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Chapter 19: Problem 49

You have two \(10 \Omega\) resistors and one \(40 \Omega\) resistor. What combination of these will produce an equivalent resistance of (a) \(45 \Omega\), (b) \(18 \Omega,(\mathrm{c}) 4.44 \Omega ?\) 19.6 Kirchhoff's Rules

Short Answer

Expert verified
(a) Two 10 惟 in parallel with 40 惟 in series, (b) 10 惟 in series with 10 惟 || 40 惟, (c) All in parallel.

Step by step solution

01

Identify available components

We have two resistors each with a resistance of \(10 \Omega\) and one resistor with a resistance of \(40 \Omega\). The problem requires us to find combinations of these resistors to achieve specific equivalent resistances.
02

Calculate combinations for equivalent resistance (a)

To achieve an equivalent resistance of \(45 \Omega\), we can connect the two \(10 \Omega\) resistors in series, which gives \(10 + 10 = 20 \Omega\). Then connect this series combination in series with the \(40 \Omega\) resistor to get \(20 + 40 = 60 \Omega\), but this is incorrect. Try a parallel combination: connect the two \(10 \Omega\) resistors in parallel, which gives \(5 \Omega\) (as \(1/10 + 1/10 = 1/5\)). Now, put this in series with the \(40 \Omega\) resistor, \(5 + 40 = 45 \Omega\).
03

Attempt combinations for equivalent resistance (b)

To achieve an equivalent resistance of \(18 \Omega\), connect\ the two \(10 \Omega\) resistors in series to form \(20 \Omega\). Now consider all three in parallel: the combination of \(20 \Omega\) and \(40 \Omega\) in parallel can be calculated using \(1/R = 1/20 + 1/40 = 3/40\), resulting in an equivalent resistance of \(13.33 \Omega\), incorrect. However, try connecting one \(10 \Omega\) in parallel with \(40 \Omega\): \(1/R = 1/10 + 1/40 = 0.125\) which gives \(R = 8 \Omega\). Now place this in series with the second \(10 \Omega\) to get \(18 \Omega\).
04

Calculate combinations for equivalent resistance (c)

For an equivalent resistance of \(4.44 \Omega\), connect all resistors in parallel. The equivalent resistance formula for parallel resistors is \(1/R = 1/R_1 + 1/R_2 + 1/R_3\). Substituting for our resistors gives \(1/R = 1/10 + 1/10 + 1/40\) which simplifies to \(R = 4.44 \Omega\).
05

Verification of correct combinations

Re-evaluate each calculation to ensure all resistor combinations are correctly achieving the specified target resistances. Verify each approach numerically to confirm they meet the target exactly.

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Key Concepts

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

Equivalent Resistance
Equivalent resistance is a way to simplify complex circuits by reducing multiple resistors into a single resistor that has the same overall effect on the circuit's current and voltage.
This simplification makes it easier to analyze and understand electrical circuits.
  • In a series circuit, the equivalent resistance is the sum of all the resistances.
  • In a parallel circuit, the reciprocal of the equivalent resistance is the sum of the reciprocals of each resistor's resistance.
Understanding equivalent resistance is crucial when designing circuits to ensure they work as intended. It helps in calculating how much current will flow through the circuit, and how voltage will be distributed across the resistors.
Kirchhoff's Rules
Kirchhoff's rules, consisting of the junction rule and the loop rule, are essential guidelines in circuit analysis.
The junction rule states that at any junction in an electrical circuit, the sum of currents flowing into that junction is equal to the sum of currents flowing out.
The loop rule states that around any closed loop in a circuit, the sum of voltage gains and drops must equal zero.
  • These rules are based on the conservation of charge and energy.
  • They allow for the determination of unknown values like currents and voltages in circuit networks.
Applying Kirchhoff's rules can simplify the process of analyzing complex circuits, particularly when dealing with multiple loops and junctions.
Series and Parallel Circuits
Understanding the difference between series and parallel circuits is crucial for determining how resistors affect the overall circuit.
In a series circuit, components are connected end-to-end so that there is only one path for current to flow.
  • In series, the total resistance is simply the sum of all resistances.
  • If one component fails, the current flow stops.
Conversely, in parallel circuits, components are connected across common points, providing multiple paths for current to flow:
  • The total resistance in parallel is calculated using the reciprocal formula for resistances.
  • If one component fails, current can still flow through other paths.
Recognizing these differences allows us to combine and manipulate circuit components to achieve desired outcomes.
Ohm's Law
Ohm鈥檚 law is a fundamental principle used to understand electrical circuits, establishing a direct relationship between voltage, current, and resistance.
The law is represented mathematically as \( V = IR \), where:
  • \( V \) is the voltage across the resistance (in volts).
  • \( I \) is the current flowing through the resistance (in amperes).
  • \( R \) is the resistance (in ohms).
This equation shows that if you know two of the values, you can easily find the third:
  • It helps calculate how changing the resistance or voltage will affect the current and vice versa.
  • By understanding Ohm's law, you can analyze and design circuits for specific requirements.
Applying Ohm鈥檚 law alongside Kirchhoff鈥檚 rules and concepts of series and parallel circuits facilitates comprehensive circuit analysis.

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Most popular questions from this chapter

This procedure is not recommended! You'll see why after you work the problem. You are on an aluminum ladder that is standing on the ground, trying to fix an electrical connection with a metal screwdriver that has a metal handle. Your body is wet because you are sweating from the exertion; therefore, it has a resistance of \(1.0 \mathrm{k} \Omega .\) (a) If you accidentally touch the "hot" wire connected to the \(120 \mathrm{~V}\) line, how much current will pass through your body? Is this amount enough to be dangerous? (The maximum safe current is about \(5 \mathrm{~mA} .\) ) (b) How much electric power is delivered to your body?

Lightning strikes. During lightning strikes from a cloud to the ground, currents as high as 25,000 A can occur and last for about \(40 \mu\) s. How much charge is transferred from the cloud to the earth during such a strike?

A nonideal \(10.0 \mathrm{~V}\) battery is connected across a resistor \(R .\) The internal resistance of the battery is \(2.0 \Omega\). Calculate the potential difference across the resistor for the following values: (a) \(R=100 \Omega\), (b) \(R=10 \Omega,\) (c) \(R=2 \Omega\). (d) Find the current through the battery for all three cases.

A fully charged \(6.0 \mu \mathrm{F}\) capacitor is connected in series with a \(1.5 \times 10^{5} \Omega\) resistor. What percentage of the original charge is left on the capacitor after 1.8 s of discharging?

Nerve cells transmit electrical signals through their long tubular axons. These signals propagate due to a sudden rush of \(\mathrm{Na}^{+}\) ions, each with charge \(+e,\) into the axon. Measurements have revealed that typically about \(5.6 \times 10^{11} \mathrm{Na}^{+}\) ions enter each meter of the axon during a time of \(10 \mathrm{~ms}\). What is the current during this inflow of charge in a meter of axon?
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