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Chapter 11: Problem 1640

A battery is used to charge a parallel plate capacitor till the potential difference between the plates becomes equal to the electromotive force of the battery. The ratio of the energy stored in the capacitor and work done by the battery will be (A) \((1 / 2)\) (B) \((2 / 1)\) (C) 1 (D) \((1 / 4)\)

Short Answer

Expert verified
The ratio of the energy stored in the capacitor and the work done by the battery is \(\frac{1}{2}\).

Step by step solution

01

Write the formula for energy stored in a capacitor

The formula for the energy stored in a capacitor is given by: \[U_{cap} = \frac{1}{2}CV^2\] where \(U_{cap}\) is the energy stored in the capacitor, \(C\) is the capacitance, and \(V\) is the potential difference across the capacitor.
02

Write the formula for work done by the battery

The formula for the work done by the battery to charge the capacitor is given by: \[W_{battery} = QV\] where \(W_{battery}\) is the work done by the battery, \(Q\) is the charge on the capacitor, and \(V\) is the potential difference (EMF of the battery in this case) across the capacitor.
03

Express charge in terms of capacitance and potential difference

Recall that the charge on a capacitor can be expressed as: \[Q = CV\] Substitute this expression for charge into the work done by the battery formula: \[W_{battery} = (CV)V\] \[W_{battery} = CV^2\]
04

Calculate the ratio of energy stored and work done

Now we can find the ratio of the energy stored in the capacitor to the work done by the battery: \[\frac{U_{cap}}{W_{battery}} = \frac{\frac{1}{2}CV^2}{CV^2}\]
05

Simplify the ratio

Simplify the expression for the ratio: \[\frac{U_{cap}}{W_{battery}} = \frac{\frac{1}{2}CV^2}{CV^2} = \frac{1}{2}\] According to our calculations, the ratio of the energy stored in the capacitor and the work done by the battery is \(\frac{1}{2}\). Therefore, the correct answer is: (A) \(\frac{1}{2}\)

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Most popular questions from this chapter

The electric potential \(\mathrm{V}\) is given as a function of distance \(\mathrm{x}\) (meter) by $\mathrm{V}=\left(5 \mathrm{x}^{2}+10 \mathrm{x}-9\right)\( volt. Value of electric field at \)\mathrm{x}=1$ is \(\ldots \ldots\) \((\mathrm{A})-20(\mathrm{v} / \mathrm{m})\) (B) \(6(\mathrm{v} / \mathrm{m})\) (C) \(11(\mathrm{v} / \mathrm{m})\) (D) \(-23(\mathrm{v} / \mathrm{m})\)

Charges \(+q\) and \(-q\) are placed at point \(A\) and \(B\) respectively which are a distance \(2 \mathrm{~L}\) apart, \(\mathrm{C}\) is the midpoint between \(\mathrm{A}\) and \(\mathrm{B}\). The work done in moving a charge \(+Q\) along the semicircle \(C R D\) is \(\ldots \ldots\) (A) $\left[(\mathrm{qQ}) /\left(2 \pi \mathrm{\epsilon}_{0} \mathrm{~L}\right)\right]$ (B) \(\left[(-q Q) /\left(6 \pi \in_{0} L\right)\right]\) (C) $\left[(\mathrm{qQ}) /\left(6 \pi \mathrm{e}_{0} \mathrm{~L}\right)\right]$ (D) \(\left[(q Q) /\left(4 \pi \in_{0} L\right)\right]\)

A thin spherical conducting shell of radius \(\mathrm{R}\) has a charge q. Another charge \(Q\) is placed at the centre of the shell. The electrostatic potential at a point p a distance \((\mathrm{R} / 2)\) from the centre of the shell is ..... (A) \(\left[(q+Q) /\left(4 \pi \epsilon_{0}\right)\right](2 / R)\) (B) $\left[\left\\{(2 Q) /\left(4 \pi \epsilon_{0} R\right)\right\\}-\left\\{(2 Q) /\left(4 \pi \epsilon_{0} R\right)\right]\right.$ (C) $\left[\left\\{(2 Q) /\left(4 \pi \in_{0} R\right)\right\\}+\left\\{q /\left(4 \pi \epsilon_{0} R\right)\right]\right.$ (D) \(\left[(2 \mathrm{Q}) /\left(4 \pi \epsilon_{0} \mathrm{R}\right)\right]\)

The capacitors of capacitance \(4 \mu \mathrm{F}, 6 \mu \mathrm{F}\) and $12 \mu \mathrm{F}$ are connected first in series and then in parallel. What is the ratio of equivalent capacitance in the two cases? (A) \(2: 3\) (B) \(11: 1\) (C) \(1: 11\) (D) \(1: 3\)

Two identical metal plates are given positive charges \(\mathrm{Q}_{1}\) and \(\mathrm{Q}_{2}\left(<\mathrm{Q}_{1}\right)\) respectively. If they are now brought close to gather to form a parallel plate capacitor with capacitance \(\mathrm{c}\), the potential difference between them is (A) \(\left[\left(Q_{1}+Q_{2}\right) /(2 c)\right]\) (B) \(\left[\left(\mathrm{Q}_{1}+\mathrm{Q}_{2}\right) / \mathrm{c}\right]\) (C) \(\left[\left(\mathrm{Q}_{1}-\mathrm{Q}_{2}\right) /(2 \mathrm{c})\right]\) (D) \(\left[\left(\mathrm{Q}_{1}-\mathrm{Q}_{2}\right) / \mathrm{c}\right]\)
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