Electrostatic Potential and Capacitance Set-3

Q1. The electrostatic potential along the $x$-axis for $0\le x\le 3.0\ \mathrm{m}$ is given by $V(x)=12-4x$ (in volts). A test charge $q=-2.0\ \mu\mathrm{C}$ is moved quasi‑statically from $x=1.0\ \mathrm{m}$ to $x=2.0\ \mathrm{m}$. The change in its electrostatic potential energy $\Delta U=U_f-U_i$ equals:

Q2. Two parallel metal plates of area $A=200\ \mathrm{cm}^2$ separated by $d=2.00\ \mathrm{mm}$ are connected to a $300\ \mathrm{V}$ battery. Assuming vacuum between the plates ($\varepsilon_0=8.85\times10^{-12}\ \mathrm{F/m}$), the magnitude of charge on each plate is closest to:

Q3. Assertion (A): For any two points lying on the same equipotential surface, the work done by the electrostatic force in moving a test charge between them is zero. Reason (R): The electric field at every point on an equipotential surface is perpendicular to the surface.

Q4. A capacitor of capacitance $C_0=10\ \mathrm{pF}$ is charged to $100\ \mathrm{V}$ and then disconnected from the battery so the charge is isolated. A dielectric slab with dielectric constant $\kappa=4$ is inserted completely between the plates. The change in stored electrostatic energy $\Delta U=U_f-U_i$ (in joules) is:

Q5. Two identical point charges $+q$ are fixed at $x=-a$ and $x=+a$ on the $x$-axis. Consider the electrostatic potential $V(x)$ along the $x$-axis. At the midpoint $x=0$, which of the following is correct?

Q6. The graph of potential near $x=0$ is symmetric and can be approximated as $V(x)=V_0+\alpha x^2$ with $\alpha>0$ (so $V$ has a local minimum at $x=0$). A small test charge $q$ is placed at $x=0$ and slightly displaced to $+x$. For which sign of $q$ will the electrostatic force be restorative (i.e., tend to bring the charge back toward $x=0$)?

Q7. Assertion (A): For an isolated charged capacitor (battery disconnected) the insertion of a dielectric slab between the plates reduces the potential difference across the plates. Reason (R): The dielectric polarises and produces an internal electric field parallel to the external field, thereby reducing the net field.

Q8. A capacitor $C=4.0\ \mu\mathrm{F}$ is connected to a $100\ \mathrm{V}$ battery. The energy stored in the capacitor is:

Q9. A parallel-plate capacitor has initial capacitance $C_0=\varepsilon_0 A/d$. It is connected to a battery of voltage $V$ and charged. A dielectric slab of $\kappa=3$ is inserted so it fills exactly half the plate separation (slab thickness $d/2$). If $C_0=5.0\ \mathrm{pF}$ and $V=100\ \mathrm{V}$, the work done by the battery during insertion (in joules) is:

Q10. Two capacitors $C_1=2.0\ \mu\mathrm{F}$ and $C_2=6.0\ \mu\mathrm{F}$ are initially isolated. $C_1$ is charged to $100\ \mathrm{V}$ while $C_2$ is uncharged. Then the two capacitors are connected in parallel by joining like plates (positive-to-positive, negative-to-negative) and allowed to come to equilibrium. The heat dissipated during this process is: