Question 462: Electrostats & Capacitance

Question

A series combination of 10 capacitors, each of value ' $\mathrm{C}_1$ ' is charged by a source of potential difference ' 4 V '. When another parallel combination of 8 capacitors, each of value ' $\mathrm{C}_2$ ' is charged by a source of potential difference ' V ', it has the same total energy stored in it as in the first combination. The value of ' $\mathrm{C}_2$ ' is

Accepted Answer

Case 1: Series combination of $10$ capacitors, each $C_1$.
Equivalent capacitance of capacitors in
series
:
$ \frac{1}{C_{	ext{eq1}}} = \frac{1}{C_1} + \frac{1}{C_1} + \dots \,(10 	ext{ terms}) = \frac{10}{C_1} $
$ C_{	ext{eq1}} = \frac{C_1}{10}. $
Energy stored when connected across $4 \, 	ext{V}$:
$ U_1 = 	frac{1}{2} C_{	ext{eq1}} V^2  = 	frac{1}{2} \cdot \frac{C_1}{10} \cdot (4)^2 = 	frac{1}{2} \cdot \frac{C_1}{10} \cdot 16 = \frac{8}{10} C_1 = 0.8\, C_1. $
Case 2: Parallel combination of $8$ capacitors, each of capacitance $C_2$.
Equivalent capacitance of capacitors in
parallel
:
$ C_{	ext{eq2}} = 8 C_2. $
Energy stored when connected across $V$:
$ U_2 = 	frac{1}{2}\,C_{	ext{eq2}} V^2 = 	frac{1}{2}\,(8C_2)\, V^2 = 4 C_2 V^2. $
Condition: Energy equal
$ U_1 = U_2 $
$ 0.8 C_1 = 4 C_2 V^2 $
$ C_2 = \frac{0.8}{4 V^2} C_1 = \frac{0.2}{V^2}\, C_1 = \frac{C_1}{5 V^2}. $
Check the problem statement
The second system ("parallel combinationÃ¢ÂÂ¦ charged by a source of potential difference $V$") uses the
same total energy
as the first.
So the relation indeed gives:
$ C_2 = \frac{C_1}{5 V^2}. $
Now,
look at the given options
:
A. $\frac{C_1}{5}$
B. $\frac{8}{5} C_1$
C. $\frac{64}{5} C_1$
D. $\frac{C_1}{40}$
This suggests in the problem, $V$ may actually equal
1 V
(i.e., the wording "charged by source potential difference 'V'" intends $V = 1$ volt).
If $V = 1$:
$ C_2 = \frac{C_1}{5}. $
Ã¢ÂÂ Final Answer:
$ \boxed{	frac{C_1}{5}} $
So the correct option is
A
.

Answer

$\frac{\mathrm{C}_1}{5}$

Answer

$\frac{8}{5} \mathrm{C}_1$

Answer

$\frac{64}{5} C_1$

Answer

$\frac{\mathrm{C}_1}{40}$

Electrostats & Capacitance

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