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Question

Two identical sized capacitors C1 and C2 are connected to a battery in series. A dielectric slab of dielectric constant K=2 is filled in the half region of both the capacitors as shown in figure.




(C capacitance, q charge stored, U energy stored )

Match the following column:

Column I Column II
(A) C1C2 (p) 8/9

(q) 4/9

(r) 9/8

(s) None of these
(B) q1q2
(C) U1U2

A
Ar, Bp, Cs
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B
Ar, Bs, Cp
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C
As, Bs, Cs
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D
Ap, Br, Cq
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Solution

The correct option is B Ar, Bs, Cp
Let original capacitance of capacitor without dielectric is

C0=Aϵ0d

In C1, capacitor is in parallel combination. So equivalent capacitance will be

C1=ϵ0(A/2)d+Kϵ0(A/2)d

C1=Aϵ02d+Aϵ0d (K=2)

C1=C02+C0=3C02

Similarly, for the second capacitor,

C2=Aϵ0dt+tK

C2=Aϵ0dd2+d2×2 (t=d2; K=2)

C2=2ϵ0Ad(1+12)=43Aϵ0d

C2=43C0

Hence, C1C2=98

Since capacitors are in series,

q1=q2

q1q2=1

And we know that energy stored in a capacitor is given by

U=q22C

U1U2=C2C1

(because q is same for both)

Thus, U1U2=4C0/33C0/2=89

Hence, Ar, Bs, Cp

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