The separation between two plates of a parallel plate condenser is filled with two dielectric media as shown in figure . The ratio of its capacities , with and without dielectric , is

Options

A. `K_(1)(K_(1)+K_(2))`
B. `2K_(1)K_(2)//(K_(1)+K_(2))`
C. `(K_(1)+K_(2))//2`
D. `K_(2)(K_(1)+K_(2))`

Answer

b

Explanation

In this arrangement , separation/thickness of `K_(1)`= thickness of `K_(2)=(d)/(2)`
Without dielectric , `C_(0)=(epsilon_(0)A)/(d)`
The two capacitors formed with `K_(1)andK_(2)` are in series
`C_(1)=(K_(1)epsilon_(0)A)/(d//2)=2K_(1)*(epsilon_(0)A)/(d)=2K_(1)C_(0)`
`C_(2)=(K_(2)epsilon_(0)A)/(d//2)=2K_(2)*(epsilon_(0)A)/(d)=2K_(2)C_(0)`
In series combination,
`C_(eq)=(C_(1)C_(2))/((C_(1)+C_(2)))=((2K_(1)C_(0))(2K_(2)C_(0)))/(2K_(1)C_(0)+2K_(2)C_(0))`
`C_(eq)=(4K_(1)K_(2)C_(0)^(2))/(2C_(0)(K_(1)+K_(2)))=(2K_(1)K_(2)*C_(0))/((K_(1)+K_(2)))`
`(C_(eq))/(C_(0))=(2K_(1)K_(2))/((K_(1)+K_(2)))`