Question 477: Electrostatics & Capacitance

Question

Seven capacitors each of capacitance $2 \mu \mathrm{~F}$ are connected in a configuration to obtain an effective capacitance $\frac{6}{13} \mu \mathrm{~F}$. The combination which will achieve this will have

Accepted Answer

To achieve an effective capacitance of $\frac{6}{13} \, \mu \mathrm{F}$ using seven capacitors, each with a capacitance of $2 \, \mu \mathrm{F}$, you should arrange them with 3 capacitors in parallel and 4 capacitors in series.
Here's how the calculation works:
Parallel Connection
: When capacitors are connected in parallel, their capacitances add up. So for 3 capacitors in parallel:
$    C_{	ext{parallel}} = 3 	imes C = 3 	imes 2 \, \mu \mathrm{F} = 6 \, \mu \mathrm{F}    $
Series Connection
: When capacitors are connected in series, the reciprocal of the total capacitance is the sum of the reciprocals of individual capacitances. For 4 capacitors, each of capacitance $C$, in series, the total capacitance is:
$    \frac{1}{C_{	ext{series}}} = \frac{1}{C} + \frac{1}{C} + \frac{1}{C} + \frac{1}{C} = \frac{4}{C}    $
Therefore:
$    C_{	ext{series}} = \frac{C}{4} = \frac{2 \, \mu \mathrm{F}}{4} = 0.5 \, \mu \mathrm{F}    $
Combining Series and Parallel
: For combining the parallel and series connections, where $C_{	ext{total}}$ is the effective capacitance:
$    \frac{1}{C_{	ext{total}}} = \frac{1}{C_{	ext{parallel}}} + \frac{1}{C_{	ext{series}}} = \frac{1}{6} + \frac{1}{0.5} = \frac{1 + 12}{6} = \frac{13}{6}    $
Consequently:
$    C_{	ext{total}} = \frac{6}{13} \, \mu \mathrm{F}    $
Therefore, the required configuration is 3 capacitors in parallel followed by 4 capacitors in series to yield an overall capacitance of $\frac{6}{13} \, \mu \mathrm{F}$.

Answer

5 capacitors in parallel and then 2 capacitors series.

Answer

4 capacitors in parallel and then 3 capacitors series.

Answer

3 capacitors in parallel and then 4 capacitors in series.

Answer

2 capacitors in parallel and then 5 capacitors in series.

Electrostatics & Capacitance

Seven capacitors each of capacitance 2 \mu \mathrm{~F} are connected in a configuration to obtain an effective capacitance \frac{6}{13} \mu \mathrm{~F}. The combination which will achieve this will have

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