Electric Potential and Capacitance | Physics JEE previous year questions with Solutions

Q 1 :
Three capacitors of capacitances $25$ $μ F$ , $30$ $μ F$ , and $45$ $μ F$ are connected in parallel to a supply of 100 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 _______ . [2024]

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(86) In series combination: $C_{1} = 25 + 30 + 45 = 100 μ F$

In parallel combination: $\frac{1}{C_{2}} = \frac{1}{25} + \frac{1}{30} + \frac{1}{45} \Rightarrow C_{2} = \frac{450}{43} μ F$

Energy $E = \frac{1}{2} C V^{2}$

$\frac{E_{2}}{E_{1}} = \frac{C_{2}}{C_{1}} = \frac{450 / 43}{100} = \frac{9}{86} \Rightarrow E_{2} = \frac{9}{86} E$

$\Rightarrow \frac{9}{86} E = \frac{9}{x} E \Rightarrow x = 86$

Q 2 :
A capacitor is made of a flat plate of area A and a second plate having a stair-like structure as shown in the figure. If the area of each stair is $\frac{A}{3}$ and the height is d, the capacitance of the arrangement is: [2024]

$\frac{11 ϵ_{0} A}{18 d}$
$\frac{13 ϵ_{0} A}{17 d}$
$\frac{11 ϵ_{0} A}{20 d}$
$\frac{18 ϵ_{0} A}{11 d}$

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(1)

All capacitors are in parallel combination.

Also, effective area is common area only

$C_{e q} = C_{1} + C_{2} + C_{3}$

$= \frac{ϵ_{0} A}{3 \times d} + \frac{ϵ_{0} A}{3 \times 2 d} + \frac{ϵ_{0} A}{3 \times 3 d}$

$= \frac{6 ϵ_{0} A + 3 ϵ_{0} A + 2 ϵ_{0} A}{18 d} = \frac{11 ϵ_{0} A}{18 d}$

Q 3 :
Two identical capacitors have same capacitance C. One of them is charged to the potential V and other to the potential 2V. The negative ends of both are connected together. When the positive ends are also joined together, the decrease in energy of the combined system is [2024].

$\frac{1}{4} C V^{2}$
$2 C V^{2}$
$\frac{1}{2} C V^{2}$
$\frac{3}{4} C V^{2}$

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(1)

Initial energy $= \frac{1}{2} C V^{2} + \frac{1}{2} C \times 4 V^{2} = \frac{5 C V^{2}}{2}$

$\frac{2 C V - q}{C} = \frac{C V + q}{C}$

$2 C V - C V = 2 q$

$q = \frac{C V}{2}$

Final charge on capacitor $= \frac{3 C V}{2}$

$E n e r g y_{final} = 2 \times {(\frac{3 C V}{2})}^{2} \times \frac{1}{2 C} = \frac{9 C V^{2}}{4}$

Decrease in Energy $= \frac{5 C V^{2}}{2} - \frac{9 C V^{2}}{4} = \frac{C V^{2}}{4}$

Q 4 :
A $16 Ω$ wire is bent to form a square loop. A 9V battery with internal resistance $1 Ω$ is connected across one of its sides. If a $4 μ F$ capacitor is connected across one of its diagonals, the energy stored by the capacitor will be $x / 2$ $μ J$ , where $x$ = ______. [2024]

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(81)

At steady state, the current supplied from the battery,

$I = \frac{V}{R_{eq}} = \frac{9}{1 + \frac{12 \times 4}{12 + 4}} = \frac{9}{4} A$

$I_{1} = \frac{9}{4} \times \frac{4}{16} = \frac{9}{16} A$

$V_{A} - V_{B} = I_{1} \times 8 = \frac{9}{16} \times 8 = \frac{9}{2} V$

$∴$ $U = \frac{1}{2} \times 4 \times \frac{81}{4} μ J = \frac{81}{2} μ J$

$∴$ $x = 81$

Q 5 :
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? [2025]

$C_{1} > C_{2}, U_{1} > U_{2}$
$C_{2} > C_{1}, U_{2} > U_{1}$
$C_{1} > C_{2}, U_{1} < U_{2}$
$C_{2} > C_{1}, U_{2} < U_{1}$

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(2)

For a capacitor at steady state

q = CV and $U = \frac{1}{2} C V^{2}$

Since $C_{1}$ and $C_{2}$ are connected in parallel, $V_{1} = V_{2}$

Also from graph $q_{1} < q_{2}$

$\Rightarrow C_{1} V_{1} < C_{2} V_{2}$

i.e., $C_{1} < C_{2}$ or $C_{2} > C_{1}$

$\frac{U_{1}}{U_{2}} = \frac{C_{1} V_{1}^{2}}{C_{2} V_{2}^{2}} = \frac{C_{1}}{C_{2}} < 1$

or $U_{1} < U_{2}$ or $U_{2} > U_{1}$

Q 6 :
A capacitor, $C_{1} = 6 μ F$ is charged to a potential difference of $V_{0} =$ 5 V using a 5 V battery. The battery is removed and another capacitor, $C_{2} = 12 μ F$ is inserted in place of the battery. When the switch 'S' is closed, the charge 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? [2025]

$q_{1} = 15 μ C, q_{2} = 30 μ C$
$q_{1} = 30 μ C, q_{2} = 15 μ C$
$q_{1} = 10 μ C, q_{2} = 20 μ C$
$q_{1} = 20 μ C, q_{2} = 10 μ C$

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(3)

Initially

Finally

q_{1}^{'} = 6 \times 5 = 30 μ C

$6 V_{C} + 12 V_{C} = 30 + 0$

$18 V_{C} = 30$

$\Rightarrow V_{C} = \frac{30}{18} = \frac{5}{3} Volt$

$\Rightarrow q_{1} = \frac{6 \times 5}{3} = 10 μ C$

and $q_{2} = \frac{12 \times 5}{3} = 20 μ C$

Q 7 :
Using a battery, a 100 pF capacitor is charged to 60 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 V, its capacitance is (in pF) [2025]

600
200
400
100

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(2)

New potential $= \frac{C_{0} V_{0}}{C_{0} + C} = \frac{V_{0}}{3}$

$3 C_{0} V_{0} = C_{0} V_{0} + C V_{0}$

$2 C_{0} V_{0} = C V_{0}$

$C = 2 C_{0} = 2 \times 100 = 200 pF$

Q 8 :
Four capacitors each of capacitance 16 $μ$ F are connected as shown in the figure. The capacitance between points A and B is : ______ (in $μ$ F). [2025]

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(64)

$C_{eq} = 4 C = 64$

Q 9 :
Space between the plates of a parallel plate capacitor of plate area 4 ${cm}^{2}$ and separation of (d) 1.77 mm, is filled with uniform dielectric materials with dielectric constants (3 and 5) as shown in figure. Another capacitor of capacitance 7.5 pF is connected in parallel with it. The effective capacitance of this combination is ______ pF. (Given $ε_{0} = 8.85 \times 10^{- 12} F / m$ ) [2025]

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(15)

$\frac{1}{C} = \frac{1}{C_{1}} + \frac{1}{C_{2}} = \frac{d / 2}{A k_{1} ε_{0}} + \frac{d / 2}{A k_{2} ε_{0}}$

$= (\frac{1}{k_{1}} + \frac{1}{k_{2}}) \frac{d}{2 A ε_{0}} = (\frac{1}{3} + \frac{1}{5}) \frac{d}{2 A ε_{0}}$

$\frac{1}{C} = \frac{4}{15} \frac{d}{A ε_{0}}$

$C = \frac{15}{4} \times \frac{A ε_{0}}{d}$

$= \frac{15}{4} \times \frac{4 \times 10^{- 4} \times 8.85 \times 10^{- 12}}{1.77 \times 10^{- 3}} = 7.5 p F$

Finally equivalent capacitance ( $C_{eq}$ ) = 7.5 + 7.5 = 15 pF

Topic Question Set

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Q 2 :
A capacitor is made of a flat plate of area A and a second plate having a stair-like structure as shown in the figure. If the area of each stair is $\frac{A}{3}$ and the height is d, the capacitance of the arrangement is: [2024]

Q 3 :
Two identical capacitors have same capacitance C. One of them is charged to the potential V and other to the potential 2V. The negative ends of both are connected together. When the positive ends are also joined together, the decrease in energy of the combined system is [2024].

Q 4 :
A $16 Ω$ wire is bent to form a square loop. A 9V battery with internal resistance $1 Ω$ is connected across one of its sides. If a $4 μ F$ capacitor is connected across one of its diagonals, the energy stored by the capacitor will be $x / 2$ $μ J$ , where $x$ = ______. [2024]

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Q 5 :
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? [2025]

Q 7 :
Using a battery, a 100 pF capacitor is charged to 60 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 V, its capacitance is (in pF) [2025]

Q 8 :
Four capacitors each of capacitance 16 $μ$ F are connected as shown in the figure. The capacitance between points A and B is : ______ (in $μ$ F). [2025]

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Topic Question Set

Q 1 : Three capacitors of capacitances 25 μF, 30 μF, and 45 μF are connected in parallel to a supply of 100 V. Energy stored in the above combination is E. When these capacitors are connected in series to the same supply, the stored energy is 9xE. The value of x is _______ . [2024]

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Q 2 : A capacitor is made of a flat plate of area A and a second plate having a stair-like structure as shown in the figure. If the area of each stair is A3 and the height is d, the capacitance of the arrangement is: [2024]

Q 3 : Two identical capacitors have same capacitance C. One of them is charged to the potential V and other to the potential 2V. The negative ends of both are connected together. When the positive ends are also joined together, the decrease in energy of the combined system is [2024].

Q 4 : A 16Ω wire is bent to form a square loop. A 9V battery with internal resistance 1Ω is connected across one of its sides. If a 4μF capacitor is connected across one of its diagonals, the energy stored by the capacitor will be x/2 μJ, where x = ______. [2024]

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Q 5 : Two capacitors C1 and C2 are connected in parallel to a battery. Charge-time graph is shown below for the two capacitors. The energy stored with them are U1 and U2, respectively. Which of the given statements is true? [2025]

Q 7 : Using a battery, a 100 pF capacitor is charged to 60 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 V, its capacitance is (in pF) [2025]

Q 8 : Four capacitors each of capacitance 16 μF are connected as shown in the figure. The capacitance between points A and B is : ______ (in μF). [2025]

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Q 2 :
A capacitor is made of a flat plate of area A and a second plate having a stair-like structure as shown in the figure. If the area of each stair is $\frac{A}{3}$ and the height is d, the capacitance of the arrangement is: [2024]

Q 3 :
Two identical capacitors have same capacitance C. One of them is charged to the potential V and other to the potential 2V. The negative ends of both are connected together. When the positive ends are also joined together, the decrease in energy of the combined system is [2024].

Q 4 :
A $16 Ω$ wire is bent to form a square loop. A 9V battery with internal resistance $1 Ω$ is connected across one of its sides. If a $4 μ F$ capacitor is connected across one of its diagonals, the energy stored by the capacitor will be $x / 2$ $μ J$ , where $x$ = ______. [2024]

Q 5 :
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? [2025]

Q 7 :
Using a battery, a 100 pF capacitor is charged to 60 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 V, its capacitance is (in pF) [2025]

Q 8 :
Four capacitors each of capacitance 16 $μ$ F are connected as shown in the figure. The capacitance between points A and B is : ______ (in $μ$ F). [2025]