Electrostatic Potential and Capacitance Set-5

Q1. A parallel‑plate capacitor of plate area $A$ and separation $d$ is connected to a battery of potential difference $V$ and fully charged. The battery is then disconnected and the plate separation is increased quasi‑statically to $2d$. What is the final potential difference between the plates and the ratio of final stored energy to the initial stored energy?

Q2. A parallel‑plate capacitor of area $A$ has its gap filled by two dielectric slabs of thicknesses $d_1$ and $d_2$ ($d_1+d_2=d$) and dielectric constants $k_1$ and $k_2$ respectively (no air gaps). The plates are connected to a battery providing potential difference $V$. The magnitude of the potential difference across the slab of thickness $d_1$ is

Q3. A capacitor of capacitance $C$ is charged to potential $V_0$ and then disconnected from the battery. It is then connected in parallel with an uncharged capacitor of capacitance $2C$ so that charges redistribute. The final common potential and the energy dissipated in the redistribution are:

Q4. Assertion (A): When a neutral dielectric slab is allowed to enter between the plates of an isolated (battery disconnected), charged parallel‑plate capacitor, the slab is pulled into the region between the plates spontaneously.
Reason (R): Insertion of the dielectric increases the capacitance so that the stored electrostatic energy $U=\dfrac{Q^2}{2C}$ decreases; the dielectric is therefore attracted into the region which lowers the system's energy.

Q5. A point charge $q$ is held fixed at a distance $d$ from an infinite grounded conducting plane. The magnitude and direction of the electrostatic force on the charge due to induced charges on the plane is

Q6. A parallel-plate capacitor of capacitance $C$ is charged to a potential difference $V$ and then disconnected from the battery so that the charge remains fixed. A dielectric slab of dielectric constant $K$ is now fully inserted between the plates (no charge leakage). The new potential difference across the capacitor is

Q7. Two isolated conducting spheres of radii $R_1=5\ \text{cm}$ and $R_2=10\ \text{cm}$ carry charges $q_1=+6\ \mu\text{C}$ and $q_2=-2\ \mu\text{C}$ respectively. They are then connected by a long thin conducting wire and reach electrostatic equilibrium (assume distance between spheres is large enough that mutual induction before connection is negligible). What is the final charge on the smaller sphere?

Q8. Capacitors $C_1=3\ \mu\text{F}$ and $C_2=6\ \mu\text{F}$ are involved in the following process: $C_1$ is charged to $100\ \text{V}$ while $C_2$ is uncharged; both are then disconnected from any source and connected in parallel (positive to positive) so charges redistribute. The energy dissipated during redistribution is

Q9. Two concentric conducting spherical shells of radii $a$ (inner conductor) and $b$ (outer conductor) form a spherical capacitor. The region $a\le r\le c$ (with $a<c<b$) is filled with a dielectric of constant $K$, and the region $c\le r\le b$ is vacuum ($\varepsilon_0$). The capacitance $C$ of this arrangement is

Q10. A parallel-plate capacitor has plate area $A$ and plate separation $d$. The space between plates is filled with a dielectric whose permittivity varies with position as $\varepsilon(x)=\varepsilon_0\!\left(1+\alpha\,\dfrac{x}{d}\right)$ for $0\le x\le d$, where $\alpha$ is a dimensionless constant. Treating the capacitor as a stack of infinitesimal layers in series, the capacitance $C$ of the capacitor is

Q11. A parallel-plate capacitor of capacitance $2\,\mu\text{F}$ stores energy $18\,\mu\text{J}$. What is the potential difference across its plates?

Q12. A parallel-plate capacitor has plate area $A$ and separation $d$ in vacuum. A dielectric slab of relative permittivity $\kappa=4$ and thickness $d/2$ is inserted between the plates, filling the entire area. If the original capacitance is $C_0=\dfrac{\epsilon_0 A}{d}$, the new capacitance $C'$ equals:

Q13. A capacitor of capacitance $C$ is charged to a potential difference $V$, disconnected from the battery, and then connected in parallel to an identical uncharged capacitor. Immediately after connection, the total energy stored in the two-capacitor system is:

Q14. Two isolated conducting spheres, one of radius $R$ carrying charge $Q$ and the other of radius $2R$ uncharged, are initially far apart. They are connected by a thin conducting wire and then disconnected after equilibrium. Consider the statements:
Assertion A: The final common potential of each sphere (w.r.t. infinity) equals one-third of the initial potential of the charged sphere.
Reason R: The total electrostatic self-energy of the system after connection decreases to one-third of its initial value.
Which of the following is correct?

Q15. A parallel-plate capacitor (plate area $A$, plate separation $d$) in vacuum is connected to a battery that maintains a constant potential difference $V$. A dielectric slab of relative permittivity $\kappa$ is slowly inserted fully between the plates. Let $C_0=\dfrac{\epsilon_0 A}{d}$. The work done by the external agent on the system during the slow insertion (positive if done on the system) is: