You measure the capacitance C1 of a capacitor by doing the...

You measure the capacitance $C_{1}$ of a capacitor by doing the following: First connect capacitors $C_{1}$ and $C_{2}$ in series to a power supply that provides a voltage $V$ that can be varied. The capacitance of $C_{2}$ is known to be $3.00 \mu \mathrm{F}$. Then vary the applied voltage $V,$ and for each value of $V$ measure the voltage $V_{2}$ across $C_{2}$. After plotting your data as $V_{2}$ versus $V,$ you find that the data fall close to a straight line that has slope $0.650 .$ What is the capacitance $C_{1} ?$

Key Concepts

Graphical Analysis in Experimental Measurements

Graphing experimental data, such as plotting the voltage across one circuit element versus the total applied voltage, can reveal linear relationships that simplify the extraction of circuit parameters. The slope of the resulting straight line can correspond directly to ratios of electrical quantities, allowing one to solve for unknown values by comparing experimental results with theoretical predictions. This method exemplifies how linear data analysis is applied to validate and quantify physical relationships in experimental physics.

Capacitors in Series

When capacitors are connected in series, they share the same charge but divide the total voltage across them. The overall or equivalent capacitance of a series combination is less than any of the individual capacitances. The relationship is defined by the reciprocal sum, where 1/C_equiv = 1/C1 + 1/C2 + ... . This concept is key in understanding how voltage distributes among components and how unknown values can be deduced when paired with known capacitances.

Voltage Division in Capacitive Circuits

In a series circuit involving capacitors, the voltage across each capacitor is inversely proportional to its capacitance. This voltage division results because the same charge appears on all capacitors, making the voltage across each one equal to the charge divided by its capacitance (V = Q/C). Understanding this principle allows one to relate measured voltage drops to the corresponding capacitances, which is useful in both circuit analysis and experimental determination of unknown component values.

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