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Chapter 27: Problem 65

A total resistance of \(5.00 \Omega\) is to be produced by connecting an unknown resistance to a \(15.0 \Omega\) resistance. (a) What must be the value of the unknown resistance, and (b) should it be connected in series or in parallel? (c) What is the total resistance if the unknown resistance is connected the other way?

Short Answer

Expert verified
The unknown resistance must be 7.50 鈩� and connected in parallel; otherwise, in series, it would equal 22.50 鈩�.

Step by step solution

01

Define Resistances in Series

When resistances are connected in series, the total resistance, \( R_{total} \), is the sum of the individual resistances, \( R_1 \) and \( R_2 \). For a series combination: \[ R_{total} = R_1 + R_2 \]. We know \( R_{total} = 5.00 \ \Omega \) and \( R_1 = 15.0 \ \Omega \). So, \( R_2 \) (unknown resistance) is calculated as: \[ R_2 = R_{total} - R_1 = 5.00 \ \Omega - 15.0 \ \Omega \] This results in a negative resistance, which is not possible, indicating the series connection does not produce the desired total resistance.
02

Define Resistances in Parallel

For resistances in parallel, the total resistance \( R_{total} \) is given by the formula: \[ \frac{1}{R_{total}} = \frac{1}{R_1} + \frac{1}{R_2} \]. So, \( R_2 \) is calculated as: \[ \frac{1}{5.00} = \frac{1}{15.0} + \frac{1}{R_2} \]. Solving for \( \frac{1}{R_2} \), we get: \[ \frac{1}{R_2} = \frac{1}{5.00} - \frac{1}{15.0} \]. Calculate \( \frac{1}{R_2} \): \( \frac{1}{5.00} = 0.200 \) and \( \frac{1}{15.0} = 0.0667 \), leading to \( \frac{1}{R_2} = 0.200 - 0.0667 = 0.1333 \). Thus, \( R_2 = \frac{1}{0.1333} \approx 7.50 \ \Omega \).
03

Verify Parallel Total Resistance

Verify that \( R_2 = 7.50 \ \Omega \) results in a total parallel resistance of \( 5.00 \ \Omega \). Recalculate: \[ \frac{1}{R_{total}} = \frac{1}{15.0} + \frac{1}{7.50} = 0.0667 + 0.1333 = 0.200 \]. Thus, \( R_{total} = \frac{1}{0.200} = 5.00 \ \Omega \), confirming our calculation is correct.
04

Calculate Series Total Resistance

If the resistances were connected in series, \( R_{total} = R_1 + R_2 = 15.0 \ \Omega + 7.50 \ \Omega = 22.50 \ \Omega \).

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Key Concepts

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

Resistance Calculations
Effectively calculating resistance involves a good understanding of Ohm's Law and circuit combinations. Resistance is a measure of how much a component opposes the flow of electrical current. Ohm's Law states that the voltage across a resistor is directly proportional to the current flowing through it, represented by the formula \( V = IR \), where \( V \) is voltage, \( I \) is current, and \( R \) is resistance.

When calculating resistance, one must consider the layout of the resistors in the circuit: whether they are in series or parallel. Knowing how these configurations alter the total resistance is key to solving related problems efficiently.

- For resistors in **series**, the total resistance is the sum of the individual resistances. This is expressed as \( R_{total} = R_1 + R_2 + \ldots + R_n \).
- For resistors in **parallel**, the total resistance can be found using: \( \frac{1}{R_{total}} = \frac{1}{R_1} + \frac{1}{R_2} + \ldots + \frac{1}{R_n} \). This method results in a total resistance that is less than any of the individual resistances, except if all resistors have the same value.

These calculations enable you to determine how resistors affect electrical paths in circuits.
Series and Parallel Circuits
Understanding series and parallel circuits is crucial in electronics. A series circuit refers to a configuration where components are connected along a single path. If one component fails in a series circuit, the current flow in the entire circuit stops.

**Characteristics of Series Circuits:**
  • All components share the same current.
  • Total resistance is the sum of individual resistors (\( R_{total} = R_1 + R_2 + ... + R_n \)).
  • Voltage across the circuit is the sum of the voltages across each component.

A parallel circuit, however, connects components so that there are multiple paths for the current to travel. An advantage of this setup is if one component fails, current can still pass through other paths.

**Characteristics of Parallel Circuits:**
  • Voltage across each component is the same.
  • Total resistance is reduced, calculated via \( \frac{1}{R_{total}} = \frac{1}{R_1} + \frac{1}{R_2} + ... + \frac{1}{R_n} \).
  • Components can be added or removed without affecting the current flow in the circuit.
These circuit types are essential for designing efficient electronic devices.
Resistivity
Resistivity is a property of a material that indicates how strongly it resists current flow. It is denoted by \( \rho \) and is dependent on the material's characteristics. Materials with low resistivity are conductors, while high resistivity materials serve as insulators.

Understanding resistivity helps in selecting proper materials for building circuits and designing electronic components. It is related to resistance through the formula \( R = \rho \frac{L}{A} \), where \( R \) is resistance, \( L \) is the length of the material, and \( A \) is the cross-sectional area.

Key points to remember about resistivity:
  • It is intrinsic to the material, not dependent on size or shape.
  • Helps determine how much a given material will resist electrical current over its length.
  • Units are ohm-meter (\( \Omega \cdot m \)).
With resistivity knowledge, you can better predict how changing materials or dimensions of a conductor will affect overall resistance in circuits.

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Most popular questions from this chapter

A wire of resistance \(5.0 \Omega\) is connected to a battery whose emf 8 is \(12 \mathrm{~V}\) and whose internal resistance is \(0.70 \Omega\). In \(5.0 \mathrm{~min}\), how much energy is (a) transferred from chemical form in the battery, (b) dissipated as thermal energy in the wire, and (c) dissipated as thermal energy in the battery?

Assume that \(8=13.2 \mathrm{~V}, r=100 \Omega, R_{1}=250 \Omega\), and \(R_{2}=300 \Omega\). If the voltmeter resistance \(R_{\mathrm{V}}\) is \(5.0 \mathrm{k} \Omega\), what percent error does it introduce into the measurement of the potential difference across \(R_{1}\) ? Ignore the presence of the ammeter.

A \(3.00 \mathrm{M} \Omega\) resistor and a \(1.00 \mu \mathrm{F}\) capacitor are connected in series with an ideal battery of emf \(8=4.00 \mathrm{~V}\). At \(6.00 \mathrm{~s}\) after the connection is made, what is the rate at which (a) the charge of the capacitor is increasing, (b) energy is being stored in the capacitor, (c) thermal energy is appearing in the resistor, and (d) energy is being delivered by the battery?

A car battery with a \(12 \mathrm{~V}\) emf and an internal resistance of \(0.030 \Omega\) is being charged with a current of \(40 \mathrm{~A}\). What are (a) the potential difference \(V\) across the terminals, (b) the rate \(P_{r}\) of energy dissipation inside the battery, and (c) the rate \(P_{\mathrm{cmf}}\) of energy conversion to chemical form? When the battery is used to supply 40 A to the starter motor, what are (d) \(V\) and (e) \(P_{r}\) ?

A \(1.0 \mu \mathrm{F}\) capacitor with an initial stored energy of \(0.60 \mathrm{~J}\) is discharged through a \(1.0 \mathrm{M} \Omega\) resistor. (a) What is the initial charge on the capacitor? (b) What is the current through the resistor when the discharge starts? Find an expression that gives, as a function of time \(t\), (c) the potential difference \(V_{c}\) across the capacitor, (d) the potential difference \(V_{R}\) across the resistor, and (e) the rate at which thermal energy is produced in the resistor.
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