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 source17dbc45a f305 4071 80d8 053512197e32. For the coordinate 4b6047b9 2695 4e21 8029 69b151899022 shown, the resulting potential distribution is the linear function of 4b6047b9 2695 4e21 8029 69b151899022 shown. The constant a2095763 4dd7 495d b96a f9da2cb959ba is determined by the definition of where the potential is zero. In the case shown in Fig. 5.4.2a, if 81d247d1 788a 4ad3 8be2 c1ccd514e456 is zero at 9ec7cd74 fbfd 4c64 a826 7a8bd0353782, then b199e42d 6ff8 4a4c abf1 89d852da1a14.

(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 02cdf9be 5383 4647 89b4 d961d7abe8d8. Inside the cylinder, what must the potential be in the planes 7ffda8c3 2dd1 44bf a075 9ea14ca38159 and fd5edab9 7bf3 4b27 bc65 662a9410defe?

Fig. 5.4.2b

c47c7171 92ca 41cc bfd0 3e9fbf713b1c

(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 81d247d1 788a 4ad3 8be2 c1ccd514e456 takes the form

f31c2fe1 ea4b 4aea a226 8f8128dcf997

so that it satisfies Laplace’s equation inside the cylinder, what are the coefficients 7fefbc89 ed61 40d9 a77b 91a592b3c8a2, and 39845880 aec2 4468 8956 46e32a481a8b?

(c) Determine E for this potential.

(d) Sketch 81d247d1 788a 4ad3 8be2 c1ccd514e456 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 ca81939d e595 4c2d 98d6 202294f2fc8c. 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

314af8d7 6842 4bdf b615 3b64dd192935

Fig. P5.4.2a

5a0b8001 6979 42ab 90a5 683a36da56b6

Fig. 5.4.2a

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