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Any circulating flow constitutes a circuit. Learn how to model the logic boards in your computer, the flow of nutrients in your blood, or the daily fluctuations in the temperature of your house.

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Find \(V_A\) in volts.

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Bill has two identical lamps each of resistance 5 ohms and two 9-volt-batteries. If he wants to light the two lamps for a duration as long as possible, how should he connect the lamps and batteries?

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When a charged capacitor is disconnected from its source it will eventually discharge. This is because a small amount of charge leaks through the dielectric between the plates of the capacitor. Suppose we fill a parallel plate capacitor with a ceramic of relative permittivity \(\epsilon=2.1\) and resistivity \(\rho=1.4 \times 10^{13}~\Omega \cdot \mbox{m} \). The capacitor is charged by connecting it to a voltage source. How long will it take **in seconds** for the capacitor to lose half of the charge acquired after disconnecting it from the source?
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A 120V source is connected across the plates of a parallel-plate capacitor. The capacitor is being submerged vertically at constant speed into a container filled with water. What current **in Amps** flows through the voltage source during this process if it takes \(\tau=20 s\) to totally submerge the capacitor? The capacitance (in air) of the capacitor is \(C_{0}=12\mu F\) and water's relative permittivity is
\[ \epsilon=81.\]

Hint: If the capacitor is partially submerged at an instant of time try figuring out how to treat it as two separate capacitors at that instant.

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