46 Week 1: Discrete Charge and the Electrostatic Field
Figure 6: The electric field of a classic electric dipole in the vicinity of thecharges. Bear in mind
that this figure is a plane cross-section of a three-dimensional, cylindrically symmetric field! The
dashed lines are the projections into the plane of theequipotential surfacesof this arrangment of
charges. As we shall see later, finding the (scalar!)potentialof an electric dipole is very easy, where
finding the field inevitably involves a certain amount of vector annoyance.
the denominator on top (with a negative exponent) in the numerator, and then performing abinomial
expansionand keeping terms to any desired degree of precision. In this case the process yields:
E~tot(x,0) = − ke|~p|
(x^2 +a^2 )^3 /^2ˆy= −
ke|~p|
x^3(1 +
(a
x) 2
)−^3 /^2 yˆ≈ −ke|~p|
x^3(
1 −
3
2
(a
x) 2
+...
)
ˆy≈ −
ke|~p|
x^3yˆ+O(
1
x^5)
(24)
(where the last term is read “plus neglected terms oforder 1 /x^5 ”).
As we will see later the field of a point dipole scales like 1/r^3 whereris the distance from the
dipole to the point of observation. It thus vanishes more rapidly than the electric monopolar moment
(the field of a single bare charge, which goes like 1/r^2 ) with distance, but thatdoes not mean the field
is negligiblebecause the electric force isvery powerful, far stronger than gravity, and the strongest
force of nature outside of the nucleus of an atom. Indeed, for most problems in physics thatdon’t
involve planet-sized masses, the electromagnetic forces – whatever form or magnitude they might
have – are by far thelargestforces acting within a system. To decide whether or notanyalgebraic
expression for the field can be neglected requires specific numbers; for that reason many problems
will have you find theleading order term(s)in a binomial or taylor series expansion of the field or
potential.
Please go back to the section on math and review both the binomial and taylor series expansions,
as they will be very useful to us as we solve problems and work examples. The binomial expansion
in particular is a wonderful way to do “in your head” estimates of quantities that would otherwise