Between the plates of a parallel plate condenser, a plate of thickness ${t_1}$ and dielectric constant ${k_1}$ is placed. In the rest of the space, there is another plate of thickness ${t_2}$ and dielectric constant ${k_2}$. The potential difference across the condenser will be
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(a) Potential difference across the condenser
$V = {V_1} + {V_2} = {E_1}{t_1} + {E_2}{t_2} = \frac{\sigma }{{{K_1}{\varepsilon _0}}}{t_1} + \frac{\sigma }{{{K_2}{\varepsilon _0}}}{t_2}$
$==>$ $V = \frac{\sigma }{{{\varepsilon _0}}}\left( {\frac{{{t_1}}}{{{K_1}}} + \frac{{{t_2}}}{{{K_2}}}} \right) = \frac{Q}{{A{\varepsilon _0}}}\left( {\frac{{{t_1}}}{{{K_1}}} + \frac{{{t_2}}}{{{K_2}}}} \right)$
$V = {V_1} + {V_2} = {E_1}{t_1} + {E_2}{t_2} = \frac{\sigma }{{{K_1}{\varepsilon _0}}}{t_1} + \frac{\sigma }{{{K_2}{\varepsilon _0}}}{t_2}$
$==>$ $V = \frac{\sigma }{{{\varepsilon _0}}}\left( {\frac{{{t_1}}}{{{K_1}}} + \frac{{{t_2}}}{{{K_2}}}} \right) = \frac{Q}{{A{\varepsilon _0}}}\left( {\frac{{{t_1}}}{{{K_1}}} + \frac{{{t_2}}}{{{K_2}}}} \right)$
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