Question

Initially the space between the plates of the capacitor is filled with air, and the field strength in the gap is equal to $$E_{0}$$. Then half the gap is filled with uniform isotropic dielectric with permittivity $$\epsilon$$ as shown in Fig. Find the moduli of the vectors $$E$$ and $$D$$ in both parts parts of the gap $$(1$$ and $$2)$$ if the introduction of the dielectric
(a) does not charge the voltage across the plates;
(b) leaves the charges at the plates constant.

Answer 1

(a) We have $$D_{1} = D_{2}$$, or, $$\epsilon E_{2} = E_{1}$$

Also, $$E_{1} \dfrac {d}{2} + E_{2} \dfrac {d}{2} = E_{0} d$$ or, $$E_{1} + E_{2} = 2E_{0}$$

Hence, $$E_{2} = \dfrac {2E_{0}}{\epsilon + 1}$$ and $$E_{1} = \dfrac {2\epsilon E_{0}}{\epsilon + 1}$$ and $$D_{1} = D_{2} = \dfrac {2\epsilon \epsilon_{0} E_{0}}{\epsilon + 1}$$

(b) $$D_{1} = D_{2}$$, or, $$\epsilon E_{2} = E_{1} = \dfrac {\sigma}{\epsilon_{0}} = E_{0}$$

Thus, $$E_{1} = E_{0}, E_{2} = \dfrac {E_{0}}{\epsilon}$$ and $$D_{1} = D_{2} = \epsilon_{0}E_{0}$$.