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Chapter 25: Problem 14

Sketch a diagram for a circuit consisting of two batteries, a resistor, and a capacitor, all in series. Does the circuit description allow you any flexibility?

Short Answer

Expert verified
Yes, the described circuit does allow flexibility in the order of components. There could be a total of twelve distinct arrangements of the batteries, resistor, and capacitor.

Step by step solution

01

Sketch the Circuit

Begin by sketching a single closed loop to represent the series circuit. Place one battery in the circuit making sure to include the positive and negative terminals. A battery can be represented as a long (representing positive) and short (representing negative) parallel lines. Draw a second battery, identical to the first one. Then sketch a resistor in the series, represented by a jagged line. Lastly, add the capacitor, indicated by two parallel lines with a gap in between.
02

Assess Flexibility in the Circuit

Now, let's assess the flexibility. In a series circuit, the arrangement of the components doesn't affect the total circuit properties. Therefore, the positioning of the batteries, resistor, and capacitor in relation to each other can vary. They can be in any order in the circuit, for example, the resistor could be placed before one of the batteries or after the capacitor and the circuit will still function in the same way.
03

Final Count of Circuits

The total number of ways to arrange 4 components (2 batteries considered as identical) in a sequence is given by permutations of multicombinations formula \( n! / (n1! * n2! * ... * ni!) \), where n is the total number of items, and ni are the number of the same items. Substituting values we get \( 4! / (2! * 1! * 1!) = 12 \) possible unique arrangements.

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Most popular questions from this chapter

The voltage across a charging capacitor in an \(R C\) circuit rises to \(1-1 / e\) of the battery voltage in \(5.0 \mathrm{ms}\). (a) How long will it take to reach \(1-1 / e^{3}\) of the battery voltage? (b) If the capacitor is charging through a \(22-\mathrm{k} \Omega\) resistor, what's the capacitance?

A partially discharged car battery can be modeled as a \(9-\mathrm{V}\) emf in series with a \(0.08-\Omega\) internal resistance. Jumper cables connect this battery to a fully charged battery, modeled as a \(12-\mathrm{V}\) emf in series with a \(0.02-\Omega\) internal resistance. The cables connect \(+\) to \(+\) and \(-\) to \(-.\) What current flows through the discharged battery?

If capacitance is in \(\mu \mathrm{F}\), what will be the units of the time constant \(R C\) when resistance is in (a) \(\Omega,\) (b) \(\mathrm{k} \Omega,\) and \((\mathrm{c})\) M\Omega? (Your answers eliminate the need for tedious power- of-10 conversions.)

You're designing an external defibrillator that discharges a capacitor through the patient's body, providing a pulse that stops ventricular fibrillation. Specifications call for a capacitor storing 250 J of energy; when discharged through a body with \(40-\Omega\) transthoracic resistance, the capacitor voltage is to drop to half its initial value in \(10 \mathrm{ms}\). Determine the capacitance (to the nearest \(10 \mu \mathrm{F})\) and initial capacitor voltage (to the nearest \(100 \mathrm{V}\) ) that meet these specs.

Sketch a circuit diagram for a circuit that includes a resistor \(R_{\square}\) connected to the positive terminal of a battery, a pair of paralle resistors \(R_{2}\) and \(R_{3}\) connected to the lower-voltage end of \(R_{1}\) and then returned to the battery's negative terminal, and a capacitor across \(R_{2}\)
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