A device proposed for using the voltage to measure the angular velocity ? of a shaft is shown in…

A device proposed for using the voltage 84f23779 1b33 44ce 9d94 4e307e700bff to measure the angular velocity Ω of a shaft is shown in Fig. P5.8.3a. A cylindrical grounded electrode has radius R. (The resistance 603dcd59 1cb0 4291 9a22 2ec68bfc787e is “small.”) Outside and concentric at r = a is a rotating shell supporting the surface charge density distribution shown in Fig. P5.8.3b

Fig. P5.8.3a , Fig. P5.8.3b

 

. 25f9af0e b84e 45d7 8433 469b02821866

(a) Given c2d34043 1927 4795 bd9d 652fcaf9fe94 and b48560c3 bc1c 438f a941 15f4138d16c8, show that in regions (a) and (b), respectively, outside and inside the rotating shell,

d9f9a416 d031 4892 811b 064ef6418faa

(b) Show that the charge on the segment of the inner electrode attached to the resistor is

7eca1ff7 62c9 4c5a b07b 4af440e17dbe

where 53cdd936 f1e8 4291 b89b 58c0e0098fae is the length in the c5b5ab3c 0c02 4098 9fbc 4e5043370b87 direction.

(c) Given that c2d34043 1927 4795 bd9d 652fcaf9fe94 3ac417ed 89e6 494a 932d dc924d855bca, show that the output voltage is related to Ω by

5c39e2f7 acaf 4fe2 8acd 995f01f109ff

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