You need a capacitance of 50 μF, but you don't happen to have a 50 μF capacitor. You do have a 75 μF capacitor. What additional capacitor do you need to produce a total capacitance of 50 μF? Should you join the two capacitors in parallel or in series?

Verified step by step guidance

1. To determine the configuration (series or parallel), recall the formulas for equivalent capacitance: For capacitors in series, the reciprocal of the total capacitance is given by: $\frac{1}{C_{\mathrm{total}}}=\frac{1}{C_1}+\frac{1}{C_2}$. For capacitors in parallel, the total capacitance is the sum: $C_{\mathrm{total}}=C_1+C_2$.
2. Since the desired total capacitance (50 μF) is less than the capacitance of the given capacitor (75 μF), the capacitors must be connected in series. This is because connecting capacitors in series reduces the total capacitance.
3. Using the series formula, substitute the known values: $\frac{1}{50}=\frac{1}{75}+\frac{1}{C_{\mathrm{unknown}}}$. Here, $C_{\mathrm{unknown}}$ is the capacitance of the additional capacitor.
4. Rearrange the equation to solve for $C_{\mathrm{unknown}}$: $\frac{1}{C_{\mathrm{unknown}}}=\frac{1}{50}-\frac{1}{75}$. Perform the subtraction on the right-hand side to find the reciprocal of $C_{\mathrm{unknown}}$.
5. Finally, take the reciprocal of the result to find the value of $C_{\mathrm{unknown}}$. This is the capacitance of the additional capacitor needed to achieve a total capacitance of 50 μF when connected in series with the 75 μF capacitor.

Key Concepts

Capacitance in Series

When capacitors are connected in series, the total capacitance (C_total) is less than the smallest individual capacitor. The formula for total capacitance in series is given by 1/C_total = 1/C1 + 1/C2 + ... + 1/Cn. This means that adding capacitors in series effectively increases the voltage rating while decreasing the overall capacitance.

Capacitance in Parallel

In a parallel configuration, the total capacitance is the sum of the individual capacitances. The formula is C_total = C1 + C2 + ... + Cn. This arrangement allows the capacitors to share the same voltage across their terminals, resulting in a higher total capacitance while maintaining the same voltage rating as the individual capacitors.

Finding Equivalent Capacitance

To achieve a desired capacitance using available capacitors, one must understand how to combine them effectively. In this case, to reach a total capacitance of 50 μF using a 75 μF capacitor, one would need to connect an additional capacitor in series. The required capacitance can be calculated using the series capacitance formula to find the value of the additional capacitor needed.