A capacitor is formed by two square metal plates of edge 1m, separated by a distance 1mm. Dielectrics of dielectric constant K1 and K2 are filled in the gap as shown in figure. The new capacitance is
[ Take ln2=0.7]
A
25nF
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B
10nF
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C
20nF
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D
15nF
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Solution
The correct option is A25nF
Let us consider a small strip of width dx at x distance from the left end.
From figure,
tanϕ=1×10−31=1×10−3
∴t=xtanϕ=10−3x
We can see that both small capacitors formed by the small strip are in series.
So, 1dCeq=1dC1+1dC2
⇒1dCeq=d1K1ϵ0A1+d2K2ϵ0A2
[ Area of small plate =1×dx=dx]
⇒1dCeq=10−3−10−3x2ϵ0dx+10−3x4ϵ0dx
⇒dCeq=4ϵ0dx2×10−3−10−3x
⇒Ceq=∫dCeq=4ϵ010−31∫0dx2−x
⇒Ceq=4×8.85×10−1210−3[−ln(2−x)]10[∵∫dxx=lnx+C]
⇒Ceq=3.6×10−8[−ln1+ln2]
⇒Ceq=2.5×10−8F=25nF
Hence, option (d) is correct.
Why this question ?Tip-we use concept of integration for suchkind of problem. This question helps youto recall your concept of integration.