Formula for energy stored
Energy stored before
Energy stored after
Ratio
Energy stored =‌

1

2

CV2(=‌

1

2

‌

Q2

C

)
Net capacitance with switch S closed
‌‌‌‌‌=C+C=2C
Energy stored =‌

1

2

×2C×V2=CV2
After the switch S is opened, capacitance each capacitor of =KC
∴ Energy stored in capacitor A=‌

1

2

KCV2
For capacitor B,
Energy stored =‌

1

2

‌

Q2

KC

=‌

1

2

‌

C2V2

KC

=‌

1

2

‌

CV2

K

∴ Total Energy stored
‌=‌

1

2

KCV2+‌

1

2

‌

CV2

K

=‌

1

2

CV2(K+‌

1

K

)
‌=‌

1

2

CV2(‌

K2+1

K

)
∴ Required ratio =‌

2CV2⋅K

CV2(K2+1)

=‌

2K

(K2+1)