Find the equivalent capacitance of \(4.0\,\mu\mathrm{F}\) and \(7.0\,\mu\mathrm{F}\) connected in parallel.

Two capacitors, \(3.0\,\mu\mathrm{F}\) and \(6.0\,\mu\mathrm{F}\), are connected in series across \(18\,\mathrm{V}\). Find \(C_{\mathrm{eq}}\), the series charge, and the voltage across each capacitor.

A \(3.0\,\mu\mathrm{F}\) capacitor and a \(6.0\,\mu\mathrm{F}\) capacitor are connected in parallel. That parallel combination is then connected in series with a \(9.0\,\mu\mathrm{F}\) capacitor. Find the equivalent capacitance.

A \(2.0\,\mu\mathrm{F}\) capacitor is charged to \(12\,\mathrm{V}\). It is disconnected from the battery and then connected positive-to-positive with an initially uncharged \(4.0\,\mu\mathrm{F}\) capacitor. Find the final common voltage and the energy lost during redistribution.

A \(2.0\,\mu\mathrm{F}\) capacitor and a \(5.0\,\mu\mathrm{F}\) capacitor are connected in parallel across \(10\,\mathrm{V}\). Find the charge on each capacitor and the total charge supplied by the battery.

Three capacitors \(2.0\,\mu\mathrm{F}\), \(3.0\,\mu\mathrm{F}\), and \(6.0\,\mu\mathrm{F}\) are connected in series. Find their equivalent capacitance.

A \(2.0\,\mu\mathrm{F}\) capacitor and an \(8.0\,\mu\mathrm{F}\) capacitor are connected in series across \(20\,\mathrm{V}\). Find the voltage across each capacitor.

A \(4.0\,\mu\mathrm{F}\) capacitor is in series with a \(12\,\mu\mathrm{F}\) capacitor. That combination is in parallel with a \(6.0\,\mu\mathrm{F}\) capacitor. Find the equivalent capacitance.

A network has \(C_1\) and \(C_2\) connected in parallel, and that parallel group connected in series with \(C_3\). Derive \(C_{\mathrm{eq}}\) in terms of \(C_1,C_2,C_3\).

Two capacitors \(4.0\,\mu\mathrm{F}\) and \(12\,\mu\mathrm{F}\) are connected in series across \(48\,\mathrm{V}\). Find the charge and energy stored in each capacitor.

Derive the two-capacitor series formula \(C_{\mathrm{eq}}=C_1C_2/(C_1+C_2)\).

A \(10\,\mu\mathrm{F}\) capacitor is connected in series with an unknown capacitor. Their equivalent capacitance is \(4.0\,\mu\mathrm{F}\). Find the unknown capacitance.

A \(3.0\,\mu\mathrm{F}\) capacitor and a \(6.0\,\mu\mathrm{F}\) capacitor are connected in parallel across \(12\,\mathrm{V}\), then disconnected from the battery while remaining connected to each other. Find the total stored charge and total stored energy.

A \(3.0\,\mu\mathrm{F}\) capacitor charged to \(12\,\mathrm{V}\) is disconnected and connected positive-to-positive with a \(6.0\,\mu\mathrm{F}\) capacitor charged to \(3.0\,\mathrm{V}\). Find the final common voltage and the energy lost.

A \(6.0\,\mu\mathrm{F}\) capacitor rated at \(30\,\mathrm{V}\) is connected in series with a \(3.0\,\mu\mathrm{F}\) capacitor rated at \(20\,\mathrm{V}\). What is the maximum safe voltage across the series pair?

Three identical capacitors \(C\) are arranged with two in parallel and the result in series with the third. Derive the equivalent capacitance and compare it with three identical capacitors all in series.

An infinite ladder has a series capacitor \(C\) followed by a node where one branch is a capacitor \(C\) to ground and the other branch repeats the same ladder. Derive the equivalent capacitance seen at the input.

Two capacitors \(C_1\) and \(C_2\) are initially charged to voltages \(V_1\) and \(V_2\) with the same polarity. They are disconnected from their batteries and connected positive-to-positive. Derive the final voltage and the energy lost.

Two capacitors \(C_1\) and \(C_2\) are initially charged to voltages \(V_1\) and \(V_2\), then connected positive-to-negative so their charges partially cancel. Derive the final voltage magnitude and state the condition for complete cancellation.

A bridge network has capacitors \(C_1\) and \(C_2\) in series on the top branch, \(C_3\) and \(C_4\) in series on the bottom branch, and a capacitor \(C_5\) connecting the two middle nodes. Derive the condition under which \(C_5\) carries no charge, and find the equivalent capacitance under that condition.