Question 504: Electrostats & Capacitance

Question

Two identical capacitors have the same capacitance ' $C$ '. One of them is charged to potential $\mathrm{V}_1$ and other to $\mathrm{V}_2$. The negative ends of capacitors are connected together. When positive ends are also connected, the decrease in energy of the combined system is

Accepted Answer

To determine the decrease in energy when two identical capacitors are connected, we start by calculating the initial energy of the system. Each capacitor has capacitance $ C $, with one charged to a potential $ V_1 $ and the other to $ V_2 $.
Initial Energy:
The initial energy $ U_i $ of the system is the sum of the energies stored in both capacitors:
$ U_i = \frac{1}{2} C V_1^2 + \frac{1}{2} C V_2^2 = \frac{1}{2} C (V_1^2 + V_2^2) $
When Capacitors are Joined:
When the capacitors are connected, they share charge to reach a common potential $ V $. The formula for this common potential is:
$ V = \frac{CV_1 + CV_2}{2C} = \frac{V_1 + V_2}{2} $
Final Energy:
The final energy $ U_f $ of the system is given by:
$ U_f = \frac{1}{2} (2C) V^2 = \frac{1}{2} 2C \left(\frac{V_1 + V_2}{2}ight)^2 = \frac{1}{4} C (V_1 + V_2)^2 $
Decrease in Energy:
The decrease in energy is the difference between the initial and final energies:
$ 	ext{Decrease in energy} = U_i - U_f = \frac{1}{2} C (V_1^2 + V_2^2) - \frac{1}{4} C (V_1 + V_2)^2 $
Through simplification, this can be expressed as:
$ 	ext{Decrease in energy} = \frac{1}{4} C (V_1 - V_2)^2 $

Answer

$\frac{1}{4} \mathrm{C}\left(\mathrm{V}_1^2+\mathrm{V}_2^2\right)$

Answer

$\frac{1}{4} \mathrm{C}\left(\mathrm{V}_1^2-\mathrm{V}_2^2\right)$

Answer

$\frac{1}{4} C\left(V_1+V_2\right)^2$

Answer

$\frac{1}{4} \mathrm{C}\left(\mathrm{V}_1-\mathrm{V}_2\right)^2$

Electrostats & Capacitance

Two identical capacitors have the same capacitance ' C '. One of them is charged to potential \mathrm{V}_1 and other to \mathrm{V}_2. The negative ends of capacitors are connected together. When positive ends are also connected, the decrease in energy of the combined system is

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