W9_parallel_resonance.eps

122 Week 3: Potential Energy and Potential

Example 3.4.7: Potential of an Infinite Line of Charge

r

r

λ

0

Figure 35: An “infinitely long” line of uniform charge densityλ.

Find the fieldandthe potentialrelative to the reference radiusr 0 at all points in space

around an infinite line of charge. Explore the necessity of a reference point (because the

indefinite integral is infinite at 0 and∞).

As before, we will assume that you already know and can easily show that thefieldof an infinite
straight line of charge is:

E~=^2 keλ

r

rˆ

in cylindrical coordinates, so thatrˆpoints directly away from the line. In fact, you should be able
to show thistwo ways– using Gauss’s Law (very easy) and by direct integration (much harder).

We can thus equally easily write down an expression for the potentialat a distancerfrom the
line:

V(r) =−

∫r

∞

2 keλ

r′ dr

′=− 2 keλ(ln(r)−ln(∞)) =∞− 2 keλln(r) (182)

Oops. Looks like our potential isinfinite. That’s a problem...

To solve it, we compute the potential not relative to infinity but to some particular radiusr 0 :

V(r) =−

∫r

r 0

2 keλ

r′ dr

′=− 2 keλ(ln(r)−ln(r 0 )) =− 2 keλln

(

r

r 0

)

(183)

where we use the convenient property of natural logs: ln(a) + ln(b) = ln(ab) to simplify the final
expression. If we letr 0 = 1 (in whatever units we are considering this can be further simplifiedto:

V(r) =− 2 keλln(r) (184)

but thisobscures the units– recall that the argument of any function with a power series expansion
e.g. lnmust be dimensionless, so the “r” in this is theratioofrin the units of choice to “1” in the
unit of choice. Note well that this does not matter whenever we computepotential difference, which
is the quantity that will be the most important one in the next chapter/week:

∆V(r 1 →r 2 ) =−

∫r 2

r 1

2 keλ

r′

dr′= 2keλln

(

r 1

r 2

)

(185)

where the natural log isnegative(recall) whenr 1 < r 2 sor 1 /r 2 <1. This makessense!Note well
that the potentialdecreaseswhen we moveawayfrom the line in the direction of the field (as the
potential energy decreases when we move in the direction of its associated conservative force).

On your own, show that we also get this expression if we form ∆V(r 1 →r 2 ) =V(r 2 )−V(r 1 )
usinganyof the forms forV(r) given above (even the one with∞in it, as long as we are permitted
to subtract∞−∞= 0, which of course is not necessarily or generally true but whichcanbe true
as the setting of the zero of the potential).