Because one has to integrate over the vectors, this integral is remarkably difficult. We’ll revisitit in a much more similar form when we get to electrostaticpotential, a scalar quantity.•Electric DipolesWhen two electric charges of equal magnitude and opposite sign are bound together, they formanelectric dipole.Thedipole momentof this arrangement is the source of a characteristicelectrostatic field, thedipole field. The dipole moment of the two charges is defined to be:vectorp=qvectorlwhereqis the magnitude of the charge andvectorlis the vector that points from the negative chargeto the positive charge.When an electric dipolevectorpis placed in auniformelectric fieldvectorE, the following expressionsdescribe the force and torque acting on the dipole (which tries to align itself with the appliedfield):vectorF=0vectorτ=vectorp×vectorEAssociated with this torque is the following potential energy:U=−vectorp·vectorEand from this, we can see that the force on the dipole in a more general (non-uniform) fieldshould be:vectorF=−vector∇U=vector∇(vectorp·vectorE)which is actually nontrivial to compute.This completes the chapter/week summary. The sections below illuminate these basic facts andillustrate them with examples.
34Week 1: Discrete Charge and the Electrostatic Field1.1: ChargeIn nature we can readily observe electromagnetic forces.In fact, we can do little else.In a veryfundamental sense, weareelectromagnetism. Electromagnetic forces bind electrons to atomic nuclei,bond atoms together to form molecules, mediate the interactions between molecules that allow themto change and organize and, eventually, live. The energy that is used to support life processes iselectromagnetic energy. The objects that we touch, or hear, or taste, or smell, the light that we see,the organized pattern of neural impulses that we use to think about the input from our senses – allare electromagnetic.Given its ubiquity, it should come as no surprise that the directed observation and study ofelectricity is quite ancient. It was studied, and written about, at least 3000 years ago, and artifactsthat may have been primitive electrical batteries have been discovered in the Middle East thatdate back to perhaps 250 BCE. It is revealing that the verywordelectricity and the name of theelementary particle most visibly responsible for its transport is derived from the greek word foramber,electron.One of the first recorded observations of electrical force was the static electricalforce created between amber, charged by rubbing it with wool, and small bits of wool or hair,although the observation of static sparks and electrostatic forces no doubt predates the writtenword.However, it took until the Enlightenment (roughly 1600) and the invention of physics and calculusfor the scientific method to develop to where systematic studies of the phenomenon could occur, andit wasn’t until the middle 1700s that the correct model forelectrical charge23was proposed. From