Figure P4.1.8 shows a cross-sectional view of a model for a “capacitance” probe designed to…

Figure P4.1.8 shows a cross-sectional view of a model for a “capacitance” probe designed to measure the depth ccfffba7 1bb0 4e1b acf3 8e509f993f8d of penetration of a tool into a metallic groove. Both the “tool” and the groove can be considered constant potential surfaces having the potential difference 09a690ee cd9f 425e 8690 69740cc0e6ba as shown. An insulating segment at the tip of the tool is used as a probe to measure h. This is done by measuring the charge on the surface of the segment. In the following, we start with a field distribution that can be made to fit the problem, determine the charge and complete some instructive manipulations along the way.

Figure P4.1.8

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(a) Given that the electric field intensity between the groove and tool takes the form

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show that 671196e4 4988 4ac3 a17b a140a7eb7570 is irrational and evaluate the coefficient a0dbb5d0 c00f 44f1 b0c1 71422c8cd955 by computing the integral of 671196e4 4988 4ac3 a17b a140a7eb7570 · a57c92eb 23b7 455d 8cfc d71d94d91b27 between point (a) and the origin.

(b) Find the potential function consistent with (a) and evaluate a0dbb5d0 c00f 44f1 b0c1 71422c8cd955 by inspection. Check with part (a).

(c) Using the conventions of Figs. 2.7.3 and 4.1.3, sketch lines of constant potential and electric field E for the region between the groove and the tool surfaces.

Figs. 2.7.3

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Fig. 4.1.3

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(d) Determine the total charge on the insulated segment, given 09a690ee cd9f 425e 8690 69740cc0e6ba. (Hint: Use the integral form of Gauss’ law with a convenient surface S enclosing the electrode.)

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