College Physics
11th Edition
ISBN:9781305952300
Author:Raymond A. Serway, Chris Vuille
Publisher:Raymond A. Serway, Chris Vuille
Chapter1: Units, Trigonometry. And Vectors
Section: Chapter Questions
Problem 1CQ: Estimate the order of magnitude of the length, in meters, of each of the following; (a) a mouse, (b)...
Related questions
Question
![**Finding the Magnitude of \(\oint \vec{B} \cdot d \vec{s}\) for the Closed Path (a)**
In the problem presented, we need to find the magnitude of \(\oint \vec{B} \cdot d \vec{s}\) for the closed path labeled (a) shown in the diagram.
**Diagram Explanation:**
The diagram depicts a field with three currents flowing through it: \(1A\) (1 Ampere), \(5A\) (5 Amperes), and \(2A\) (2 Amperes). These are marked by circular crosses which typically indicate currents going into the plane of the diagram. The path (a) is represented by a dashed, closed loop.
**Problem Statement:**
Based on the given diagram, we are asked to compute the magnitude of \(\oint \vec{B} \cdot d \vec{s}\), where \(\vec{B}\) is the magnetic field and \(d \vec{s}\) represents an infinitesimal vector element of the path (a).
**Options Provided:**
a. \(4 \mu_0 \) (T.m)
b. \(3 \mu_0 \) (T.m)
c. \(1 \mu_0 \) (T.m)
d. \(0\)
To solve this, we utilize Ampère's Law which states that the integral of the magnetic field \(\vec{B}\) around a closed path is equal to \(\mu_0\) times the total current enclosed by that path:
\[ \oint \vec{B} \cdot d \vec{s} = \mu_0 I_{\text{enc}} \]
Here, \( I_{\text{enc}} \) is the net current enclosed by path (a).
Let's determine \( I_{\text{enc}} \):
1. Current \(5A\) is going into the plane (positive direction as per the assumption).
2. Current \(2A\) is also going into the plane (positive direction).
3. Current \(1A\) is going into the plane (positive direction).
Thus,
\[ I_{\text{enc}} = 5A + 2A + 1A = 8A \]
Using Ampère's Law:
\[ \oint \vec{B} \cdot d \vec{s}](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2Fdd8f2877-2578-48c0-ab77-5dda294aaf93%2F1acffed3-4afb-492e-a3d9-1db412f69788%2F52w2wyf_processed.jpeg&w=3840&q=75)
Transcribed Image Text:**Finding the Magnitude of \(\oint \vec{B} \cdot d \vec{s}\) for the Closed Path (a)**
In the problem presented, we need to find the magnitude of \(\oint \vec{B} \cdot d \vec{s}\) for the closed path labeled (a) shown in the diagram.
**Diagram Explanation:**
The diagram depicts a field with three currents flowing through it: \(1A\) (1 Ampere), \(5A\) (5 Amperes), and \(2A\) (2 Amperes). These are marked by circular crosses which typically indicate currents going into the plane of the diagram. The path (a) is represented by a dashed, closed loop.
**Problem Statement:**
Based on the given diagram, we are asked to compute the magnitude of \(\oint \vec{B} \cdot d \vec{s}\), where \(\vec{B}\) is the magnetic field and \(d \vec{s}\) represents an infinitesimal vector element of the path (a).
**Options Provided:**
a. \(4 \mu_0 \) (T.m)
b. \(3 \mu_0 \) (T.m)
c. \(1 \mu_0 \) (T.m)
d. \(0\)
To solve this, we utilize Ampère's Law which states that the integral of the magnetic field \(\vec{B}\) around a closed path is equal to \(\mu_0\) times the total current enclosed by that path:
\[ \oint \vec{B} \cdot d \vec{s} = \mu_0 I_{\text{enc}} \]
Here, \( I_{\text{enc}} \) is the net current enclosed by path (a).
Let's determine \( I_{\text{enc}} \):
1. Current \(5A\) is going into the plane (positive direction as per the assumption).
2. Current \(2A\) is also going into the plane (positive direction).
3. Current \(1A\) is going into the plane (positive direction).
Thus,
\[ I_{\text{enc}} = 5A + 2A + 1A = 8A \]
Using Ampère's Law:
\[ \oint \vec{B} \cdot d \vec{s}
Expert Solution
This question has been solved!
Explore an expertly crafted, step-by-step solution for a thorough understanding of key concepts.
Step by step
Solved in 2 steps with 2 images
Knowledge Booster
Learn more about
Need a deep-dive on the concept behind this application? Look no further. Learn more about this topic, physics and related others by exploring similar questions and additional content below.Recommended textbooks for you
College Physics
Physics
ISBN:
9781305952300
Author:
Raymond A. Serway, Chris Vuille
Publisher:
Cengage Learning
University Physics (14th Edition)
Physics
ISBN:
9780133969290
Author:
Hugh D. Young, Roger A. Freedman
Publisher:
PEARSON
Introduction To Quantum Mechanics
Physics
ISBN:
9781107189638
Author:
Griffiths, David J., Schroeter, Darrell F.
Publisher:
Cambridge University Press
College Physics
Physics
ISBN:
9781305952300
Author:
Raymond A. Serway, Chris Vuille
Publisher:
Cengage Learning
University Physics (14th Edition)
Physics
ISBN:
9780133969290
Author:
Hugh D. Young, Roger A. Freedman
Publisher:
PEARSON
Introduction To Quantum Mechanics
Physics
ISBN:
9781107189638
Author:
Griffiths, David J., Schroeter, Darrell F.
Publisher:
Cambridge University Press
Physics for Scientists and Engineers
Physics
ISBN:
9781337553278
Author:
Raymond A. Serway, John W. Jewett
Publisher:
Cengage Learning
Lecture- Tutorials for Introductory Astronomy
Physics
ISBN:
9780321820464
Author:
Edward E. Prather, Tim P. Slater, Jeff P. Adams, Gina Brissenden
Publisher:
Addison-Wesley
College Physics: A Strategic Approach (4th Editio…
Physics
ISBN:
9780134609034
Author:
Randall D. Knight (Professor Emeritus), Brian Jones, Stuart Field
Publisher:
PEARSON