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Chapter 10: Problem 12

What resistance values can be created by combining a \(1 \mathrm{k} \Omega\) resistor and a \(10 \mathrm{k} \Omega\) resistor? (solution in the pdf version of the book)

Short Answer

Expert verified
Values are 909.09 惟, 1 k惟, 10 k惟, and 11 k惟.

Step by step solution

01

Understanding Series Combination

When resistors are combined in series, their resistances add up. Thus, the total resistance is given by the sum of the individual resistances: \[ R_{ ext{series}} = R_1 + R_2 \] In this case, we have a \(1 \text{k}\, \Omega\) resistor and a \(10 \text{k}\, \Omega\) resistor. Thus, \[ R_{ ext{series}} = 1 \text{k}\, \Omega + 10 \text{k}\, \Omega = 11 \text{k}\, \Omega\]
02

Understanding Parallel Combination

When resistors are combined in parallel, their reciprocal resistances add up. The formula for the equivalent resistance in parallel is: \[ \frac{1}{R_{ ext{parallel}}} = \frac{1}{R_1} + \frac{1}{R_2} \] Using the given resistances, \[ \frac{1}{R_{ ext{parallel}}} = \frac{1}{1 \text{k}\, \Omega} + \frac{1}{10 \text{k}\, \Omega} = \frac{1}{1000 \Omega} + \frac{1}{10000 \Omega} \] Calculating this gives, \[ \frac{1}{R_{ ext{parallel}}} = \frac{10}{10000} + \frac{1}{10000} = \frac{11}{10000} \] Thus, \[ R_{ ext{parallel}} = \frac{10000}{11} \approx 909.09 \Omega \]
03

Listing the Possible Combinations

With the aforementioned series and parallel formulas calculated, the possible resistances obtainable using a \(1 \text{k}\, \Omega\) resistor and a \(10 \text{k}\, \Omega\) resistor are: - 11 \(\text{k}\, \Omega\) when in series- \(\approx 909.09 \Omega\) when in parallel- In case only one resistor is used, the values possible are individually: \(1 \text{k}\, \Omega\) and \(10 \text{k}\, \Omega\).

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

Series Circuit
In a series circuit, resistors are connected end-to-end, so the current flows through each resistor one-by-one along a single path. The total or equivalent resistance of the circuit is simply the sum of all individual resistances along the path. This is due to the fact that the current has no alternative path but to pass through each resistor, encountering each resistance consecutively.
For example, if you have a 1 k惟 and a 10 k惟 resistor connected in series, their cumulative resistance would be:
  • Total Resistance, \( R_{\text{series}} = R_1 + R_2 \)
  • Thus, \( R_{\text{series}} = 1 \, \text{k} \Omega + 10 \, \text{k} \Omega = 11 \, \text{k} \Omega \)
This approach is simple because adding resistances in series is no different than adding ordinary numbers together. It allows you to predict the total resistance easily.
Parallel Circuit
On the other hand, in a parallel circuit, resistors are connected across the same two points, creating multiple paths for the current. Unlike series circuits, the voltage across each resistor is the same. However, the total or equivalent resistance of the circuit is found using the reciprocal of the sum of the reciprocals of each resistor's resistance. This means that the total equivalent resistance is always less than the smallest individual resistor in the network.
Here's how you calculate it for a 1 k惟 and a 10 k惟 resistor:
  • Use the formula: \( \frac{1}{R_{\text{parallel}}} = \frac{1}{R_1} + \frac{1}{R_2} \)
  • Applying it: \( \frac{1}{R_{\text{parallel}}} = \frac{1}{1000 \, \Omega} + \frac{1}{10000 \, \Omega} \)
  • So, \( R_{\text{parallel}} \approx 909.09 \, \Omega \)
Parallel circuits are practical in many applications due to this reduction in total resistance, allowing larger currents to flow than would be possible with the same resistors in a series circuit.
Equivalent Resistance
Equivalent resistance is a key concept when working with combinations of resistors, as it provides a way to describe how complex circuits behave using simpler models. When resistors are combined, they can be thought of as a single, larger resistor with a certain resistance value. This equivalent resistance gives insight into how the circuit will react to electrical flow in terms of voltage drop and current flow.
Equivalent resistance calculations are essential in simplifying circuits, as they help identify the effective resistance between two points, making it easier to analyze circuit behavior.
  • For series circuits, equivalent resistance is straightforward: the sum of individual resistances.
  • For parallel circuits, calculating equivalent resistance requires totaling the reciprocals, emphasizing their efficient flow management.
  • Understanding how to switch between these forms allows for better manipulation and understanding of circuits with multiple components.
By mastering equivalent resistance, engineers and hobbyists can design more efficient and functional electronic systems.

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Most popular questions from this chapter

A \(1.0 \Omega\) toaster and a \(2.0 \Omega\) lamp are connected in parallel with the 110-V supply of your house. (Ignore the fact that the voltage is \(\mathrm{AC}\) rather than \(\mathrm{DC.}\).) (a) Draw a schematic of the circuit. (b) For each of the three components in the circuit, find the current passing through it and the voltage drop across it. (answer check available at lightandmatter.com) (c) Suppose they were instead hooked up in series. Draw a schematic and calculate the same things.(answer check available at lightandmatter.com)

A silk thread is uniformly charged by rubbing it with llama fur. The thread is then dangled vertically above a metal plate and released. As each part of the thread makes contact with the conducting plate, its charge is deposited onto the plate. Since the thread is accelerating due to gravity, the rate of charge deposition increases with time, and by time \(t\) the cumulative amount of charge is \(q=c t^{2}\), where \(c\) is a constant. (a) Find the current flowing onto the plate.(answer check available at lightandmatter.com) (b) Suppose that the charge is immediately carried away through a resistance \(R .\) Find the power dissipated as heat.(answer check available at lightandmatter.com)

You are given a battery, a flashlight bulb, and a single piece of wire. Draw at least two configurations of these items that would result in lighting up the bulb, and at least two that would not light it. (Don't draw schematics.) If you're not sure what's going on, borrow the materials from your instructor and try it. Note that the bulb has two electrical contacts: one is the threaded metal jacket, and the other is the tip (at the bottom in the figure). [Problem by Arnold Arons.]

(a) Many battery-operated devices take more than one battery. If you look closely in the battery compartment, you will see that the batteries are wired in series. Consider a flashlight circuit. What does the loop rule tell you about the effect of putting several batteries in series in this way? (b) The cells of an electric eel's nervous system are not that different from ours - - - each cell can develop a voltage difference across it of somewhere on the order of one volt. How, then, do you think an electric eel can create voltages of thousands of volts between different parts of its body?

(a) Express the power dissipated by a resistor in terms of \(R\) and \(\Delta V\) only, eliminating \(I\). (answer check available at lightandmatter.com) (b) Electrical receptacles in your home are mostly \(110 \mathrm{~V}\), but circuits for electric stoves, air conditioners, and washers and driers are usually \(220 \mathrm{~V}\). The two types of circuits have differently shaped receptacles. Suppose you rewire the plug of a drier so that it can be plugged in to a \(110 \mathrm{~V}\) receptacle. The resistor that forms the heating element of the drier would normally draw 200 W. How much power does it actually draw now? (answer check available at lightandmatter.com)
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