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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

A potential difference of \(V=0.500 \mathrm{~V}\) is applied across a block of silicon with resistivity \(8.70 \cdot 10^{-4} \Omega \mathrm{m}\). As indicated in the figure, the dimensions of the silicon block are width \(a=2.00 \mathrm{~mm}\) and length \(L=15.0 \mathrm{~cm} .\) The resistance of the silicon block is \(50.0 \Omega\), and the density of charge carriers is \(1.23 \cdot 10^{23} \mathrm{~m}^{-3}\) Assume that the current density in the block is uniform and that current flows in silicon according to Ohm's Law. The total length of 0.500 -mm-diameter copper wire in the circuit is \(75.0 \mathrm{~cm},\) and the resistivity of copper is \(1.69 \cdot 10^{-8} \Omega \mathrm{m}\) a) What is the resistance, \(R_{w}\) of the copper wire? b) What are the direction and the magnitude of the electric current, \(i\), in the block? c) What is the thickness, \(b\), of the block? d) On average, how long does it take an electron to pass from one end of the block to the other? \(?\) e) How much power, \(P\), is dissipated by the block? f) In what form of energy does this dissipated power appear?

A battery has a potential difference of \(14.50 \mathrm{~V}\) when it is not connected in a circuit. When a \(17.91-\Omega\) resistor is connected across the battery, the potential difference of the battery drops to \(12.68 \mathrm{~V}\). What is the internal resistance of the battery?

A copper wire has a diameter \(d_{\mathrm{Cu}}=0.0500 \mathrm{~cm}\) is \(3.00 \mathrm{~m}\) long, and has a density of charge carriers of \(8.50 \cdot 10^{28}\) electrons \(/ \mathrm{m}^{3}\). As shown in the figure, the copper wire is attached to an equal length of aluminum wire with a diameter \(d_{\mathrm{A} \mathrm{I}}=0.0100 \mathrm{~cm}\) and density of charge carriers of \(6.02 \cdot 10^{28}\) electrons \(/ \mathrm{m}^{3}\). A current of 0.400 A flows through the copper wire. a) What is the ratio of the current densities in the two wires, \(J_{\mathrm{Cu}} / J_{\mathrm{Al}} ?\) b) What is the ratio of the drift velocities in the two wires, \(v_{\mathrm{d}-\mathrm{Cu}} / v_{\mathrm{d}-\mathrm{Al}} ?\)

Two conductors are made of the same material and have the same length \(L\). Conductor \(\mathrm{A}\) is a hollow tube with inside diameter \(2.00 \mathrm{~mm}\) and outside diameter \(3.00 \mathrm{~mm} ;\) conductor \(\mathrm{B}\) is a solid wire with radius \(R_{\mathrm{B}}\). What value of \(R_{\mathrm{B}}\) is required for the two conductors to have the same resistance measured between their ends?

Two resistors with resistances \(200 . \Omega\) and \(400 . \Omega\) are connected (a) in series and (b) in parallel with an ideal 9.00-V battery. Compare the power delivered to the \(200 .-\Omega\) resistor.
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