48 Week 1: Discrete Charge and the Electrostatic Field
F = −qE
F = qE
l E
+q
−q
Figure 8: The basic dipole consists of two equal and opposite charges±qseparated by a vector
displacement~l, in which case thedipole momentof the arrangement is defined to be~p=q~l. Note
that we are not in this figure assuming that theE-field iscreatingthe dipole; rather we are assuming
that it is fixed, with the charges rigidly separated by e.g. a massless rod in between.
changein response to any field one might put them in, although this is clearly only a model and
not the reality for most real dipoles bound together by anon-rigid force. Thedipole momentof this
arrangement is the source of a characteristic electrostatic field,thedipole field. The dipole moment
of the two charges is defined to be:
~p=q~l (25)
whereqis the magnitude of the charge and~lis the vector that points from the negative charge to
the positive charge.
In the example above and the homework, we algebraically evaluate the field produced by a dipole
along lines of symmetry where the field has a simple form, and qualitatively draw out the general
form of the field atarbitrarypoints in space as illustrated in figure 6. The electric field of a “point
like” dipole has an extremely characteristic shape and a precisely defined functional form in terms
of~p, although we will find it far simpler to evaluate the electrostaticpotentialof a dipole at an
arbitrary point when we get to the appropriate chapter.
At this point, let us consider theforceand thetorgueexerted by an electric field on a dipole.
If an electric dipole is placed in auniformelectrical field, the forces on the two poles are equal in
magnitude and opposite in direction. The net force on the dipole is therefore zero. Algebraically:
F~ = −qE~+qE~
= 0 (26)If the dipole is not aligned or antialigned with the uniform field, however, the field clearly exerts
atorqueon the dipole. The forces form a “couple” (two opposite forces that do not act along the
same line), and therefore this torque is independent of our choice of pivot (seeIntroductory Physics
Iif necessary to review this and other aspects of torque).
If we pick (say) the negative charge as the pivot, then the torqueis due to the force exerted on
the positive charge only, at positionvlrelative to the pivot. The torque is therefore:
~τ = ~r×F~
= ~l×qE~
= q~l×E~
= ~p×E~ (27)(noting that charge is a scalar quantity). This is a very important result; learn this picture and mini-
derivation well so you can easily remember and apply it. Since this is thefirst time this semester