# One way to constrain a boundary so that it has a potential distribution that is a linear function…

One way to constrain a boundary so that it has a potential distribution that is a linear function of position is shown in Fig. P5.4.2a. A uniformly resistive sheet having a length 2a is driven by a voltage source. For the coordinate  shown, the resulting potential distribution is the linear function of  shown. The constant  is determined by the definition of where the potential is zero. In the case shown in Fig. 5.4.2a, if  is zero at , then .

(a) Suppose a cylindrical region having a square cross-section of length 2a on a side, as shown in Fig. 5.4.2b, is constrained in potential by resistive sheets and voltage sources, as shown. Note that the potential is defined to be zero at the lower right-hand corner, where . Inside the cylinder, what must the potential be in the planes  and ?

Fig. 5.4.2b

(b) Find the linear combination of the potentials from the first column of Table 5.4.1 that satisfies the conditions on the potentials required by the resistive sheets. That is, if  takes the form

so that it satisfies Laplace’s equation inside the cylinder, what are the coefficients , and ?

(c) Determine E for this potential.

(d) Sketch  and E.

(e) Now the potential on the walls of the square cylinder is constrained as shown in Fig. 5.4.2c. This time the potential is zero at the location . Adjust the coefficients in (a) so that the potential satisfies these conditions. Determine E and sketch the equipotentials and field lines.

Fig. 5.4.2c

Fig. P5.4.2a

Fig. 5.4.2a

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