Electrostatic Potential and Capacitance Set-6

Q1. A parallel-plate capacitor of capacitance $2.0\ \mu\text{F}$ is charged to $100\ \text{V}$ and then disconnected from the battery. The plate separation is then doubled while keeping the plates parallel. The new energy stored in the capacitor is

Q2. A parallel-plate capacitor has plate area $200\ \text{cm}^2$ and plate separation $2.0\ \text{mm}$. The gap is filled by two dielectric slabs of equal thickness ($1.0\ \text{mm}$ each). The slab adjacent to the top plate has dielectric constant $\kappa_1=2$ and the other slab has $\kappa_2=3$. The effective capacitance of this arrangement is approximately

Q3. A parallel-plate capacitor of plate area $A$ and separation $d$ is connected to a battery supplying potential difference $V$. The electric field between the plates is non-uniform: $E(x)=E_0\,(x/d)$ for $0\le x\le d$, where $x$ is measured from the negative plate. With $C_0=\varepsilon_0 A/d$, the energy stored in the capacitor is $U=(2/3)\,C_0V^2$. The ratio $\dfrac{U}{\tfrac{1}{2}C_0V^2}$ equals

Q4. Statement-I: A parallel-plate capacitor of plate area $A$ and separation $d$ carries charge $Q$ and is disconnected from the battery. When a dielectric slab of dielectric constant $\kappa$ is inserted completely between the plates, the electric field between the plates becomes $E'=\dfrac{E}{\kappa}$ and the potential difference becomes $V'=\dfrac{V}{\kappa}$.
Statement-II: If the same slab is inserted while the capacitor remains connected to the battery (constant $V$), the slab is attracted into the gap by the capacitor; therefore, to insert it quasi-statically an external agent must apply a force opposite to the motion (i.e., perform negative work).

Q5. A spherical capacitor consists of an inner conducting sphere of radius $a=1.0\ \text{cm}$ and a concentric outer conducting shell whose inner radius is $b=4.0\ \text{cm}$. The region $a<r<c$ (with $c=2.0\ \text{cm}$) is filled with a dielectric of constant $\kappa_1=2$ and the region $c<r<b$ is filled with dielectric $\kappa_2=4$. The capacitance of this layered spherical capacitor is closest to

Q6. A conducting sphere of radius $4\ \text{cm}$ is isolated and held at a potential $200\ \text{V}$ with respect to infinity. What is the magnitude of the electric field at a point $8\ \text{cm}$ from the centre of the sphere?

Q7. A parallel-plate capacitor of plate area $A$ and separation $d$ is charged to $V_0=100\ \text{V}$ and then disconnected from the battery. A dielectric slab with dielectric constant $k=4$ is inserted so that it fills exactly half the area between the plates (covering area $A/2$) while spanning the full separation $d$; fringing effects are negligible. What is the new potential difference across the plates?

Q8. Two capacitors $C_1=2.0\ \mu\text{F}$ and $C_2=3.0\ \mu\text{F}$ are connected in series across a $100\ \text{V}$ battery. After they are fully charged the battery is disconnected and the capacitors are then reconnected in parallel with like terminals together (positive to positive, negative to negative). What is the final common potential difference across the pair?

Q9. A parallel-plate capacitor with plate separation $d=1.0\times 10^{-3}\ \text{m}$ is connected to a battery of $V=200\ \text{V}$. The plates have width $w=0.10\ \text{m}$ perpendicular to the insertion direction. A dielectric slab of dielectric constant $k=5$ and thickness equal to $d$ is slowly inserted so that it overlaps the plates by a length $x$ (the slab always fills the gap). Assuming edge effects negligible, the steady magnitude of force required to hold the slab is $F=\dfrac{1}{2}V^{2}\dfrac{\varepsilon_0}{d}(k-1)w$. Using $\varepsilon_0=8.85\times 10^{-12}\ \text{F m}^{-1}$, what is $F$?

Q10. A parallel-plate capacitor of plate area $A$ and plate separation $d=3.0\ \text{mm}$ is charged to $V_0=120\ \text{V}$ and then disconnected from the battery so that the charge $Q$ remains constant. A thin conducting slab of thickness $t=1.0\ \text{mm}$ (less than $d$) is inserted fully between the plates without touching either plate and is centrally placed. Neglecting edge effects, what is the new potential difference between the plates?

Q11. A parallel-plate capacitor of plate area $A$ and separation $d$ is charged to a potential $V$ and then disconnected from the battery. The initial electrostatic energy stored is $U$. The plate separation is slowly increased to $2d$ while the plates remain isolated and parallel. Assuming vacuum between the plates, the final stored energy is:

Q12. A parallel-plate capacitor of plate area $A$ and separation $d$ is connected to a battery that maintains a constant potential $V$. A dielectric slab of dielectric constant $k$ and thickness equal to the plate separation is inserted slowly so that it covers exactly half the area of the plates (one half of the area has dielectric, the other half remains vacuum). The magnitude of charge that flows from the battery during the insertion is:

Q13. A parallel-plate capacitor of plate area $A$ and separation $d$ is charged to a potential $V$ and then disconnected (isolated). A dielectric slab of dielectric constant $k$ and thickness $d/2$ is inserted so that it fills half the separation (one dielectric layer of thickness $d/2$ and one vacuum layer of thickness $d/2$ in series). The new potential difference between the plates is:

Q14. Two concentric conducting spherical shells have radii $a<c<b$. The region $a<r<c$ is filled with dielectric of constant $k_1$ and the region $c<r<b$ is filled with dielectric of constant $k_2$. Treating the two dielectric regions as capacitors in series, the equivalent capacitance between the inner conductor ($r=a$) and outer conductor ($r=b$) is:

Q15. A spherical capacitor is formed by two concentric conducting spheres with inner radius $R$ and outer radius $3R$. The capacitor is connected to a battery and charged so that the potential difference between the spheres is $V$. The battery is then disconnected, isolating the charges, and the outer spherical conductor is removed to infinity (carrying its charge with it). The final potential of the inner sphere with respect to infinity is: