Answer
64.8k+ views
Hint: Kirchhoff’s Voltage Law is that the second of his elementary laws we will use for circuit analysis. His voltage law states that for a closed-loop system series path the algebraically total of all the voltages around any closed-loop system in an exceeding circuit is adequate zero. This is often as a result of a circuit loop that could be a closed conducting path thus no energy is lost.
Complete step by step solution: Here in the question it is given that the capacitor is charged initially with the battery. For this, the diagram is shown below.
![](https://www.vedantu.com/question-sets/46d39468-26e0-4844-b638-ec4e4a04b3c21265405751357374768.png)
![](https://www.vedantu.com/question-sets/2791205a-0ea4-403e-ac15-c15410372066474408443093521590.png)
The total charge is equal to ${Q_0}$;
Therefore from the figure, we can write,
$ \Rightarrow {Q_1} + {Q_2} = {Q_0}$
Now applying KVL,
And as we know in KVL for a closed-loop system series path the algebraically total of all the voltages around any closed-loop system in an exceeding circuit is adequate zero.
Therefore,
\[ \Rightarrow \dfrac{{{Q_1}}}{C} = \dfrac{{{Q_2}}}{C} = 0\]
On further solving the equation, we can write it as;
$ \Rightarrow {Q_1} = {Q_2}$
And since ${Q_1} + {Q_2} = {Q_0}$
Therefore
$ \Rightarrow {Q_1} = {Q_2} = \dfrac{{{Q_0}}}{2}$
Since the initial electrostatic charge is,
\[ \Rightarrow {E_i} = \dfrac{1}{2}C{V^2} = \dfrac{{Q_0^2}}{{2C}}\]
Therefore, the final electrostatic charge will be
\[ \Rightarrow {E_f} = \dfrac{1}{2}C{V^2} + \dfrac{1}{2}C{V^2}\]
Now putting the values which we have calculated initially in the above equation, we get
\[ \Rightarrow \dfrac{{Q_1^2}}{{2C}} + \dfrac{{Q_2^2}}{{2C}}\]
On further solving,
\[ \Rightarrow \dfrac{{Q_0^2}}{{8C}} + \dfrac{{Q_0^2}}{{8C}}\]
Adding both the terms, we get
\[ \Rightarrow \dfrac{{Q_0^2}}{{4C}}\]
So now we will calculate the ratio of ${E_f}$and ${E_i}$
We get,
$\dfrac{{{E_f}}}{{{E_i}}} = \dfrac{{\dfrac{{Q_0^2}}{{4C}}}}{{\dfrac{{Q_0^2}}{{2C}}}}$
On solving this, we get
$\dfrac{{{E_f}}}{{{E_i}}} = \dfrac{1}{2}$
Therefore, the total electrostatic energy of the resulting system decrease by a factor $2$.
Note: The theory behind Kirchhoff’s second law is additionally called the law of conservation of voltage, and this is often notably helpful for us once managing series circuits, as series circuits conjointly act as voltage dividers and therefore the voltage divider circuit is a very important application of the many series circuits.
Complete step by step solution: Here in the question it is given that the capacitor is charged initially with the battery. For this, the diagram is shown below.
![](https://www.vedantu.com/question-sets/46d39468-26e0-4844-b638-ec4e4a04b3c21265405751357374768.png)
![](https://www.vedantu.com/question-sets/2791205a-0ea4-403e-ac15-c15410372066474408443093521590.png)
The total charge is equal to ${Q_0}$;
Therefore from the figure, we can write,
$ \Rightarrow {Q_1} + {Q_2} = {Q_0}$
Now applying KVL,
And as we know in KVL for a closed-loop system series path the algebraically total of all the voltages around any closed-loop system in an exceeding circuit is adequate zero.
Therefore,
\[ \Rightarrow \dfrac{{{Q_1}}}{C} = \dfrac{{{Q_2}}}{C} = 0\]
On further solving the equation, we can write it as;
$ \Rightarrow {Q_1} = {Q_2}$
And since ${Q_1} + {Q_2} = {Q_0}$
Therefore
$ \Rightarrow {Q_1} = {Q_2} = \dfrac{{{Q_0}}}{2}$
Since the initial electrostatic charge is,
\[ \Rightarrow {E_i} = \dfrac{1}{2}C{V^2} = \dfrac{{Q_0^2}}{{2C}}\]
Therefore, the final electrostatic charge will be
\[ \Rightarrow {E_f} = \dfrac{1}{2}C{V^2} + \dfrac{1}{2}C{V^2}\]
Now putting the values which we have calculated initially in the above equation, we get
\[ \Rightarrow \dfrac{{Q_1^2}}{{2C}} + \dfrac{{Q_2^2}}{{2C}}\]
On further solving,
\[ \Rightarrow \dfrac{{Q_0^2}}{{8C}} + \dfrac{{Q_0^2}}{{8C}}\]
Adding both the terms, we get
\[ \Rightarrow \dfrac{{Q_0^2}}{{4C}}\]
So now we will calculate the ratio of ${E_f}$and ${E_i}$
We get,
$\dfrac{{{E_f}}}{{{E_i}}} = \dfrac{{\dfrac{{Q_0^2}}{{4C}}}}{{\dfrac{{Q_0^2}}{{2C}}}}$
On solving this, we get
$\dfrac{{{E_f}}}{{{E_i}}} = \dfrac{1}{2}$
Therefore, the total electrostatic energy of the resulting system decrease by a factor $2$.
Note: The theory behind Kirchhoff’s second law is additionally called the law of conservation of voltage, and this is often notably helpful for us once managing series circuits, as series circuits conjointly act as voltage dividers and therefore the voltage divider circuit is a very important application of the many series circuits.
Recently Updated Pages
Write a composition in approximately 450 500 words class 10 english JEE_Main
![arrow-right](/cdn/images/seo-templates/arrow-right.png)
Arrange the sentences P Q R between S1 and S5 such class 10 english JEE_Main
![arrow-right](/cdn/images/seo-templates/arrow-right.png)
What is the common property of the oxides CONO and class 10 chemistry JEE_Main
![arrow-right](/cdn/images/seo-templates/arrow-right.png)
What happens when dilute hydrochloric acid is added class 10 chemistry JEE_Main
![arrow-right](/cdn/images/seo-templates/arrow-right.png)
If four points A63B 35C4 2 and Dx3x are given in such class 10 maths JEE_Main
![arrow-right](/cdn/images/seo-templates/arrow-right.png)
The area of square inscribed in a circle of diameter class 10 maths JEE_Main
![arrow-right](/cdn/images/seo-templates/arrow-right.png)
Other Pages
A boat takes 2 hours to go 8 km and come back to a class 11 physics JEE_Main
![arrow-right](/cdn/images/seo-templates/arrow-right.png)
Electric field due to uniformly charged sphere class 12 physics JEE_Main
![arrow-right](/cdn/images/seo-templates/arrow-right.png)
In the ground state an element has 13 electrons in class 11 chemistry JEE_Main
![arrow-right](/cdn/images/seo-templates/arrow-right.png)
According to classical free electron theory A There class 11 physics JEE_Main
![arrow-right](/cdn/images/seo-templates/arrow-right.png)
Differentiate between homogeneous and heterogeneous class 12 chemistry JEE_Main
![arrow-right](/cdn/images/seo-templates/arrow-right.png)
Excluding stoppages the speed of a bus is 54 kmph and class 11 maths JEE_Main
![arrow-right](/cdn/images/seo-templates/arrow-right.png)