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Chapter 25: Problem 9

How would you connect a pair of equal resistors across an ideal battery in order to get the greatest power dissipation?

Short Answer

Expert verified
To attain the greatest power dissipation, the two resistors should be connected in parallel across the battery.

Step by step solution

01

Analyze resistor configurations

Consider the two Basic Configurations for connecting resistors: Series and Parallel. We will analyze each one by formulas and compare them.
02

Calculating Resistance and Power in Series

In a series circuit, the total resistance is the sum of the individual resistances. If we take \(R\) as the resistance of one resistor, the total resistance is \(R_{series} = R + R = 2R\). Using Ohm's law, the current in this situation is \(I_{series} = V/R_{series} = V/2R\). Then calculaate the power dissipated using \(P_{series} = V * I_{series} = V^2/2R\).
03

Calculating Resistance and Power in Parallel

In a parallel configuration, the reciprocal of the total resistance is the sum of the reciprocals of the individual resistances. So the overall resistance is \(R_{parallel} = 1/ ((1/R) + (1/R)) = R/2\). The current in this situation is \(I_{parallel} = V/R_{parallel} = 2V/R\). The power can now be calculated with \(P_{parallel} = V * I_{parallel} = 2V^2/R\).
04

Comparison and Conclusion

Comparing the power dissipation in both cases, \(P_{parallel}\) and \(P_{series}\), it can be seen that the parallel configuration provides a greater power dissipation than the series configuration because \(2V^2/R > V^2/2R\). So, to get the greatest power dissipation, connect the two resistors in parallel across the battery.

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Most popular questions from this chapter

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

You have a battery whose voltage and internal resistance are unknown. Using an ideal voltmeter and an ideal ammeter, how would you determine each of these characteristics?

Can the voltage across a battery's terminals differ from the battery's rated voltage? Explain.

Two identical resistors in series dissipate equal power. How can this be, when electric charge loses energy in flowing through the first resistor?

When a large electric load such as a washing machine or oven comes on, lights throughout a house often dim. Why?
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