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251.

Consider the situation shown in the figure. The capacitor `A` has a charge `q` on it whereas `B` is uncharged. The charge appearing on the capacitor `B` a long `7` time after the switch is closed is : .A. zeroB. `q//2`C. `q`D. `2q`

Answer» Correct Answer - B
Since the other plate of `B` is free, charge cannot flow from `A` to `B`. Hence the correct choice is (a).
Discussion
252.

Calculate the capacitance of a parallel plate capacitor, with plate area `A` and distance between the plates `d`, when filled with a dielectric whose permittivity varies as `in(x)=in_(0)+kx(0 lt x lt (d)/(2)), in(x)=in_(0) +k(d-x)((d)/(2) lt x le d)`.

Answer» Correct Answer - A::B::D
`(1)/(C_(1)) = int (1)/(dC_(1)) = overset(d//2)underset(0)int (dx)/((in_(0)+betax)A)`
`= (1)/(betaA) [l n(in_(0)+betax)]_(0)^(d//2)`
`(1)/(C_(1))=(1)/(betaA)[l n(in_(0)+(betad)/(2))-l nin_(0)]`
`= (1)/(betaA) ln ((2in_(0)+betad)/(2in_(0)))`
`(1)/(C_(2)) = int (1)/(dC_(2)) = (1)/(A) overset(d)underset(d//2)int (dx)/(in_(0)+beta(d-x))`
`=- (1)/(betaA) [l n{in_(0)+beta(d-x)}]_(d//2)^(d)`
`(1)/(C_(1))=(1)/(betaA)[l n{in_(0)+beta(d-d)}]-l n[in_(0)+(betad)/(2)]`
`=- (1)/(betaA) [l nin_(0) -l n(2in_(0) +betad)/(2)]`
`(1)/(C_(2)) =- (1)/(betaA) ln (2in_(0))/(2in_(0)+betad)`
`(1)/(C_(2)) = (1)/(betaA) l n (2in_(0) +betad)/(2in_(0))`
`(1)/(C ) = (1)/(C_(1))+(1)/(C_(2)) =(1)/(betaA)[2l n(2in_(0)+betad)/(2in_(0))]`
`C =(betaA//2)/(l n(2in_(0)+betad)/(2in_(0)))`
Discussion
253.

A capacitor is filled with an insulator and a certain potential difference is applied to its pltaes. The energy stored in the capacitor is `U`. Now the capacitor is disconnected from the source and the insulator is pulled out of the capacitor. The work performed against the forces of electric field in pulling out the insulator is `4 U`. Then dielectric constant of the insulator is.A. 4B. 8C. 5D. 3

Answer» Correct Answer - C
`U=(1)/(2)(q^(2))/(C),U+4U=(1)/(2)(q^(2))/(C_(0))`
`5U=(1)/(2)(q^(2))/(C_(0)),k=(C)/(C_(0))=5`.
Discussion