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Two parallel plate air capacitors of same capacity C are connected in series to a battery of emf E. Then one of the capacitors is completely filled with dielectric material of constant K. The change in the effective capacity of the series combination is 

Answer:

Explanation:

initial effective capacity of the series combination is 

$\frac{1}{C_{1}}=\frac{1}{C_{}}+\frac{1}{C_{}}=\frac{2}{C_{}}$

$\Rightarrow$    $C_{1}=\frac{C}{2}$

 Effective capacity of the series combination  with dielectric material is 

 $\frac{1}{C_{1}}=\frac{1}{C}+\frac{1}{KC}$

 $\frac{1}{C_{2}}=\frac{1}{C_{}}\left[1+\frac{1}{K}\right]$

$\therefore$     $C_{2}=\frac{C}{\left(1+\frac{1}{K}\right)}=\frac{CK}{(K+1)}$

The change in the effective capacitance  is $\triangle C= C_{2}-C_{1}$

$=\frac{CK}{\left(1+K\right)}-\frac{C}{2}=C\left[\frac{K}{K+1}-\frac{1}{2}\right]$

$=C\left[\frac{2K-K-1}{( 2(K+1)}\right]=\frac{C}{2}\left[\frac{K-1}{K+1}\right]$