Ash: Import converted Year 1 Semester 2 lecture notes from 443937a7d37850a3c57dfa863f35e80d07f2ccbc

2026-09-11T14:49:07Z

Import converted Year 1 Semester 2 lecture notes from 443937a7d37850a3c57dfa863f35e80d07f2ccbc

New page

<div class="pn-lecture-meta">
<b>Lecture 8: Electrostatics VIII: Electric Dipoles in E-Fields</b><br>
Thu 12 Feb 2026 11:00
</div>

[[Year 1 Sem 2/LC Electromagnetism I/Lecture 07|← Previous lecture]] · [[Year 1 Sem 2/LC Electromagnetism I|Module]] · [[Year 1 Sem 2/LC Electromagnetism I/Lecture 09|Next lecture →]]

In this lecture:

<ul>
<li>

Electric dipoles in uniform external E-fields:

<ul>
<li>

Torque

</li>
<li>

Potential energy

</li>
<li>

Work done

</li>
<li>

Right-hand rule

</li>
<li>

Examples

</li>
</ul>

</li>
</ul>

== Torque ==

Consider a dipole in a uniform external E-field:

[[File:EM-Y1S2-lec08-01.png|thumb|center|360px]]

As the two charges have equal and opposite forces applied by the field, they do not have a net force. The torque <nowiki>$\pmb{\tau}$</nowiki> rotates the whole couple.

The magnitude of this torque is given by: <div class="pn-import-equation"><nowiki>$$| \pmb{\tau} | = Fd$$</nowiki></div>

Were <nowiki>$F$</nowiki> is one of the two forces and <nowiki>$d$</nowiki> is the perpendicular distance. This is given by: <div class="pn-import-equation"><nowiki>$$\tau = | \pmb{\tau} | = Fd = qEd = qE a \sin \theta = pE \sin \theta$$</nowiki></div>

In vector form its given as a cross product: <div class="pn-import-equation"><nowiki>$$\pmb{\tau} = \pmb{r} \wedge \pmb{F} = q \pmb{a} \wedge \pmb{E} = \pmb{p} \wedge \pmb{E}$$</nowiki></div>

== Potential Energy ==

The electric potential at the positive charge be <nowiki>$V_+$</nowiki> and hence the electric potential energy is <nowiki>$qV_+$</nowiki>, and likewise <nowiki>$V_-$</nowiki> and <nowiki>$qV_-$</nowiki> at the negative charge.

[[File:EM-Y1S2-lec08-02.png|thumb|center|600px]]

The net potential energy on the system is therefore given by: <div class="pn-import-equation"><nowiki>$$U = q(V_+ - V_-) = -q \int_{0}^{d} \pmb{E} \cdot \, d \pmb{x}$$</nowiki></div> <div class="pn-import-equation"><nowiki>$$= - qEd$$</nowiki></div> <div class="pn-import-equation"><nowiki>$$= -qE a \cos \theta = -qa E \cos \theta = -p E \cos \theta$$</nowiki></div> Hence in vector notation: <div class="pn-import-equation"><nowiki>$$U = - \pmb{p} \cdot \pmb{E}$$</nowiki></div>

=== Directionality ===

What is the direction of this setup:

[[File:EM-Y1S2-lec08-03.png|thumb|center|600px]]

Here, as the dipole is aligned with the field, there’s no movement. This (as p goes from the negative to positive charge) is the lowest potential energy configuration. The opposite, with p antialigned to the field would still have zero torque, but is the maximum potential energy configuration.

=== Microwaves ===

Dipoles explain how microwave ovens work. For foods containing water, the alternating electric field causes the water molecule’s dipole to align itself with the E-field.

Alternating the microwaves cause the water molecules to oscillate in phase. They gain kinetic energy, so heat.

This works for microwaves because a wavelength too low would cause too little a rate of oscillation and a wavelength too high would oscillate the direction of the E-field before the water dipoles have time to fully align. Microwaves fit this region.

== Work Done ==

Over some infinitesimal change in angular displacement, <nowiki>$\delta \theta$</nowiki>: <div class="pn-import-equation"><nowiki>$$\delta W = \tau \delta \theta$$</nowiki></div>

And takes place in the direction which minimises the angle, hence: <div class="pn-import-equation"><nowiki>$$\tau = - pE \sin \theta \implies \delta W = -p E \sin \theta \delta \theta$$</nowiki></div>

So, over a displacement from <nowiki>$\theta_1$</nowiki> to <nowiki>$\theta_2$</nowiki>: <div class="pn-import-equation"><nowiki>$$W = \int dW = \int_{\theta_1}^{\theta_2} -pE \sin \theta \, d \theta$$</nowiki></div> <div class="pn-import-equation"><nowiki>$$W = pE \cos \theta_2 - pE \cos \theta_1 = -(U_2 - U_1) = U_1 - U_2$$</nowiki></div>

== Direction of Torque ==

We can use the Right Hand Rule to determine the direction of the torque:

[[File:EM-Y1S2-lec08-04.png|thumb|center|600px]]

Here, <nowiki>$\pmb{a}$</nowiki> represents <nowiki>$\pmb{p}$</nowiki> and <nowiki>$\pmb{b}$</nowiki> represents <nowiki>$\pmb{E}$</nowiki>.

----
[[Year 1 Sem 2/LC Electromagnetism I/Lecture 07|← Previous lecture]] · [[Year 1 Sem 2/LC Electromagnetism I|Module]] · [[Year 1 Sem 2/LC Electromagnetism I/Lecture 09|Next lecture →]]

Year 1 Sem 2/LC Electromagnetism I/Lecture 08 - Revision history

Ash: Import converted Year 1 Semester 2 lecture notes from 443937a7d37850a3c57dfa863f35e80d07f2ccbc