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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4

(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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4

(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,

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

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

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

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

$∴$ $x = 81$

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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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 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]