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

Videos

Question
Book Icon
Chapter 7, Problem 45P

(a)

To determine

Find the magnetic flux density B for the given magnetic vector potential.

(a)

Expert Solution
Check Mark

Answer to Problem 45P

The magnetic flux density B for the given magnetic vector potential is (6xz+4x2y+3xz2)ax+(y+6yz4xy2)ay+(y2z32x2z)azWb/m2.

Explanation of Solution

Calculation:

Given the magnetic vector potential,

A=(2x2y+yz)ax+(xy2xz3)ay(6xyz2x2y2)azWb/m

Write the expression to calculate the magnetic flux density.

B=×A        (1)

Here,

A is the magnetic vector potential.

Substitute (2x2y+yz)ax+(xy2xz3)ay(6xyz2x2y2)azWb/m for A in Equation (1).

B=×A=|axayazxyz(2x2y+yz)(xy2xz3)(6xyz2x2y2)|=[(y((6xyz2x2y2))z(xy2xz3))ax(x((6xyz2x2y2))z(2x2y+yz))ay+(x(xy2xz3)y(2x2y+yz))az]=[((6xz(1)2x2(2y))(0x(3z2)))ax((6yz(1)2(2x)y2)(0+y(1)))ay+((y2(1)z3(1))(2x2(1)+(1)z))az]

Simplify the above equation.

B=[(6xz+4x2y+3xz2)ax(6yz+4xy2y)ay+(y2z32x2z)az]=(6xz+4x2y+3xz2)ax+(y+6yz4xy2)ay+(y2z32x2z)azWb/m2

Conclusion:

Thus, the magnetic flux density B for the given magnetic vector potential is (6xz+4x2y+3xz2)ax+(y+6yz4xy2)ay+(y2z32x2z)azWb/m2.

(b)

To determine

Find the magnetic flux for the given magnetic vector potential.

(b)

Expert Solution
Check Mark

Answer to Problem 45P

The magnetic flux (ψ) for the given magnetic vector potential is 8Wb.

Explanation of Solution

Calculation:

Write the expression to calculate the magnetic flux through a surface.

ψ=SBdS

Substitute (6xz+4x2y+3xz2)ax+(y+6yz4xy2)ay+(y2z32x2z)azWb/m2 for B in above equation.

ψ=S((6xz+4x2y+3xz2)ax+(y+6yz4xy2)ay+(y2z32x2z)az)dS=z=02y=02[(6xz+4x2y+3xz2)ax+(y+6yz4xy2)ay+(y2z32x2z)az]dydzax {dS=dydzax}=z=02y=02[(6xz+4x2y+3xz2)axdydzax+(y+6yz4xy2)aydydzax+(y2z32x2z)azdydzax]=z=02y=02[(6xz+4x2y+3xz2)dydz+0+0] {axax=1,ayax=0,azax=0}

Simplify the above equation.

ψ=z=02y=02(6xz+4x2y+3xz2)dydz=z=02[6xzy+4x2(y22)+3xz2y]02dz=z=02[6xyz+2x2y2+3xyz2]02dz=z=02[(6x(2)z+2x2(2)2+3x(2)z2)(6x(0)z+2x2(0)2+3x(0)z2)]dz

Simplify the above equation.

ψ=z=02[(12xz+8x2+6xz2)(0)]dz=z=02(12xz+8x2+6xz2)dz=[12x(z22)+8x2z+6x(z33)]02=[6xz2+8x2z+2xz3]02

Simplify the above equation.

ψ=[(6x(2)2+8x2(2)+2x(2)3)(6x(0)2+8x2(0)+2x(0)3)]=[(24x+16x2+16x)(0)]=24(1)+16(1)2+16(1) {x=1}=8Wb

Conclusion:

Thus, the magnetic flux (ψ) for the given magnetic vector potential is 8Wb.

(c)

To determine

Show that the relation A and B is equal to zero.

(c)

Expert Solution
Check Mark

Explanation of Solution

Calculation:

Substitute (2x2y+yz)ax+(xy2xz3)ay(6xyz2x2y2)azWb/m for A to find A.

A=xAx+yAy+zAz=x(2x2y+yz)+y(xy2xz3)+z((6xyz2x2y2))=(2(2x)y+0)+(x(2y)0)(6xy(1)0)=4xy+2xy6xy

Simplify the above equation.

A=6xy6xy=0

Substitute (6xz+4x2y+3xz2)ax+(y+6yz4xy2)ay+(y2z32x2z)azWb/m2 for B to find B.

B=xBx+yBy+zBz=x(6xz+4x2y+3xz2)+y(y+6yz4xy2)+z(y2z32x2z)=(6(1)z+4(2x)y+3(1)z2)+((1)+6(1)z4x(2y))+(03z201)=(6z+8xy+3z2)+(1+6z8xy)+(3z21)

Simplify the above equation.

B=6z+8xy+3z2+1+6z8xy3z21=0

Conclusion:

Thus, the relation A and B is equal to zero and it is shown.

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!
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
Unit Conversion the Easy Way (Dimensional Analysis); Author: ketzbook;https://www.youtube.com/watch?v=HRe1mire4Gc;License: Standard YouTube License, CC-BY