Two parallel plate capacitors A and B have the same separation `d=8.85xx10^-4m` between the plates. The plate area of A and B are `0.04m^2` and `0.02m^2` respectively. A slab of dielectric constant (relative permittivity) `K=9` has dimensions such that it can exactly fill the space between the plates of capacitor B.

(i) The dielectric slab is placed inside. A as shown in figure (a). A is then charged to a potential difference of 110V. Calculate the capacitance of A and the energy stored in it.
The battery is disconnected and then the dielectric slab is moved from A. Find the work done by the external agency in removing the slab from A.
(iii) The same dielectric slab is now placed inside B, filling it completely, The two capacitors A and B are then connected as shown in figure(c). Calculate the energy stored in the system.

Answer

A, B, D

Explanation

(i) The combined capacitance is

`C=C_1+C_2`
`=((A//2)epsilon_0)/(d)+((A//2)epsilon_0epsilon_r)/(d)=A/2epsilon_0/d[1+epsilon_r]`
`=(0.4xx8.85xx10^-(12))/(2xx8.85xx10^-4)[1+9]=2xx10^-9F`
`:.` Energy stored `=1/2CV^2=1/2xx2xx10^-9xx(110)^2`
`=1.21xx10^-5J`
(ii) Work done in removing the dielectric slab = (Energy stored in capacitor without dielectric)-(Energy stored in capacitor with dielectric).
`:.` `W=(q^2)/(2C^')-(q^2)/(2C)` Here, `C=2xx10^-9F`,
`C^'=(Aepsilon_0)/(d)=(0.04xx8.85xx10^-14)/(8.85xx10^-4)=0.4xx10^-9F`
`q=CV=2xx10^-9xx110=2.2xx10^-7C`
`:.` `W=((2.2xx10^-7)^2)/(2)[(1)/(0.4xx10^-9)-(1)/(2xx10^-9)]`
`=4.84xx10^-5J`
(iii) The capacitance of `B=(epsilon_0epsilon_rA_B)/(d)`
`=(8.85xx10^(-12)xx9xx0.2)/(8.85xx10^-4)`
`C_B=1.8xx10^-9F`
The charge on A, `q_A=2.2xx10^-7C` gets distributed into two parts.
`:.` `q_1+q_2=2.2xx10^-7C`
also the potential difference across A=p.d. across B
`q_1/C_A=q_2/C_B`
`implies q_1=C_A/C_Bq_2=((0.4xx10^-9)/(1.8xx10^-9)q_2=0.22q_2`
`:.` `0.22q_2+q_2=2.2xx10^-7`
`implies q_2=(2.2)/(1.22)xx10^-7=1.8xx10^-7C`
`implies q_1=0.4xx10^-7C`
Total energy stored `=(q_1^2)/(2C_A)+(q_2^2)/(2C_B)`
`=(0.4xx0.4xx10^(-14))/(2xx0.4xx10^-9)+(1.8xx1.8xx10^-14)/(2xx1.8xx10^-8)`
`=0.2xx10^-5+0.9xx10^-5=1.1xx10^-5J`