Elements Of Electromagnetics
Elements Of Electromagnetics
7th Edition
ISBN: 9780190698614
Author: Sadiku, Matthew N. O.
Publisher: Oxford University Press
bartleby

Concept explainers

bartleby

Videos

Question
Book Icon
Chapter 8, Problem 13P

(a)

To determine

Find the force on side 1 of the triangular loop.

(a)

Expert Solution
Check Mark

Answer to Problem 13P

The force on side 1 of the triangular loop is 2.197azμN.

Explanation of Solution

Calculation:

Write the general expression to calculate the force.

F=LIdl×B        (1)

Here,

Idl is the current element, and

B is the magnetic flux density.

Refer to the mentioned Figure given in the textbook.

Using equation (1), the force on side 1 of the triangular loop is calculated as follows with integration limits along ρ-direction.

F1=I1dρaρ×(μoI22πρ)aϕ{dl=dρaρ,B=μoI22πρaϕ}=μoI1I22π261ρdρaz{aρ×aϕ=az}=μoI1I22π(lnρ)26az=μoI1I22π(ln6ln2)az

Substitute 4π×107 for μo, 5 for I1, and 2 for I2 in above equation.

F1=(4π×107)(5)(2)2π(1.0986)az=2.197×106azN=2.197azμN{1μ=106}

Conclusion:

Thus, the force on side 1 of the triangular loop is 2.197azμN.

(b)

To determine

Find the total force on the loop.

(b)

Expert Solution
Check Mark

Answer to Problem 13P

The total force on the loop is 0.575aρμN.

Explanation of Solution

Calculation:

Refer to the mentioned Figure given in the textbook.

Write the general expression to calculate the force for side 2.

F2=LI2dl2×B1        (2)

Using equation (2), the force on the triangular loop is calculated as follows.

F2=I2(dρaρ+dzaz)×(μoI12πρ)aϕ{dl=(dρaρ+dzaz),B1=μoI12πρaϕ}=μoI1I22π1ρ(dρaρ+dzaz)×aϕ=μoI1I22π1ρ(dρaρ×aϕ+dzaz×aϕ)

F2=μoI1I22π1ρ(dρazdzaρ){aρ×aϕ=az,az×aϕ=aρ}        (3)

Consider,

ρ=z+2

On differentiating the above equation,

dρ=dz

Substitute 4π×107 for μo, 5 for I1, and 2 for I2, and dρ for dz in equation (3) with integration limits.

F2=(4π×107)(5)(2)2π421ρ(dρazdρaρ)=20×107421ρdρ(azaρ)=20×107(lnρ)42(azaρ)=20×107(ln2ln4)(azaρ)

Reduce the equation as follows,

F2=20×107(ln2ln4)(azaρ)=1.386×106(azaρ)N=1.386(azaρ)μN{1μ=106}=1.386aρ1.386azμN

Likewise using equation (3), the force on the triangular loop for side 3 is calculated as follows.

F3=μoI1I22π1ρ(dρazdzaρ)        (4)

Consider,

z=ρ+6

On differentiating the above equation,

dz=dρ

Substitute 4π×107 for μo, 5 for I1, and 2 for I2, and dρ for dz in equation (4) with integration limits.

F3=(4π×107)(5)(2)2π641ρ(dρaz+dρaρ)=20×107641ρdρ(az+aρ)=20×107(lnρ)64(az+aρ)=20×107(ln4ln6)(az+aρ)

Reduce the equation as follows,

F3=20×107(ln4ln6)(az+aρ)=0.8109×106(az+aρ)=0.8109×106az0.8109×106aρ=0.8109aρ0.8109azμN{1μ=106}

The total force on the loop is calculated as follows,

F=F1+F2+F3

Substitute 1.386aρ1.386azμN for F2, 0.8109aρ0.8109azμN for F3, and 2.197azμN for F1 in above equation.

F=(2.197azμN)+(1.386aρ1.386azμN)+(0.8109aρ0.8109azμN)=0.575aρμN

Conclusion:

Thus, the total force on the loop is 0.575aρμN.

Want to see more full solutions like this?

Subscribe now to access step-by-step solutions to millions of textbook problems written by subject matter experts!

Chapter 8 Solutions

Elements Of Electromagnetics

Knowledge Booster
Background pattern image
Mechanical Engineering
Learn more about
Need a deep-dive on the concept behind this application? Look no further. Learn more about this topic, mechanical-engineering and related others by exploring similar questions and additional content below.
Recommended textbooks for you
Text book image
Elements Of Electromagnetics
Mechanical Engineering
ISBN:9780190698614
Author:Sadiku, Matthew N. O.
Publisher:Oxford University Press
Text book image
Mechanics of Materials (10th Edition)
Mechanical Engineering
ISBN:9780134319650
Author:Russell C. Hibbeler
Publisher:PEARSON
Text book image
Thermodynamics: An Engineering Approach
Mechanical Engineering
ISBN:9781259822674
Author:Yunus A. Cengel Dr., Michael A. Boles
Publisher:McGraw-Hill Education
Text book image
Control Systems Engineering
Mechanical Engineering
ISBN:9781118170519
Author:Norman S. Nise
Publisher:WILEY
Text book image
Mechanics of Materials (MindTap Course List)
Mechanical Engineering
ISBN:9781337093347
Author:Barry J. Goodno, James M. Gere
Publisher:Cengage Learning
Text book image
Engineering Mechanics: Statics
Mechanical Engineering
ISBN:9781118807330
Author:James L. Meriam, L. G. Kraige, J. N. Bolton
Publisher:WILEY
Understanding Conduction and the Heat Equation; Author: The Efficient Engineer;https://www.youtube.com/watch?v=6jQsLAqrZGQ;License: Standard youtube license