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Chapter 18: Problem 36

Suppose a collection of five batteries is connected as shown. (a) What is the equivalent emf of the collection? Treat them as ideal sources of emf. (b) What is the current through the resistor if its value is \(3.2 \Omega ?\)

Short Answer

Expert verified
Question: Given an arrangement of five ideal batteries with emf values \(E_1\), \(E_2\), \(E_3\), \(E_4\), and \(E_5\), find the equivalent emf and the current flowing through a \(3.2 \Omega\) resistor connected to the arrangement. Solution: (a) The equivalent emf is ______. (b) The current through the resistor is ______.

Step by step solution

01

Analyze the Battery Configuration

First, we need to observe the given configuration of the batteries and note down their emf values and arrangement (whether they are connected in series or parallel). The arrangement might look like this: - Battery 1, emf: \(E_1\) - Battery 2, emf: \(E_2\) - Battery 3, emf: \(E_3\) - Battery 4, emf: \(E_4\) - Battery 5, emf: \(E_5\)
02

Calculate the Equivalent EMF

The formula for equivalent emf in series is the algebraic sum of the individual emf values, and in parallel, it is the ratio of the sum of the product of the emf values and their internal resistances to the sum of the internal resistances. As we are asked to treat them as ideal sources of emf, we can consider their internal resistance to be zero. The problem can be solved more easily by treating each part of the combination separately. Based on the specific arrangement given, determine the equivalent emf values for the individual parts and then find the total equivalent emf of the collection.
03

Apply Ohm's Law to Find the Current Through the Resistor

Ohm's law states that \(I = \frac{E_{eq}}{R}\), where \(I\) is the current through the resistor, \(E_{eq}\) is the equivalent emf, and \(R\) is the resistance. Use the equivalent emf found in step 2 and the given resistance value (\(3.2 \Omega\)) to calculate the current through the resistor.
04

Write the Final Answer

After performing the calculations in steps 2 and 3, report the equivalent emf of the battery collection and the current flowing through the resistor. The final answer will be in the format: (a) The equivalent emf is ______. (b) The current through the resistor is ______.

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Most popular questions from this chapter

Near Earth's surface the air contains both negative and positive ions, due to radioactivity in the soil and cosmic rays from space. As a simplified model, assume there are 600.0 singly charged positive ions per \(\mathrm{cm}^{3}\) and 500.0 singly charged negative ions per \(\mathrm{cm}^{3} ;\) ignore the presence of multiply charged ions. The electric field is $100.0 \mathrm{V} / \mathrm{m},$ directed downward. (a) In which direction do the positive ions move? The negative ions? (b) What is the direction of the current due to these ions? (c) The measured resistivity of the air in the region is $4.0 \times 10^{13} \Omega \cdot \mathrm{m} .$ Calculate the drift speed of the ions, assuming it to be the same for positive and negative ions. [Hint: Consider a vertical tube of air of length \(L\) and cross-sectional area \(A\) How is the potential difference across the tube related to the electric field strength?] (d) If these conditions existed over the entire surface of the Earth, what is the total current due to the movement of ions in the air?
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