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Chapter 25: Problem 19

An infinite number of resistors are connected in parallel. If \(R_{1}=10 \Omega, R_{2}=10^{2} \Omega, R_{3}=10^{3} \Omega,\) and so on, show that \(R_{e q}=9 \Omega\).

Short Answer

Expert verified
Answer: The equivalent resistance of the infinite parallel resistors circuit is 9Ω.

Step by step solution

01

Write the formula for resistors in parallel

The formula for the equivalent resistance of resistors in parallel is given by: \( \frac{1}{R_{eq}} = \frac{1}{R_1} + \frac{1}{R_2} + \frac{1}{R_3} + \cdots \)
02

Substitute the given values

Given that \(R_1 = 10\Omega\), \(R_2 = 10^2\Omega\), \(R_3 = 10^3\Omega\), and so on, we can rewrite the formula as: \( \frac{1}{R_{eq}} = \frac{1}{10} + \frac{1}{10^2} + \frac{1}{10^3} + \cdots \)
03

Recognize the geometric series

The given series is a geometric series with a first term \(a = \frac{1}{10}\) and a common ratio \(r = \frac{1}{10}\). To find the sum of an infinite geometric series, we can use the formula: \( S = \frac{a}{1 - r} \)
04

Calculate the sum of the infinite geometric series

Applying the formula for the sum of an infinite geometric series to our problem, we have: \( S = \frac{\frac{1}{10}}{1 - \frac{1}{10}} = \frac{\frac{1}{10}}{\frac{9}{10}} = \frac{1}{9} \)
05

Find the equivalent resistance

We know that the sum of the geometric series is equal to the reciprocal of the equivalent resistance: \( \frac{1}{R_{eq}} = S = \frac{1}{9} \) To find \(R_{eq}\), take the reciprocal of the sum: \( R_{eq} = \frac{1}{\frac{1}{9}} = 9\Omega \) Therefore, the equivalent resistance of the infinite parallel resistors circuit is 9Ω.

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Most popular questions from this chapter

Show that the power supplied to the circuit in the figure by the battery with internal resistance is maximum when the resistance of the resistor in the circuit, \(R\), is equal to \(R_{i}\). Determine the power supplied to \(R\). For practice, calculate the power dissipated by a \(12.0-\mathrm{V}\) battery with an internal resistance of \(2.00 \Omega\) when \(R=1.00 \Omega, R=2.00 \Omega,\) and \(R=3.00 \Omega\)

A constant electric field is maintained inside a semiconductor. As the temperature is lowered, the magnitude of the current density inside the semiconductor a) increases. c) decreases. b) stays the same. d) may increase or decrease.

What is the current density in an aluminum wire having a radius of \(1.00 \mathrm{~mm}\) and carrying a current of \(1.00 \mathrm{~mA}\) ? What is the drift speed of the electrons carrying this current? The density of aluminum is \(2.70 \cdot 10^{3} \mathrm{~kg} / \mathrm{m}^{3},\) and 1 mole of aluminum has a mass of \(26.98 \mathrm{~g}\). There is one conduction electron per atom in aluminum.

A \(12.0 \mathrm{~V}\) battery with an internal resistance \(R_{\mathrm{j}}=4.00 \Omega\) is attached across an external resistor of resistance \(R\). Find the maximum power that can be delivered to the resistor.

Should light bulbs (ordinary incandescent bulbs with tungsten filaments) be considered ohmic resistors? Why or why not? How would this be determined experimentally?
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