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JEE Mains | PYQ

In the given circuit, \( C_1 = 2 \, \mu \text{F}, \; C_2 = 0.2 \, \mu \text{F}, \; C_3 = 2 \, \mu \text{F}, \; C_4 = 4 \, \mu \text{F}, \; C_5 = 2 \, \mu \text{F}, \; C_6 = 2 \, \mu \text{F} ,\) The charge stored on capacitor \( C_4 \) is ________ \( \mu \text{C} .\)

A parallel plate capacitor was made with two rectangular plates, each with a length of \( l = 3 \, \text{cm} \) and breadth of \( b = 1 \, \text{cm} \). The distance between the plates is \( 3 \, \mu\text{m} \). Out of the following, which are the ways to increase the capacitance by a factor of \( 10 \)?
(A) \( l = 30 \, \text{cm}, \ b = 1 \, \text{cm}, \ d = 1 \, \mu\text{m} \)
(B) \( l = 3 \, \text{cm}, \ b = 1 \, \text{cm}, \ d = 30 \, \mu\text{m} \)
(C) \( l = 6 \, \text{cm}, \ b = 5 \, \text{cm}, \ d = 3 \, \mu\text{m} \)
(D) \( l = 1 \, \text{cm}, \ b = 1 \, \text{cm}, \ d = 10 \, \mu\text{m} \)
(E) \( l = 5 \, \text{cm}, \ b = 2 \, \text{cm}, \ d = 1 \, \mu\text{m} \)
Choose the correct answer from the options given below:

The reading of the ammeter (\( A \)) in steady state in the following circuit (assuming negligible internal resistance of the ammeter) is __________________ \( A \).

A capacitor of \( 50 \, \mu \text{F} \) is connected in a circuit as shown in figure. The charge on the upper plate of the capacitor is ________ \( \mu \text{C} \).

Consider a parallel plate capacitor of area \( A \) (of each plate) and separation \( 'd' \) between the plates. If \( E \) is the electric field and \( \varepsilon_0 \) is the permittivity of free space between the plates, then potential energy stored in the capacitor is:

Two capacitors, each having capacitance \( 40\,\mu \mathrm{F} \) are connected in series. The space between one of the capacitors is filled with dielectric material of dielectric constant \( K \) such that the equivalence capacitance of the system became \( 24\,\mu \mathrm{F} \). The value of \( K \) will be:

A parallel plate capacitor is formed by two plates each of area \( 30\pi \, \mathrm{cm^{2}} \) separated by \( 1 \, \mathrm{mm} \). A material of dielectric strength \( 3.6 \times 10^{7} \, \mathrm{V \, m^{-1}} \) is filled between the plates. If the maximum charge that can be stored on the capacitor without causing any dielectric breakdown is \( 7 \times 10^{-6} \, \mathrm{C} \), the value of dielectric constant of the material is:
[Use \( \dfrac{1}{4\pi \varepsilon_{0}} = 9 \times 10^{9} \, \mathrm{N \, m^{2} \, C^{-2}} \) ]

As shown in the figure, two parallel plate capacitors having equal plate area of \( 200 \, \text{cm}^{2} \) are joined in such a way that \( a \neq b \). The equivalent capacitance of the combination is \( x \epsilon_{0} \, \text{F} \). The value of \( x \) is __________.

A parallel plate capacitor of capacitance \( 12.5 \, \text{pF} \) is charged by a battery connected between its plates to potential difference of \( 12.0 \, \text{V} \). The battery is now disconnected and a dielectric slab \( (\varepsilon_r = 6) \) is inserted between the plates. The change in its potential energy after inserting the dielectric slab is ____________ \( \times 10^{-12} \, \text{J} \).

The charge on capacitor of capacitance \( 15\,\mu\text{F} \) in the figure given below is:

A galvanometer (\( G \)) of \( 2\,\Omega \) resistance is connected in the given circuit. The ratio of charge stored in \( C_{1} \) and \( C_{2} \) is:

Three parallel plate capacitors \( C_1 \), \( C_2 \) and \( C_3 \) each of capacitance \( 5\,\mu \mathrm{F} \) are connected as shown in figure. The effective capacitance between points \( A \) and \( B \), when the space between the parallel plates of \( C_1 \) capacitor is filled with a dielectric medium having dielectric constant of \( 4 \), is:

A parallel plate capacitor with plate separation \( 5 \, \text{mm} \) is charged by a battery. On introducing a mica sheet of \( 2 \, \text{mm} \) and maintaining the connections of the plates with the terminals of the battery, it is found that it draws \( 25\% \) more charge from the battery. The dielectric constant of mica is _______.

A \( 600 \, \text{pF} \) capacitor is charged by \( 200 \, \text{V} \) supply. It is then disconnected from the supply and is connected to another uncharged \( 600 \, \text{pF} \) capacitor. Electrostatic energy lost in the process is __________ \( \mu \text{J} \).

A capacitor has capacitance \( 5\,\mu F \) when it’s parallel plates are separated by air medium of thickness \( d \). A slab of material of dielectric constant \( 1.5 \) having area equal to that of plates but thickness \( \dfrac{d}{2} \) is inserted between the plates. Capacitance of the capacitor in the presence of slab will be (in \( \mu F \)) ____.

Three capacitors \( C_1 = 2\,\mu \text{F} \), \( C_2 = 6\,\mu \text{F} \) and \( C_3 = 12\,\mu \text{F} \) are connected as shown in figure. Find the ratio of the charge on capacitors \( C_1 \), \( C_2 \) and \( C_3 \) respectively:

Two metallic plates form a parallel plate capacitor. The distance between the plates is \( d \). A metal sheet of thickness \( \frac{d}{2} \) and of area equal to area of each plate is introduced between the plates. What will be the ratio of the new capacitance to the original capacitance of the capacitor?

A parallel plate capacitor is filled equally (half) with two dielectrics of dielectric constants \( \varepsilon_{1} \) and \( \varepsilon_{2} \), as shown in figures. The distance between the plates is \( d \) and area of each plate is \( A \). If capacitance in first configuration and second configuration are \( C_{1} \) and \( C_{2} \) respectively, then \( \frac{C_{1}}{C_{2}} \) is:

A parallel plate capacitor has plate area \( 40 \, \text{cm}^2 \) and plates separation \( 2 \, \text{mm} \). The space between the plates is filled with a dielectric medium of a thickness \( 1 \, \text{mm} \) and dielectric constant \( 5 \). The capacitance of the system is:

A capacitor is made of a flat plate of area \( A \) and a second plate having a stair-like structure as shown in figure. If the area of each stair is \( \frac{A}{3} \) and the height is \( d \), the capacitance of the arrangement is:

The total charge on the system of capacitors \( C_1 = 1\,\mu F \), \( C_2 = 2\,\mu F \), \( C_3 = 4\,\mu F \) and \( C_4 = 3\,\mu F \) connected in parallel is:
(Assume a battery of \( 20\,\text{V} \) is connected to the combination)

A capacitor \( P \) with capacitance \( 10 \times 10^{-6} \, \text{F} \) is fully charged with a potential difference of \( 6.0 \, \text{V} \) and disconnected from the battery. The charged capacitor \( P \) is connected across another capacitor \( Q \) with capacitance \( 20 \times 10^{-6} \, \text{F} \). The charge on capacitor \( Q \) when equilibrium is established will be \( \alpha \times 10^{-5} \, \text{C} \) (assume capacitor \( Q \) does not have any charge initially), the value of \( \alpha \) is __________.

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:
\( \displaystyle \varepsilon(x) = \varepsilon_0 + kx, \text{ for } \left(0 < x \le \frac{d}{2}\right) \)
\( \varepsilon(x) = \varepsilon_0 + k(d-x), \text{ for }\left(\dfrac{d}{2} \le x \le d \right)\)

In a parallel plate capacitor set up, the plate area of capacitor is \( 2 \ \text{m}^{2} \) and the plates are separated by \( 1 \ \text{m} \). If the space between the plates are filled with a dielectric material of thickness \( 0.5 \ \text{m} \) and area \( 2 \ \text{m}^{2} \) (see figure) the capacitance of the set-up will be ________ \( \varepsilon_{0} \). (Dielectric constant of the material \( = 3.2 \)) (Round off to the Nearest Integer)

In the given figure, the charge stored in \( 6 \, \mu \text{F} \) capacitor. when points \( A \) and \( B \) are joined by a connecting wire is _____ \( \mu \text{C} \).

A \( 16 \, \Omega \) wire is bend to form a square loop. A \( 9 \, \text{V} \) battery with internal resistance \( 1 \, \Omega \) is connected across one of its sides. If a \( 4 \, \mu\text{F} \) capacitor is connected across one of its diagonals, the energy stored by the capacitor will be \( \frac{x}{2} \, \mu \text{J} \), where \( x = \) __________.

The electric field between the two parallel plates of a capacitor of \( 1.5 \, \mu \text{F} \) capacitance drops to one third of its initial value in \( 6.6 \, \mu \text{s} \) when the plates are connected by a thin wire. The resistance of this wire is ________ \( \Omega \). (Given, \( \log 3 = 1.1 \))

A parallel plate capacitor with plate separation \( 5 \, \text{mm} \) is charged up by a battery. It is found that on introducing a dielectric sheet of thickness \( 2 \, \text{mm} \), while keeping the battery connections intact, the capacitor draws \( 25\% \) more charge from the battery than before. The dielectric constant of the sheet is \( \_\_\_\_\_\_\_\_ \).

Given below two statements: One is labelled as Assertion (A) and other is labelled as Reason (R).
Assertion (A): Non-polar materials do not have any permanent dipole moment.
Reason (R): When a non-polar material is placed in an electric field, the centre of the positive charge distribution of it's individual atom or molecule coincides with the centre of the negative charge distribution.
In the light of above statements, choose the most appropriate answer from the options given below.

A force of \( 10 \, \text{N} \) acts on a charged particle placed between two plates of a charged capacitor. If one plate of capacitor is removed, then the force acting on that particle will be:

A capacitor is connected to a \( 20 \, \text{V} \) battery through a resistance of \( 10 \, \Omega \). It is found that the potential difference across the capacitor rises to \( 2 \, \text{V} \) in \( 1 \, \mu \text{s} \). The capacitance of the capacitor is_____ \( \mu \text{F} \).
Given $\ell n\left(\frac{10}{9}\right) = 0.105$

A parallel plate air capacitor has a capacitance \( C \). When it is half filled as show in figure with a dielectric constant \( K = 5 \), the percentage increase in the capacitance is ______.

A composite parallel plate capacitor is made up of two different dielectric materials with different thickness (\( t_1 \) and \( t_2 \)) as shown in figure. The two different dielectric materials are separated by a conducting foil \( F \). The voltage of the conducting foil is ________ \( \text{V} \).

A capacitor of capacitance \( C = 1 \, \mu\text{F} \) is suddenly connected to a battery of \( 100 \, \text{volt} \) through a resistance \( R = 100 \, \Omega \). The time taken for the capacitor to be charged to get \( 50 \, \text{V} \) is: [Take \( \ln 2 = 0.69 \)]

The charge accumulated on the capacitor connected in the following circuit is ___________ $\mu\text{C}$. (given $C = 150\ \mu\text{F}$)

A parallel plate capacitor with width \( 4 \, \text{cm} \), length \( 8 \, \text{cm} \) and separation between the plates of \( 4 \, \text{mm} \) is connected to a battery of \( 20 \, \text{V} \). A dielectric slab of dielectric constant \( 5 \) having length \( 1 \, \text{cm} \), width \( 4 \, \text{cm} \) and thickness \( 4 \, \text{mm} \) is inserted between the plates of parallel plate capacitor. The electrostatic energy of this system will be ________ \( \varepsilon_{0} \, \text{J} \).
(Where \( \varepsilon_{0} \) is the permittivity of free space).

A parallel-plate capacitor of capacitance \( 40 \, \mu \, \text{F} \) is connected to a \( 100 \, \text{V} \) power supply. Now the intermediate space between the plates is filled with a dielectric material of dielectric constant \( K = 2 \). Due to the introduction of dielectric material, the extra charge and the change in the electrostatic energy in the capacitor, respectively, are:

A parallel plate capacitor with plate area \( A \) and plate separation \( d = 2\,\text{m} \) has a capacitance of \( 4\,\mu\text{F} \). The new capacitance of the system if half of the space between them is filled with a dielectric material of dielectric constant \( K = 3 \) (as show in figure) will be:

The capacitor of capacitance \( 50 \, \text{pF} \) is charged by \( 100 \, \text{V} \) source. It is then connected to another uncharged identical capacitor. Electrostatic energy loss in the process is __________ \( \text{nJ} \).

Two parallel plate capacitors of capacity \( C \) and \( 3C \) are connected in parallel combination and charged to a potential difference \( 18 \, \text{V} \). The battery is then disconnected and the space between the plates of the capacitor of capacity \( C \) is completely filled with a material of dielectric constant \( 9 \). The final potential difference across the combination of capacitors will be __________ \( \text{V} \).

Two capacitors of capacities \( 2C \) and \( C \) are joined in parallel and charged up to potential \( V \). The battery is removed and the capacitor capacity \( C \) is filled completely with a medium of dielectric constant \( K \). Potential difference across the capacitors will now be:

Two capacitors $C_1$ and $C_2$ are connected in parallel to a battery. Charge-time graph is shown below for the two capacitors. The energy stored with them are $U_1$ and $U_2$, respectively. Which of the given statements is true?

If \( q_f \) is the free charge on the capacitor plates and \( q_b \) is the bound charge on the dielectric slab of dielectric constant \( k \) placed between the capacitor plates, then bound charge \( q_b \) can be expressed as:

A capacitor is discharging through a resistor \( R \). Consider in time \( t_1 \), the energy stored in the capacitor reduces to half its initial value and in time \( t_2 \), the charge stored reduces to one eight of its initial values. The ratio \( t_1 / t_2 \) will be:

A parallel plate capacitor is having separation between plates \( 0.885 \, \text{mm} \). It has a capacitance of \( 1 \, \mu\text{F} \) when the space between the plates is filled with an insulating material of resistivity \( 1 \times 10^{13} \, \Omega \text{m} \) and resistance \( 17.7 \times 10^{14} \, \Omega \). Relative permittivity of the insulating material is \( \alpha \times 10^{7} \). The value of \( \alpha \) is ______.
[Take permittivity of free space \( = 8.85 \times 10^{-12} \, \text{F/m} \)]

The stored charge in the capacitor in steady state of the following circuit is ______ \( \mu \mathrm{C} \).

A capacitor of \( 10 \, \mu \text{F} \) capacitance whose plates are separated by \( 10 \, \text{mm} \) through air and each plate has area \( 4 \, \text{cm}^2 \) is now filled equally with two dielectric media of \( K_1 = 2, K_2 = 3 \) respectively as shown in figure. If new force between the plates is \( 8 \, \text{N} \). The supply voltage is ____ \( V \).

Two identical thin metal plates has charge \( q_1 \) and \( q_2 \) respectively such that \( q_1 > q_2 \). The plates were brought close to each other to form a parallel plate capacitor of capacitance \( C \). The potential difference between them is:

The equivalent capacitance of the combination shown is:

A parallel plate capacitor with air between the plate has a capacitance of \( 15 \, \text{pF} \). The separation between the plate becomes twice and the space between them is filled with a medium of dielectric constant \( 3.5 \). Then the capacitance becomes \( \frac{x}{4} \, \text{pF} \). The value of \( x \) is ________ .

Identify the valid statements relevant to the given circuit at the instant when the key is closed.
A. There will be no current through resistor \( R \).
B. There will be maximum current in the connecting wires.
C. Potential difference between the capacitor plates \( A \) and \( B \) is minimum.
D. Charge on the capacitor plates is minimum.
Choose the correct answer from the options given below:

A parallel plate capacitor has plate area \( 100 \, \text{m}^{2} \) and plate separation of \( 10 \, \text{m} \). The space between the plates is filled up to a thickness \( 5 \, \text{m} \) with a material of dielectric constant of \( 10 \). The resultant capacitance of the system is \( x \, \text{pF}\).
The value of \( \varepsilon_{0} = 8.85 \times 10^{-12} \, \text{F·m}^{-1} \)
The value of \( x \) to the nearest integer is __________ .

Three parallel plate capacitors each with area \( A \) and separation \( d \) are filled with two dielectric \( (k_1 \text{ and } k_2) \) in the following fashion. Which of the following is true? \( (k_1 > k_2) \).

The distance between two plates of a capacitor is \( d \) and its capacitance is \( C_1 \), when air is the medium between the plates. If a metal sheet of thickness \( {2d}/{3} \) and of the same area of plate is introduced between the plates, the capacitance of the capacitor becomes \( C_2 \). The ratio \( {C_2}/{C_1} \) is:

A radioactive sample has an average life of \( 30 \, \text{ms} \) and is decaying. A capacitor of capacitance \( 200 \, \mu \text{F} \) is first charged and later connected with resistor \( 'R' \). If the ratio of charge on capacitor to the activity of radioactive sample is fixed with respect to time then the value of \( 'R' \) should be ________ \( \Omega \).

Two capacitors having capacitance \( C_1 \) and \( C_2 \) respectively are connected as shown in figure. Initially, capacitor \( C_1 \) is charged to a potential difference \( V \) volt by a battery. The battery is then removed and the charged capacitor \( C_1 \) is now connected to uncharged capacitor \( C_2 \) by closing the switch \( S \). The amount of charge on the capacitor \( C_2 \), after equilibrium, is:

A parallel plate air capacitor is connected to a battery. The plates are pulled apart at uniform speed \( v \). If \( x \) is the separation between the plates at any instant, then the time rate of change of electrostatic energy of the capacitor is proportional to \( x^{\alpha} \), where \( \alpha \) is ______.

A parallel-plate capacitor with plate area \( A \) has separation \( d \) between the plates. Two dielectric slabs of dielectric constant \( K_1 \) and \( K_2 \) of same area \( \dfrac{A}{2} \) and thickness \( \dfrac{d}{2} \) are inserted in the space between the plates. The capacitance of the capacitor will be given by:
 

A capacitor has air as dielectric medium and two conducting plates of area \( 12 \,\text{cm}^{2} \) and they are \( 0.6 \,\text{cm} \) apart. When a slab of dielectric having area \( 12 \,\text{cm}^{2} \) and \( 0.6 \,\text{cm} \) thickness is inserted between the plates. One of the conducting plates has to be moved by \( 0.2 \,\text{cm} \) to keep the capacitance same as in previous case. The dielectric constant of the slab is:
(Given \( \varepsilon_{0} = 8.834 \times 10^{-12} \,\text{F/m} \))

Four identical rectangular plates with length, \( l = 2 \, \text{cm} \) and breadth, \( b = \dfrac{3}{2} \, \text{cm} \) are arranged as shown in figure. The equivalent capacitance between A and C is \( \dfrac{x \, \varepsilon_{0}}{d} \). The value of \( x \) is ______. (Round off to the Nearest Integer)

A parallel plate capacitor of capacitance \( 1 \,\mu\text{F} \) is charged to a potential difference of \( 20 \,\text{V} \). The distance between plates is \( 1 \,\mu\text{m} \). The energy density between plates of capacitor is:

The space between the plates of a parallel plate capacitor of capacitance \( C \) (without any dielectric) is now filled with three dielectric slabs of dielectric constants \( K_1 = 2 \), \( K_2 = 3 \) and \( K_3 = 5 \) (as shown in figure). If new capacitance is \( \frac{n}{3} C \) then the value of \( n \) is ______.

A capacitor of capacitance \( C \) and potential \( V \) has energy \( E \). It is connected to another capacitor of capacitance \( 2C \) and potential \( 2V \). Then the loss of energy is \( \frac{x}{3} E \), where \( x \) is________.

From the circuit given below, the capacitance between terminals \( A \) and \( B \) shown in the circuit is _______ \( \mu \text{F} \).
(take \( C_1 = C_2 = C_3 = 1 \, \mu \text{F} \) and \( C_4 = 2 \, \mu \text{F} \).)

A parallel plate capacitor has charge \( 5 \times 10^{-6} \, \text{C} \). A dielectric slab is inserted between the plates and almost fills the space between the plates. If the induced charge on one face of the slab is \( 4 \times 10^{-6} \, \text{C} \) then the dielectric constant of the slab is _______.

In an electrical circuit drawn below the amount of charge stored in the capacitor is ______ \( \mu\text{C} \).

The equivalent capacitance between points \( A \) and \( B \) is below shown figure will be________ \( \mu \text{F} \).

A parallel plate capacitor whose capacitance \( C \) is \( 14 \, \text{pF} \) is charged by a battery to a potential difference \( V = 12 \, \text{V} \) between its plates. The charging battery is now disconnected and a porcelain plate with \( k = 7 \) is inserted between the plates, then the plate would oscillate back and forth between the plates with a constant mechanical energy of ______ \( \text{pJ} \). (Assume no friction)

A \( 2\,\mu F \) capacitor \( C_1 \) is first charged to a potential difference of \( 10\,V \) using a battery. Then the battery is removed and the capacitor is connected to an uncharged capacitor \( C_2 \) of \( 8\,\mu F \). The charge in \( C_2 \) on equilibrium condition is ________ \( \mu C \). (Round off to the Nearest Integer)

The material filled between the plates of a parallel plate capacitor has resistivity \( 200 \, \Omega \text{m} \). The value of capacitance of the capacitor is \( 2 \, \text{pF} \). If a potential difference of \( 40 \, \text{V} \) is applied across the plates of the capacitor, then the value of leakage current flowing out of the capacitor is: (given the value of relative permittivity of material is \( 50 \)).

A capacitor of capacitance \( 500\,\mu\text{F} \) is charged completely using a \( dc \) supply of \( 100\,\text{V} \). It is now connected to an inductor of inductance \( 50\,\text{mH} \) to form an \( LC \) circuit. The maximum current in \( LC \) circuit will be ______ \( \text{A} \).

Capacitance of an isolated conducting sphere of radius \( R_1 \) becomes \( n \) times when it is enclosed by a concentric conducting sphere of radius \( R_2 \) connected to earth. The ratio of their radii \( \left( \dfrac{R_2}{R_1} \right) \) is:

In the given figure the total charge stored in the combination of capacitor is \( 100 \, \mu \text{C} \). The value of ‘x’ is __________.

A capacitor of capacitance \( C \) is charged to a potential \( V \). The flux of the electric field through a closed surface enclosing the positive plate of the capacitor is:

A slab of dielectric constant \( K \) has the same cross-sectional area as the plates of a parallel plate capacitor and thickness \( \dfrac{3}{4} d \), where \( d \) is the separation of the plates. The capacitance of the capacitor when the slab is inserted between the plates will be:
(Given \( C_{0} = \) capacitance of capacitor with air as medium between plates.)

Three capacitors of capacitances \( 25 \, \mu \text{F} \), \( 30 \, \mu \text{F} \) and \( 45 \, \mu \text{F} \) are connected in parallel to a supply of \( 100 \, \text{V} \). Energy stored in the above combination is \( E \). When these capacitors are connected in series to the same supply, the stored energy is \( \frac{9}{x} E \). The value of \( x \) is __________.

A parallel plate capacitor has capacitance $C$, when there is vacuum within the parallel plates. A sheet having thickness $\left(\frac{1}{3}\right)^{\text{rd}}$ of the separation between the plates and relative permittivity $K$ is introduced between the plates. The new capacitance of the system is:

The net current flowing in the given circuit is __________ \( \text{A} \).

A capacitor, \( C_{1} = 6\,\mu\text{F} \) is charged to a potential difference of \( V_{0} = 5\,\text{V} \) using a \( 5\,\text{V} \) battery. The battery is removed and another capacitor, \( C_{2} = 12\,\mu\text{F} \) is inserted in place of the battery. When the switch ‘S’ is closed, the charged flows between the capacitors for some time until equilibrium condition is reached. What are the charges \( (q_{1} \) and \( q_{2}) \) on the capacitors \( C_{1} \) and \( C_{2} \) when equilibrium condition is reached.

In the reported figure, a capacitor is formed by placing a compound dielectric between the plates of parallel plate capacitor. The expression for the capacity of the solid capacitor will be: (Given area of plate \( = A \))

A parallel plate capacitor of capacitance \( 2 \, \text{F} \) is charged to a potential \( V \), The energy stored in the capacitor is \( E_{1} \). The capacitor is now connected to another uncharged identical capacitor in parallel combination. The energy stored in the combination is \( E_{2} \). The ratio \( {E_{2}}/{E_{1}} \) is:

The circuit shown in the figure consists of a charged capacitor of capacity \( 3 \, \mu\text{F} \) and a charge of \( 30 \, \mu\text{C} \). At time \( t = 0 \), when the key is closed, the value of current flowing through the \( 5 \, \text{M}\Omega \) resistor is \(' x' \, \mu\text{A} \).
The value of \( 'x' \) to the nearest integer is __________ .

Using a battery, a \( 100 \, \text{pF} \) capacitor is charged to \( 60 \, \text{V} \) and then the battery is removed. After that, a second uncharged capacitor is connected to the first capacitor in parallel. If the final voltage across the second capacitor is \( 20 \, \text{V} \), its capacitance is: (in pF)

Two parallel plate capacitors \( C_{1} \) and \( C_{2} \) each having capacitance of \( 10 \, \mu \text{F} \) are individually charged by a \( 100 \, \text{V} \) D.C. source. Capacitor \( C_{1} \) is kept connected to the source and a dielectric slab is inserted between it plates. Capacitor \( C_{2} \) is disconnected from the source and then a dielectric slab is inserted in it. Afterwards the capacitor \( C_{1} \) is also disconnected from the source and the two capacitors are finally connected in parallel combination. The common potential of the combination will be ______ V.

A parallel plate capacitor filled with a medium of dielectric constant \( 10 \), is connected across a battery and is charged. The dielectric slab is replaced by another slab of dielectric constant \( 15 \). Then the energy of capacitor will:

A parallel plate capacitor with plate area \( A \) and plate separation \( d \) is filled with a dielectric material of dielectric constant \( K = 4 \). The thickness of the dielectric material is \( x \), where \( x < d \). 

Let \( C_{1} \) and \( C_{2} \) be the capacitance of the system for \( x = \dfrac{1}{3} d \) and \( x = \dfrac{2d}{3} \), respectively. If \( C_{1} = 2 \, \mu \text{F} \) the value of \( C_{2} \) is __________ \( \mu \text{F} \)

A \( 5 \, \mathrm{mg} \) particle carrying a charge of \( 5\pi \times 10^{-6} \, \mathrm{C} \) is moving with velocity of \( (3\hat{\imath} + 2\hat{k}) \times 10^{-2} \, \mathrm{m\, /s} \) in a region having magnetic field \( |\vec{B}| = 0.1 \hat{k} \, \mathrm{Wb\, /m^{2}} \). It moves a distance of \( \alpha \) meter along \( \hat{k} \) when it completes \( 5 \) revolutions. The value of \( \alpha \) is ______.

A parallel plate capacitor of capacitance \( 200 \, \mu \text{F} \) is connected to a battery of \( 200 \, \text{V} \). A dielectric slab of dielectric constant \( 2 \) is now inserted into the space between plates of capacitor while the battery remain connected. The change in the electrostatic energy in the capacitor will be _______ \( \text{J} \).

Under steady state condition the potential difference across the capacitor in the circuit is ________ \( \text{V} \).

Space between the plates of a parallel plate capacitor of plate area \( 4 \, \text{cm}^2 \) and separation of \( (d) \) \( 1.77 \, \text{mm} \), is filled with uniform dielectric materials with dielectric constants \( (3 \text{ and } 5) \) as shown in figure. Another capacitor of capacitance \( 7.5 \, \text{pF} \) is connected in parallel with it. The effective capacitance of this combination is ______ pF. [Given \( \varepsilon_0 = 8.85 \times 10^{-12} \, \text{F} / \text{m} \)]

A source of potential difference \( V \) is connected to the combination of two identical capacitors as shown in the figure. When key ‘\( K \)’ is closed, the total energy stored across the combination is \( E_1 \). Now key ‘\( K \)’ is opened and dielectric of dielectric constant \( 5 \) is introduced between the plates of the capacitors. The total energy stored across the combination is now \( E_2 \). The ratio \( {E_1}/{E_2} \) will be: 

Four capacitors each of capacitance \( 16 \, \mu \text{F} \) are connected as shown in the figure. The capacitance between points \( A \) and \( B \) is ______ (in \( \mu \text{F} \)).

A parallel plate capacitor consisting of two circular plates of radius \( 10 \, \text{cm} \) is being charged by a constant current of \( 0.15 \, \text{A} \). If the rate of change of potential difference between the plates is \( 7 \times 10^{8} \, \text{V/s} \) then the integer value of the distance between the parallel plates is \( \left( \text{Take, } \epsilon_{0} = 9 \times 10^{-12} \, \frac{F}{m} , \\ \pi = \frac{22}{7} \right) \) _______ \( \mu \text{m} \) .

A condenser of \( 2 \, \mu \text{F} \) capacitance is charged steadily from \( 0 \) to \( 5 \, \text{C} \). Which of the following graph represents correctly the variation of potential difference (\( V \)) across it’s plates with respect to the charge (\( Q \)) on the condenser?