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Question

A cylindrical capacitor is filled with two cylindrical layers of dielectric with permittivities $$\epsilon_{1}$$ and $$\epsilon_{2}$$. The inside radii of the layers are equal to $$R_{1}$$ and $$R_{2} > R_{1}$$. The maximum permissible values of electric field strength are equal to $$E_{1m}$$ and $$E_{2m}$$ for these dielectrics. At what relationship between $$\epsilon, R$$, and $$E_{m}$$ will the voltage increase result in the field strength reaching the breakdown value for both dielectrics simultaneously?

Solution
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Let $$\lambda$$ be the linear charge density then,
$$E_{1m} = \dfrac {\lambda}{2\pi \epsilon_{0} R_{1} \epsilon_{1}} .... (1)$$
and, $$E_{2m} = \dfrac {\lambda}{2\pi \epsilon_{0} R_{2} \epsilon_{2}} .... (2)$$
The breakdown in either case will occur at the smaller value of $$r$$ for a simultaneous breakdown of both dielectrics.
From (1) and (2)
$$E_{1m} R_{1} \epsilon_{1} = E_{2m} R_{2} \epsilon_{2}$$, which is the sought relationship.

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