png|300px|thumb|Dipole moment of water]]

==Mathematical Models==

===An Exact Model===
[[File:Phys2212 dipole image.PNG|300px|thumb|Polarization by an electric field]]
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.

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>.

<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})^2+p_y^2}}</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})^2+p_y^2}}</math>, where once again <math>\sqrt{(p_x + \frac{d}{2})^2+p_y^2} = |\vec r_-|</math>.

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 general formula for electric field strength from a [[Point Charge]] is <math>E = \frac{1}{4\pi\epsilon_0} \frac{q}{|\vec r|^2} \hat r</math>. Then <math>|E_+| = \frac{1}{4\pi\epsilon_0} \frac{q_+}{|\vec r_+|^2}</math> and <math>|E_-| = \frac{1}{4\pi\epsilon_0} \frac{q_-}{|\vec r_-|^2}</math>. We want solely the magnitude in this case because we can calculate direction and component forces using sin and cosine. Its worth noting that we can expand <math>r_+, r_-</math> to the form in the denominator of the sine and cosine. We will use this later.

First we calculate <math>E_{net_y}</math>. <math>E_{net_y} = E_{+_y} + E_{-_y} = \frac{1}{4\pi\epsilon_0} \frac{q_+}{|\vec r_+|^2} sin(\theta_+) + \frac{1}{4\pi\epsilon_0} \frac{q_-}{|\vec r_-|^2} sin(\theta_-)</math>.

Then we combine some terms, noting that <math> q_+ = -q_-</math>. <math>E_{net_y} = \frac{q_+}{4\pi\epsilon_0} * \Bigg(\frac{1}{|\vec r_+|^2}sin(\theta_+) + \frac{-1}{|\vec r_-|^2}sin(\theta_-)\Bigg)</math>

Now it gets ugly, we expand our radii and sines. To recap, <math>sin(\theta_+) = \frac{p_y}{\sqrt{(p_x - \frac{d}{2})^2+p_y^2}}</math>, <math>sin(\theta_-) = \frac{p_y}{\sqrt{(p_x + \frac{d}{2})^2+p_y^2}}</math>, <math>|r_+| = \sqrt{(p_x - \frac{d}{2})^2 +p_y^2}</math> and <math>|r_-| = \sqrt{(p_x + \frac{d}{2})^2 +p_y^2}</math>, giving us

<math>E_{net_y} =
\frac{q_+}{4\pi\epsilon_0} *
\Bigg(
\frac{1}{
(p_x - \frac{d}{2})^2 +p_y^2
}
\frac{p_y}{\sqrt{(p_x - \frac{d}{2})^2+p_y^2}} +
\frac{-1}{
(p_x + \frac{d}{2})^2 +p_y^2
}
\frac{p_y}{\sqrt{(p_x + \frac{d}{2})^2+p_y^2}}
\Bigg)</math>

Finally we can combine more terms, the denominators of the expanded sines are the square roots of the radii. We can also pull out the negative sign.

<math>E_{net_y} =
\frac{q_+}{4\pi\epsilon_0}
\Bigg(
\frac{p_y}{
\Big((p_x - \frac{d}{2})^2 +p_y^2 \Big)^\frac{3}{2}
}
-
\frac{p_y}{
\Big((p_x + \frac{d}{2})^2 +p_y^2 \Big)^\frac{3}{2}
}

\Bigg)</math> That's as simplified as possible.

Much of the derivation for the x direction is similar. The major difference is that instead of calculating the sine, opposite over hypotenuse, we want cosine, adjacent over hypotenuse. That is, where <math>sin(\theta_+) = \frac{p_y}{\sqrt{(p_x - \frac{d}{2})^2+p_y^2}}</math>, <math>cos(\theta_+) = \frac{p_x - \frac{d}{2}}{\sqrt{(p_x - \frac{d}{2})^2+p_y^2}}</math>. By using this and its counterpart for <math>\theta_-</math>, the result is that

<math>E_{net_x} =
\frac{q_+}{4\pi\epsilon_0}
\Bigg(
\frac{p_x - \frac{d}{2}}{
\Big((p_x - \frac{d}{2})^2 +p_y^2 \Big)^\frac{3}{2}
}
-
\frac{p_x + \frac{d}{2}}{
\Big((p_x + \frac{d}{2})^2 +p_y^2 \Big)^\frac{3}{2}
}

\Bigg)</math>. These provide exact formulae for the electric field due to an electric dipole anywhere on the two-dimensional plane, and they translate easily into 3-dimensions.

==Special Cases==
We can simplify the solution for many cases

===On the Parallel Axis===
On the parallel axis, we begin with the now known formula <math>E_{net_x} =
\frac{q_+}{4\pi\epsilon_0}
\Bigg(
\frac{p_x - \frac{d}{2}}{
\Big((p_x - \frac{d}{2})^2 +p_y^2 \Big)^\frac{3}{2}
}
-
\frac{p_x + \frac{d}{2}}{
\Big((p_x + \frac{d}{2})^2 +p_y^2 \Big)^\frac{3}{2}
}

\Bigg)</math>. Since we are on the parallel axis, we know that <math>E_{net_y} = 0</math>, and <math>p_y = 0</math>.

Simplifies to

<math>E_{net_x} =
\frac{q_+}{4\pi\epsilon_0}
\Bigg(
\frac{p_x - \frac{d}{2}}{
\Big((p_x - \frac{d}{2})^2 \Big)^\frac{3}{2}
}
-
\frac{p_x + \frac{d}{2}}{
\Big((p_x + \frac{d}{2})^2 \Big)^\frac{3}{2}
}

\Bigg)</math>.

Then, combining exponents and reducing the fraction:
<math>E_{net_x} =
\frac{q_+}{4\pi\epsilon_0}
\Bigg(
\frac{1}{
(p_x - \frac{d}{2})^2
}
-
\frac{1}{
(p_x + \frac{d}{2})^2
}

\Bigg)</math>.

Then, we can combine these fractions. to simplify the calculations, replace <math>\frac{d}{2}</math> with <math>a</math>.

<math>E_{net_x} =
\frac{q_+}{4\pi\epsilon_0}
\Bigg(
\frac{1}{
(p_x - a)^2
}
-
\frac{1}{
(p_x + a)^2
}

\Bigg) =

\frac{q_+}{4\pi\epsilon_0}
\Bigg(\frac{4p_x a}{(p_x^2 + a^2)^2}
\Bigg)

=

\frac{q_+ 4 a}{4\pi\epsilon_0}
\Bigg(\frac{p_x}{(p_x^2 + a^2)^2}
\Bigg)
</math>.

This is the formula. When <math>p_x >> a</math>, we can assume that <math>p_x^2 + a^2</math> is very close to <math>p_x^2</math>. Then

<math>E_{net_x} \approx
\frac{q_+ 4 a}{4\pi\epsilon_0}
\Bigg(\frac{p_x}{(p_x^2)^2}
\Bigg) =

\frac{q_+ 4 a}{4\pi\epsilon_0}
\Bigg(\frac{p_x}{p_x^4}
\Bigg)
=
\frac{1}{4\pi\epsilon_0}
\Bigg(\frac{4 a q_+}{p_x^3}
\Bigg)</math>

===On the Perpendicular Axis===
We can do a similar simplification for the perpendicular axis. We know that <math>E_{net_y} = 0</math> because the vertical forces from both point charges cancel, leaving only horizontal forces.

<math>
E_{net_x} =
\frac{q_+}{4\pi\epsilon_0}
\Bigg(
\frac{p_x - \frac{d}{2}}{
\Big((p_x - \frac{d}{2})^2 +p_y^2 \Big)^\frac{3}{2}
}
-
\frac{p_x + \frac{d}{2}}{
\Big((p_x + \frac{d}{2})^2 +p_y^2 \Big)^\frac{3}{2}
}

\Bigg)</math>

In this case though, <math>p_x = 0</math>

<math>
E_{net_x} =
\frac{q_+}{4\pi\epsilon_0}
\Bigg(
\frac{- \frac{d}{2}}{
\Big(( - \frac{d}{2})^2 +p_y^2 \Big)^\frac{3}{2}
}
-
\frac{\frac{d}{2}}{
\Big((\frac{d}{2})^2 +p_y^2 \Big)^\frac{3}{2}
}

\Bigg)</math>

Once again, we say <math>a = \frac{d}{2}</math>.

<math>
E_{net_x} =
\frac{q_+}{4\pi\epsilon_0}
\Bigg(
\frac{-a}{
\Big(( - a)^2 +p_y^2 \Big)^\frac{3}{2}
}
-
\frac{a}{
\Big(a^2 +p_y^2 \Big)^\frac{3}{2}
}

\Bigg)
=
\frac{q_+}{4\pi\epsilon_0}
\Bigg(
\frac{-a}{
\Big(a^2 +p_y^2 \Big)^\frac{3}{2}
}
-
\frac{a}{
\Big(a^2 +p_y^2 \Big)^\frac{3}{2}
}

\Bigg)
=\frac{q_+}{4\pi\epsilon_0}
\Bigg(
\frac{-2a}{
\Big(a^2 +p_y^2 \Big)^\frac{3}{2}
}
\Bigg)
</math>

And this is our result.

Once again, when <math>d</math> is much smaller than <math> p_y</math>, <math>a</math> is also small, so we can assume that the denominator is just <math>p_y</math>. This allows us to simplify the resulting equation to

<math>E_{net_x} \approx \frac{q_+}{4\pi\epsilon_0} \frac{-2a}{p_y^3} </math>

==Examples==

===Simple===

A dipole is located at the origin, and is composed of charged particles with charge <math>+e</math> and <math>-e</math>, separated by a distance <math>9 \times10^{-10}</math> along the <math>y</math> axis. The <math>+e</math> charge is on the <math>+y</math> axis. Calculate the force on a proton due to this dipole at a location <math>< 0, 0, 3 \times 10^{-8} ></math> meters.

<div class="toccolours mw-collapsible mw-collapsed">
===Click for Solution===
<div class="mw-collapsible-content">
The center of the dipole is at the origin and there is a proton along the z axis. In this case, we apply the perpendicular from of the electric field equation. In this case, since <math>r >> d</math>, we can also use an approximate solution. Therefore, we apply the formula <math>E_{net} = \frac{q}{4\pi\epsilon_0} \frac{-2a}{r^3}</math>. Since <math>a = \frac{d}{2}</math>, and r is the distance to the proton, we can plug in the values and solve for the net electric field.

[[File:Phys2212 sample simple.PNG | 300px]]

<math>1.6\times 10^{-19} \times 9 \times 10^9
\frac{-9 \times 10^{-10}}
{3 \times 10^{-8^3}} = -48000 \frac{N}{C}</math> on the y axis, as a vector: <math><0, -48000, 0></math>.

However, we aren't done since we want to know the force. We know that <math>F = qE</math> and in this case, both <math>q</math>, the charge on the proton and <math>E</math>, the electric field, are known. Thus the solution is <math>-48000 \times 1.6 \times 10^{-19} = -7.68 \times 10^{-15}</math> on the y axis, or <math><0, -7.68 \times 10^{-15}, 0></math>.
</div>
</div>

===Middling===
A ball of mass <math>M</math> and radius <math>R</math> is given an unknown negative charge spread uniformly over its surface. The ball is hanging from a thread and can move freely. A distance <math>L</math> directly below the center of the ball, a small permanent dipole is oriented such that the dipole axis is parallel with the center of the ball. The dipole has a dipole moment <math>p = qs</math>, with a distance <math>s</math> between the positive and negative charges of the dipole, and a mass <math>m</math>. The positive charge of the dipole is oriented closer to the center of the ball.

a) calculate the charge on the ball to levitate the dipole

b) the dipole is turned 90 degrees clockwise, without changing its position relative to the ball, what effect does this have on the ball?

<div class="toccolours mw-collapsible mw-collapsed">

===Click for Solutions===
<div class="mw-collapsible-content">
a) Because the dipole is small, we can assume that <math> s << L </math>. We wish to find the force on the dipole such that it can equal the force due to gravity. Once again, <math>F = qE</math> since by newton's third law, for a force exerted on the ball by the dipole, there is an equal and opposite for exerted on the dipole by the ball. That is <math>F_G = F_E</math>, so <math>qE = mg</math> (where <math>g</math> is the acceleration due to gravity). Therefore, in this case we wish to find the force on the ball, meaning the electric field from the dipole and the charge on the ball, <math>Q</math>. The field from the dipole is, since we are on the parallel axis, <math>E = \frac{1}{4\pi\epsilon_0} \frac{2p}{L^3}</math>. Putting this together, we get <math>mg = |Q| \frac{1}{4\pi\epsilon_0} \frac{2p}{L^3}</math>

Solving for <math>|Q|</math>: <math>|Q| = \Bigg(\frac{1}{4\pi\epsilon_0}\Bigg)^{-1} \frac{mgL^3}{2p}</math>.

However, we know that since the positive charge of the dipole is closer to the ball, the charge on the ball must be negative to create an attractive force. <math>|Q| > 0</math>, so our final answer is <math>Q = -\Bigg(\frac{1}{4\pi\epsilon_0}\Bigg)^{-1} \frac{mgL^3}{2p}</math>

b) By rotating the dipole clockwise the direction of the electric field at the location of the ball changes. Since the positive end of the dipole is to the right, and the negative end to the left of the dipole, the electric field from the dipole acting on the ball is oriented toward the left. However, since the ball has negative charge, this results in a force on the ball to the right.
</div>
</div>

===Concept Question===
Is it possible for a permanent electric dipole to have a net (total) charge of zero?

<div class="toccolours mw-collapsible mw-collapsed">

===Click for Solution===
<div class="mw-collapsible-content">
Permanent dipoles occur when two atoms in a molecule have a great difference in their electronegativity; one atom attracts electrons more than the other, becoming more negative, while the other atom becomes more positive. A permanent magnet, such as a bar magnet, owes its magnetism to the magnetic dipole moment of the electron. A molecule with a permanent dipole moment is called a polar molecule. A molecule is polarized when it carries an induced dipole. A non-degenerate (S-state) atom can have only a zero permanent dipole.

</div>
</div>

===Practice Test Problem===

[[File:Exampleprac.jpg|400px]]

==Electric Field of an Electric Dipole==
The electric field of an electric dipole can be constructed as a vector sum of the point charge fields of the two charges. As can be seen in the graphics, the electric field always points towards the negative particle and points away from the positive particle. This is an important characteristic which can be used to determine which end is positive and which is negative in a dipole.

Direction of electric dipole:

[[File:dipd.gif]]

Electric Field:

[[File:edip2.gif]]

In this class, most of the questions with either be on axis or perpendicular to the dipole, so learning the direction of the electric field at these locations is important. This is done by decomposing the two vectors from each charged particle of the dipole, as shown in the positive x-axis location in the picture below. Where the positive particle has an electric field pointing away, and the negative particle electric field points inward.

[[File:Phys2212 dipole electric field.PNG]]

==Torque==
===Derivation===
Consider a dipole with the following arbitrary orientation in a uniform electric field:

[[File:dipole_torque_2.gif]]

Note how the electric field will exert an an electric force on each of the point charges. Since the electric field is uniform and charge of each point charge is equal and opposite, the electric force exerted on each point charge will be equal and opposite. The magnitude of each force is simply the force on a point charge, or <math> F = qE </math>. The component of this force perpendicular to the dipole axis can be written as <math> F_{\bot}= qE\sin \theta</math>, where theta is the angle between the electric field and the dipole. It is this perpendicular force which causes rotational motion, and thus is the force component of the applied torque. Since the forces are separated by the distance of the dipole, it can be generalized that an electric field produces the following torque on an electric dipole:

<math> \tau\ = p \times E = Eqd\sin\theta</math>

===Direction===
The direction of the torque can be found using the right hand rule, as it will always be perpendicular to the dipole axis and applied force. The cross product relationship also indicates that when the dipole is parallel to the electric field, no torque will be acting on it.
[[File:dipole_t.gif]]

It is important to note that since the forces on each of the point charges are equal and opposite, the net force on the dipole in a uniform field is 0, which also explains why there is no linear motion by the dipole. Also, the torque from the electric field will align the orientation of the dipole to be parallel with the electric field. This is because at this point the force applied to each of the point charges is 0 because <math> F = Eqd \sin 0 = 0 </math>.

===Energy and Work===

The torque that rotates the dipole moves the orientation of the dipole from a higher energy configuration to a lower energy configuration as shown below.

[[File:dipole_torque.gif]]

This also means that any rotation done against this energy gradient requires work. The amount of work required to rotate a dipole from its low energy configuration to an angle <math>\theta_0</math> can be derived as:

<math>\int_{0}^{\theta_0}\tau d\theta = \int_{0}^{\theta_0}Eqd\sin\theta d\theta = \int_{0}^{\theta_0}Ep\sin\theta d\theta = Ep(1-\cos\theta_0)</math>

Calculating potential energy is a matter of convention. The standard convention it the potential energy is zero when the dipole is perpendicular to the electric field. With this in mind, the potential energy of a dipole can be calculated as:

<math> U = Ep \cos\theta = -p \dot\ E </math>

Again, this would indicate the potential energy is maximized when <math> \cos\theta = 1 </math>, or when the dipole is in the low energy configuration.

===Nonuniform Electric Field===
When the electric field is not uniform, the two point charges will not feel equal and opposite forces at all times, meaning that the net force on the dipole will not be zero. The force on the dipole will be in the direction of where the electric field has the steepest increase. The derivation for the net force on a dipole in a non-uniform electric field is extremely complex [http://bolvan.ph.utexas.edu/~vadim/classes/17f/dipole.pdf], but can be explained in simple terms. Consider an non-uniform electric field E with a gradient <math>\nabla</math>. For any dipole placed in this system, the difference between the electric field at small intervals between the two point charges can be expanded as a power series to calculate the total difference in electric field. It can be proven that in an ideal dipole, all the subleading terms vanish, and all that is left is the leading term of this series. Thus, the general formula for net force on a dipole in a non-uniform electric field can be written as:

<math> F = ( p \dot\ \nabla ) E(r) </math>

where <math> E(r) </math> is the electric field at the center of the dipole.

==Electric Dipole Concept Map==
This concept map illustrates the other fields and forces caused by the electric dipole.

[[File:dipolecon.gif]]

==Connectedness==
Dipoles are incredibly common in physics, chemistry, and other natural sciences. While not specific to electric dipoles, much of the mathematics taught in advanced algorithms is relevant to the study of dipoles in nature, specifically certain randomized algorithms useful in computer science can be used to effectively simulate and predict natural phenomena having to do with dipole forces and the arrangement of many dipoles.

Dipoles are useful in determining the behavior of certain molecules with each other. Polar molecules can act as electric dipoles, such as the water molecule mentioned earlier. This gives polar molecules certain properties when in a solution. Dipoles are the basis for polarity in molecules, which leads to other important properties such as hydrophilicity, which is very important in industry as well as in your body. The cells in the body are surrounded by a selectively permeable membrane. The outer and inner ends of this membrane are polar, while the middle part is non polar. This polarity is very important in determining which molecules enter and exit the cells in our body and therefore how the cells maintain homeostasis. Dipoles are also very common in regards to magnets, which have several applications including Maglev trains and even metal detectors.

To specifically Biomedical Engineering, dipole interactions can help break down molecules as a result of ion polarization and separation.

==History==

Electric dipoles have been understood since the mid to late 1800s. However, atomic dipoles could only be understood after the discovery of the correct model of the atom by Bohr in 1913. Based on this knowledge, atomic dipoles were used in a lot of technology. Even though electric dipoles are a newer concept, the human understanding of magnetic dipoles goes way back to the ancient Greeks who discovered magnetite, which had magnetic properties.

== See also ==

[http://www.physicsbook.gatech.edu/Magnetic_Dipole Magnetic Dipole]

===External links===

[https://en.wikipedia.org/wiki/Electric_charge Electric Charge]

[https://en.wikibooks.org/wiki/Physics_Exercises/Electrostatics Additional Dipole Derivations]

[https://en.wikipedia.org/wiki/Electric_dipole_moment Electric Dipole Moment]

[https://en.wikipedia.org/wiki/Dipole Dipole]

[http://hyperphysics.phy-astr.gsu.edu/hbase/electric/diptor.html Electric Dipole Torque]

==References==

[http://education.jlab.org/qa/historymag_01.html Magnet History]

[https://en.wikipedia.org/wiki/Bohr_model Bohr Model]

[http://hyperphysics.phy-astr.gsu.edu/hbase/electric/diph2o.html Electric Dipole]

[[Category:Fields]]