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Physics LibreTexts

5.8.6: Rods

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Refer to figure V.5. The potential at P due to the element δx is Gλδxr=Gλsecθδθ. The total potential at P is therefore

ψ=Gλβαsecθdθ=Gλln[secβ+tanβsecα+tanα].

Figure 5.24.png
FIGURE V.24

Refer now to figure V.24, in which A=90+α and B=90β.

secβ+tanβsecα+tanα=cosα(1+sinβ)cosβ(1+sinα)=sinA(1+cosB)sinB(1cosA)=2sin12Acos12A.2cos212B2sin12Bcos12B.2sin212A=cot12Acot12B=s(sr2)(sr1)(s2l)s(sr1)(s2l)(sr2),

where s=12(r1+r2+2l). (You may want to refer here to the formulas on pp. 37 and 38 of Chapter 2.)

Hence ψ=Gλln[r1+r2+2lr1+r22l].

If r1 and r2 are very large compared with l, they are nearly equal, so let’s put r1+r2=2r and write Equation 5.8.17 as

ψ=Gm2lln[2r(1+2l2r)2r(12l2r)]=Gm2l[ln(1+lr)ln(1lr)].

Maclaurin expand the logarithms, and you will see that, at large distances from the rod, the potential is, expected, Gm/r.

Let us return to the near vicinity of the rod and to Equation 5.8.16. We see that if we move around the rod in such a manner that we keep r1+r2 constant and equal to 2a, say − that is to say if we move around the rod in an ellipse (see our definition of an ellipse in Chapter 2, Section 2.3) − the potential is constant. In other words the equipotentials are confocal ellipses, with the foci at the ends of the rod. Equation 5.8.16 can be written

ψ=Gλln(a+lal).

For a given potential ψ, the equipotential is an ellipse of major axis

2a=2l(eψ/(Gλ)+1eψ/(Gλ)1),

where 2l is the length of the rod. This knowledge is useful if you are exploring space and you encounter an alien spacecraft or an asteroid in the form of a uniform rod of length 2l.


This page titled 5.8.6: Rods is shared under a CC BY-NC 4.0 license and was authored, remixed, and/or curated by Jeremy Tatum via source content that was edited to the style and standards of the LibreTexts platform.

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