Electric potential and capacitance practice · Hard

Statement-1: A parallel plate capacitor is charged to a potential difference of $100 \mathrm{~V}$, and disconnected from the voltage source. A slab of dielectric is then slowly inserted between the plates. Compared to the energy before the slab was inserted, the energy stored in the capacitor with the dielectric is decreased.
Statement-2: When we insert a dielectric between the plates of a capacitor, the induced charges tend to draw in the dielectric into the field (just as neutral objects are attracted by charged objects due to induction). We resist this force while slowly inserting the dielectric, and thus do negative work on the system, removing electrostatic energy from the system.

A parallel plate capacitor consisting of two circular plates of radius 10 cm is being charged by a constant current of 0.15 A. If the rate of change of potential difference between the plates is $7 \times 10^8 \mathrm{V} / \mathrm{s}$ then the integer value of the distance between the parallel plates is ____ $\mu \mathrm{m}.$

$\left(\text{Take }\epsilon_0=9 \times 10^{-12} \frac{\mathrm{~F}}{\mathrm{~m}}, \pi=\frac{22}{7}\right)$

A large insulated sphere of radius $r$ charged with $Q$ units of electricity is placed in contact with a small insulated uncharged sphere of radius $r^{\prime}$ and in then separated. The charge on smaller sphere will now be

A system consists of four equal charges - $\mathrm{Q}$ placed at the four corners of a square and another charge $\mathrm{q}$ is placed at the centre of the square, then the value of $\mathrm{q}$ for which the system is equilibrium, is

A charge $Q$ is uniformly distributed through out the volume of a dielectric sphere of radius $R$. If potential at its surface is 24 volts. Then potential at $r=\frac{R}{2}$ and $r=\frac{3 R}{2}$ respectively are. (where $r$ distance from the centre of sphere) :-

Assertion (A): At a point electric potential is decreasing along $x$-axis at a rate of $10 \mathrm{~V} / \mathrm{m}$. Therefore, $x$-component of electric field at this point should be $10 \mathrm{~V} / \mathrm{m}$ along $x$-axis.

Reason (R) : Magnitude of $E_x=\frac{\partial V}{\partial x}$

A parallel plate capacitor with plate area $A$ and distance of separation $d$ is filled with a dielectric. What is the capacity of the capacitor when permittivity of the dielectric varies as
$\varepsilon \,(X) = {\varepsilon _0} + \,\,kx,$ for $\left( {0 < x \le \frac{d}{2}} \right)$
$\varepsilon \,(X)\,\, = \,\,{\varepsilon _0} + k\,(d - x)$ for $\left( {\frac{d}{2} \le x \le d} \right)$

The energy stored in a parallel plate capacitor can be treated as the energy of the electric field. The energy per unit volume, due to the electric field is

Given below are two statements.
Statement I : Electric potential is constant within
and at the surface of each conductor.
Statement II : Electric field just outside a charged
conductor is perpendicular to the surface of the
conductor at every point.
In the light of the above statements, choose the
most appropriate answer from the options given
below.

While a capacitor remains connected to a battery and dielectric slab is applied between the plates, then

Assertion (A): Two charges $-q$ each are fixed at points $A$ and $B$. When a third charge $-q$ is moved from $A$ to $B$, electrical potential energy first decreases than increases.

Reason (R) : Along the line joining $A$ and $B$, the third charge is in stable equilibrium position at centre.

A point charge $+ Q$ is located at a distance of $'r'$ from the centre of an uncharged conducting sphere of radius $R$. The electric potential of that sphere is

Statement-1: If temperature is increased, the dielectric constant of a polar dielectric decreases whereas that of a non-polar dielectric does not change significantly.
Statement-2: The magniude of dipole moment of individual polar molecule decreases significantly with increase in temperature.

If the distance between the plates of a capacitor is $\mathrm{d}$ and potential difference is $\mathrm{V}$ then what is the energy density between the plates?

Assertion (A) : If we see along the axis of a charged ring, the magnitude of electric field is minimum at centre and magnitude of electric potential is maximum.

Reason (R): Electric field is a vector quantity while electric potential is scalar.

A capacitor of capacitance $C$ has charge $Q$ and stored energy is $W.$ If the charge is increased to $3Q,$ the stored energy will be

An electric field $(-30 \hat{\mathrm{i}}+20 \hat{\mathrm{j}}) \mathrm{Vm}^{-1}$ exists in the space. If potential at the origin is zero then potential at $(5 \mathrm{~m}, 3 \mathrm{~m})$ is given by

Statement-1: The heat produced by a resistor in any time $t$ during the charging of a capacitor in a series circuit is half the energy stored in the capacitor by that time.
Statement-2: Current in the circuit is equal to the rate of increase in charge on the capacitor.

Assertion (A): Electric potential on the surface of a charged sphere of radius $R$ is $V$. Then electric field at a distance $r=\frac{R}{2}$ from centre is $\frac{V}{2 R}$. Charge is distributed uniformly over the volume.

Reason (R) : From centre to surface, electric field varies linearly with $r$. Here, $r$ is distance from centre.

If the plates of a parallel plate capacitor connected to a battery are moved close to each other, then

A. the charge stored in it, increases.

B. the energy stored in it, decreases.

C. its capacitance increases.

D. the ratio of charge to its potential remains the same.

E. the product of charge and voltage increases.

Choose the most appropriate answer from the options given below:

Two particles of charges 4 nC and Q are kept in air with a separation of 10 cm between them. If the electrostatic potential energy of the system is $1.8 \, \mu \mathrm{J},$ then Q =

Statement-1: A dielectric is inserted between the plates of an isolated fully-charged capacitor. The dielectric completely fills the space between the plates. The magnitude of electrostatic force on either metal plate decreases, as it was before the insertion of dielectric medium.
Statement-2: Due to insertion of dielectric slab in an isolated parallel plate capacitor (the dielectric completely fills the space between the plates), the electrostatic potential energy of the capacitor decreases.

Assertion (A): In the electric field $\mathbf{E}=(4 \hat{\mathbf{i}}+4 \hat{\mathbf{j}}) \mathrm{N} / \mathrm{C}$, electric potential at $A(4 \mathrm{~m}, 0)$ is more than the electric potential at $B(0,4 \mathrm{~m})$.

Reason (R): Electric lines of forces always travel from higher potential to lower potential.