There is a double-layer cylindrical capacitor whose parameters are shoen in Fig. The breakdown field strength values for these dielectrics are equal to `E_(1)` and `E_(2)` respectively. What is the breakdown voltage of this capacitor if `epsilon_(1) R_(1) E_(1) lt epsilon_(2) R_(2) E_(2)` ?

Explanation

Let, `lambda` be the inear charge density then, the sought potentail difference,
`varphi_(+) - varphi_(-) = int_(R_(1))^(R_(2)) (lambda)/(2pi epsilon_(0) epsilon_(1) r) dr + int_(R_(1))^(R_(2)) (lambda)/(2pi epsilon_(0) epsilon_(2) r) dr`
`= (lambda)/(2pi epsilon_(0)) [(1)/(epsilon_(1)) In R_(2)//R_(1) + (1)/(epsilon_(2)) In R_(3)//R_(2)]`
Now, as, `E_(1) R_(1) epsilon lt E_(2) R_(2) epsilon_(2)`, so
`(lambda)/(2pi epsilon_(0)) = E_(1) R_(1) epsilon_(1)`
is the maximum aacceptable value, and for values greater than `E_(1) R_(1) epsilon_(1)`, dielectric breakdowns will take place,
Hence, the maximum potential difference between teh plates,
`varphi_(+) - varphi_(-) = E_(1) R_(1) epsilon_(1) [(1)/(epsilon_(1)) In R_(2)//R_(1) + (1)/(epsilon_(2)) In R_(3)//R_(2)] = E_(1) R_(1) [In R_(2)//R_(1) + (epsilon_(1))/(epsilon_(2)) In R_(3)//R_(2)]`