Chapter 26 Problem 40

Problem

Resistance is futile!

Twelve resistors, each of resistance R, are connected as the edges of a cube. Determine the equivalent resistance

(a) between points a and b, the ends of a side;

(b) between points a and c, the ends of a face diagonal;

(c) between points a and d, the ends of the volume diagonal.

[Hint: Apply an emf and determine currents; use symmetry at junctions.]

solution

(a)

Insert a probe battery between points (a) and (b). Let's label the current drawn from this probe battery as I

This setup has a symmetry plane between abed, so the current split at junction a will be

$I=I_{1}+I_{1}+I_{2}$

Another symmetry is that the afhe plane is the mirror cdgb (since one has the current I entering, and the other has the current leaving) thus we have the same currents in erverse directions in those planes.

• $I=2I_{1}+I_{2}$
• $I_{3}+I_{4}=I_{1}$
• $I_{5}=2I_{4}$
• $0=-2I_{1}R-I_{3}R+I_{2}R$
• $0=-2I_{4}R-I_{5}R+I_{3}R$
• $0={\mathcal {E}}-2I_{2}R$

(b)

Insert a probe battery between points (a) and (c). Let's label the current drawn from this probe battery as I

(c)

Insert a probe battery between points (a) and (d). Let's label the current drawn from this probe battery as I