/*! This file is auto-generated */ .wp-block-button__link{color:#fff;background-color:#32373c;border-radius:9999px;box-shadow:none;text-decoration:none;padding:calc(.667em + 2px) calc(1.333em + 2px);font-size:1.125em}.wp-block-file__button{background:#32373c;color:#fff;text-decoration:none} Problem 21 Four \(18.0 \Omega\) resistors a... [FREE SOLUTION] | 91影视

91影视

Log out

Learning Materials

Features

Discover

Chapter 27: Problem 21

Four \(18.0 \Omega\) resistors are connected in parallel across a \(25.0 \mathrm{~V}\) ideal battery. What is the current through the battery?

Short Answer

Expert verified
The current through the battery is approximately 5.56 A.

Step by step solution

01

Understanding the Problem

We have four resistors of the same resistance, each having a resistance of \(18.0 \Omega\), connected in parallel, and they are powered by a \(25.0 \mathrm{~V}\) battery. We need to find out the total current that flows through the battery.
02

Calculate the Equivalent Resistance

When resistors are connected in parallel, the equivalent resistance \( R_{eq} \) can be calculated as: \[\frac{1}{R_{eq}} = \frac{1}{R_1} + \frac{1}{R_2} + \frac{1}{R_3} + \frac{1}{R_4}\]Given that all resistors \( R_1, R_2, R_3, \text{ and } R_4 \) are \(18.0 \Omega\), we have:\[\frac{1}{R_{eq}} = \frac{1}{18} + \frac{1}{18} + \frac{1}{18} + \frac{1}{18} = \frac{4}{18} = \frac{2}{9}\]Therefore, the equivalent resistance \( R_{eq} \) is:\[R_{eq} = \frac{9}{2} = 4.5 \Omega\]
03

Determine the Total Current Using Ohm's Law

To find the total current \( I \) flowing through the battery, we use Ohm's Law:\[V = IR\]Where \( V \) is the voltage across the battery (\(25.0 \mathrm{~V}\)) and \( R \) is the equivalent resistance (\(4.5 \Omega\)). Rearrange the formula to solve for \( I \):\[ I = \frac{V}{R} = \frac{25.0}{4.5} \approx 5.56 \mathrm{~A}\]
04

Conclusion

The total current flowing through the battery, when four \( 18.0 \Omega \) resistors are connected in parallel across a \( 25.0 \mathrm{~V} \) battery, is approximately \( 5.56 \mathrm{~A} \).

Unlock Step-by-Step Solutions & Ace Your Exams!

  • Full Textbook Solutions

    Get detailed explanations and key concepts

  • Unlimited Al creation

    Al flashcards, explanations, exams and more...

  • Ads-free access

    To over 500 millions flashcards

  • Money-back guarantee

    We refund you if you fail your exam.

Over 30 million students worldwide already upgrade their learning with 91影视!

Key Concepts

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

Resistors in Parallel
When resistors are placed in a parallel configuration, they share the same two nodes, and each resistor provides a separate path for current to flow. This is different from a series connection, where resistors are connected end to end. Thus, in a parallel arrangement, the voltage across each resistor is the same. This is a key point to remember: every resistor in a parallel circuit has the same voltage across it as the source voltage.
  • In the problem given, all four resistors are connected in such a parallel structure.
  • Each resistor has a value of \( 18.0\, \Omega \), and the voltage from the battery is \( 25.0\, V \).
Putting resistors in parallel effectively reduces the total resistance of the circuit, allowing more current to flow compared to a series circuit with the same resistors.
Equivalent Resistance
Calculating the equivalent resistance for resistors in parallel can initially seem complex, but it's quite simple with some practice. For parallel resistors, the reciprocal of the total, or equivalent resistance \( R_{eq} \), is the sum of the reciprocals of each individual resistance:
  • This can be mathematically expressed as: \[ \frac{1}{R_{eq}} = \frac{1}{R_1} + \frac{1}{R_2} + \frac{1}{R_3} + \frac{1}{R_4} \]
  • In this problem, since all resistors are \( 18.0\, \Omega \): \[ \frac{1}{R_{eq}} = \frac{1}{18} + \frac{1}{18} + \frac{1}{18} + \frac{1}{18} = \frac{4}{18} = \frac{2}{9} \]
  • Flipping it gives \( R_{eq} \): \[ R_{eq} = \frac{9}{2} = 4.5\, \Omega \]
When resistors of equal resistance are placed in parallel, the equivalent resistance can be quickly calculated. Simply divide the single resistor's value by the number of resistors to get the equivalent resistance.
Total Current Calculation
To find the total current flowing in a circuit with resistors in parallel, Ohm's Law is your best friend. Ohm鈥檚 Law states that the current \( I \) through a pathway in an electric circuit is directly proportional to the voltage \( V \) and inversely proportional to the resistance \( R \). The formula is represented as:
  • \[ V = IR \]
  • If you rearrange this formula to solve for the total current \( I \): \[ I = \frac{V}{R} \]
  • Here, the voltage \( V \) is \( 25.0\, \mathrm{V} \), and the equivalent resistance \( R \) is \( 4.5\, \Omega \) giving:\[ I = \frac{25.0}{4.5} \approx 5.56\, \mathrm{A} \]
This means that approximately \( 5.56\, \mathrm{A} \) of current flows through the battery. Ohm鈥檚 Law is not only useful in analyzing simple circuits but is a fundamental tool in handling more complex circuitry with various components.

One App. One Place for Learning.

All the tools & learning materials you need for study success - in one app.

Get started for free

Most popular questions from this chapter

A wire of resistance \(5.0 \Omega\) is connected to a battery whose emf \(\mathscr{E}\) is \(2.0 \mathrm{~V}\) and whose internal resistance is \(1.0 \Omega\). In \(2.0 \mathrm{~min}\), 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?

What multiple of the time constant \(\tau\) gives the time taken by an initially uncharged capacitor in an \(R C\) series circuit to be charged to \(99.0 \%\) of its final charge?

Thermal energy is to be generated in a \(0.10 \Omega\) resistor at the rate of \(10 \mathrm{~W}\) by connecting the resistor to a battery whose emf is \(1.5 \mathrm{~V} .\) (a) What potential difference must exist across the resistor? (b) What must be the internal resistance of the battery?

Nine copper wires of length \(l\) and diameter \(d\) are connected in parallel to form a single composite conductor of resistance \(R\). What must be the diameter \(D\) of a single copper wire of length \(l\) if it is to have the same resistance?

A controller on an electronic arcade game consists of a variable resistor connected across the plates of a \(0.220 \mu \mathrm{F}\) capacitor. The capacitor is charged to \(5.00 \mathrm{~V}\), then discharged through the resistor. The time for the potential difference across the plates to decrease to \(0.800 \mathrm{~V}\) is measured by a clock inside the game. If the range of discharge times that can be handled effectively is from \(10.0 \mu \mathrm{s}\) to \(6.00 \mathrm{~ms}\), what should be the (a) lower value and (b) higher value of the resistance range of the resistor?
See all solutions

Recommended explanations on Physics Textbooks

Conservation of Energy and Momentum

Read Explanation

Energy Physics

Read Explanation

Dynamics

Read Explanation

Force

Read Explanation

Mechanics and Materials

Read Explanation

Rotational Dynamics

Read Explanation
View all explanations

What do you think about this solution?

We value your feedback to improve our textbook solutions.

Study anywhere. Anytime. Across all devices.