(a) Explanation Given data: The vectors are given as, A = x ^ 2 − y ^ 3 + z ^ 1 (1) B = x ^ B x + y ^ 2 + z ^ B z (2) Calculation: The general form of a vector is given as, R = ( x ^ R x + y ^ R y + z ^ R z ) (3) Compare equation (1) and (3), the values are written as, R x = 2 R y = − 3 R z = 1 The magnitude of the vector R is calculated as, R = ( R x ) 2 + ( R y ) 2 + ( R z ) 2 Here, R is the magnitude of the vector R . Substitute 2 for R x , − 3 for R y , 1 for R z and A for R in the above expression. A = ( 2 ) 2 + ( − 3 ) 2 + ( 1 ) 2 = 4 + 9 + 1 = 14 The unit vector in the direction of A is given as, a ^ A = A A Here, a ^ A is the unit vector in the direction of A , A is the magnitude of vector A . Substitute ( x ^ 2 − y ^ 3 + z ^ 1 ) for A and 14 for A in the above expression. a ^ A = ( x ^ 2 − y ^ 3 + z ^ 1 ) 14 Compare equation (2) and (3), the values are written as, R x = B x R y = 2 R z = B z The magnitude of the vector R is calculated as, R = ( R x ) 2 + ( R y ) 2 + ( R z ) 2 Here, R is the magnitude of the vector R . Substitute B x for R x , 2 for R y , B z for R z and B for R in the above expression. B = ( B x ) 2 + ( 2 ) 2 + ( B z ) 2 = ( B x ) 2 + 4 + ( B z ) 2 The unit vector in the direction of B is given as, a ^ B = B B Here, a ^ B is the unit vector in the direction of B , B is the magnitude of vector B . Substitute x ^ B x + y ^ 2 + z ^ B z for B and ( B x ) 2 + 4 + ( B z ) 2 for B in the above expression. a ^ B = ( x ^ B x + y ^ 2 + z ^ B z ) ( B x ) 2 + 4 + ( B z ) 2 It is given that if A is parallel to B , then their directions are equal or opposite and it is written as, a ^ A = ± a ^ B Here, a ^ A is the direction of vector A , a ^ B is the direction of vector B (b) To determine A relation between B x and B z if A is perpendicular to B .

Fundamentals of Applied Electromagnetics (7th Edition)

7th Edition

ISBN: 9780133356816

Author: Fawwaz T. Ulaby, Umberto Ravaioli

Publisher: PEARSON

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Solenoidal Vector Field

In Physics and Mathematics vector calculus attached to each point in a subset of space, there is an assignment of a vector in a field called a vector field.

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Chapter 3, Problem 4P

(a)

To determine

The values of Bx and Bz if A is parallel to B.

(b)

To determine

A relation between Bx and Bz if A is perpendicular to B.

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Chapter 3, Problem 3P

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Chapter 3 Solutions

Fundamentals of Applied Electromagnetics (7th Edition)

Ch. 3.1 - When are two vectors equal and when are they...Ch. 3.1 - Prob. 2CQ Ch. 3.1 - If AB = 0, what is AB?Ch. 3.1 - If AB = 0, what is AB?Ch. 3.1 - Is A(BC) a vector triple product?Ch. 3.1 - If AB = AC, does it follow that B = C?Ch. 3.1 - Find the distance vector between P1 = (1, 2, 3)...Ch. 3.1 - Find the angle AB between vectors A and B of...Ch. 3.1 - Prob. 3E Ch. 3.1 - Vectors A and B lie in the y-z plane and both have...

Ch. 3.1 - If AB=AC, does it follow that B = C?Ch. 3.2 - A circular cylinder of radius r = 5 cm is...Ch. 3.3 - Why do we use more than one coordinate system?Ch. 3.3 - Prob. 8CQ Ch. 3.3 - Prob. 9CQ Ch. 3.3 - How is the position vector of a point in...Ch. 3.3 - Prob. 7E Ch. 3.3 - Prob. 8E Ch. 3.4 - Prob. 9E Ch. 3.4 - Find the directional derivative of V=rz2cos2 along...Ch. 3.4 - Prob. 11E Ch. 3.4 - Prob. 12E Ch. 3.5 - Given A=e2y(xsin2x+ycos2x), find A.Ch. 3.5 - Given A=rrcos+rsin+z3z , find A at (2,0, 3).Ch. 3.5 - If E=RAR in spherical coordinates, calculate the...Ch. 3.5 - Verify the divergence theorem by calculating the...Ch. 3.5 - Prob. 17E Ch. 3.6 - Find A at (2, 0, 3) in cylindrical coordinates for...Ch. 3.6 - Find A at (3, /6, 0) in spherical coordinates for...Ch. 3.7 - What do the magnitude and direction of the...Ch. 3.7 - Prob. 12CQ Ch. 3.7 - Prob. 13CQ Ch. 3.7 - Prob. 14CQ Ch. 3.7 - What is the meaning of the transformation provided...Ch. 3.7 - Prob. 16CQ Ch. 3.7 - Prob. 17CQ Ch. 3.7 - When is a vector field conservative?Ch. 3 - Prob. 1P Ch. 3 - Given vectors A=x2y3+z, B=x2y+z3, and C=x4+y2+z2,...Ch. 3 - Prob. 3P Ch. 3 - Prob. 4P Ch. 3 - Given vectors A=x+y2z3, B=x2y4, and C=y2z4, find...Ch. 3 - Given vectors A=x2y+z3 and B=x3z2, find a vector C...Ch. 3 - Given A=x(x+2y)y(y+3z)+z(3xy), determine a unit...Ch. 3 - By expansion in Cartesian coordinates, prove: (a)...Ch. 3 - Find an expression for the unit vector directed...Ch. 3 - Prob. 10P Ch. 3 - Prob. 11P Ch. 3 - Prob. 12P Ch. 3 - A given line is described by x+2y=4. Vector A...Ch. 3 - Prob. 14P Ch. 3 - Prob. 15P Ch. 3 - Given B=x(z3y)+y(2x3z)z(x+y), find a unit vector...Ch. 3 - Find a vector G whose magnitude is 4 and whose...Ch. 3 - A given line is described by the equation: y=x1....Ch. 3 - Vector field E is given by E=R5Rcos12Rsincos+3sin....Ch. 3 - Prob. 20P Ch. 3 - Prob. 21P Ch. 3 - Prob. 22P Ch. 3 - Prob. 23P Ch. 3 - Prob. 24P Ch. 3 - Use the appropriate expression for the...Ch. 3 - Prob. 26P Ch. 3 - A section of a sphere is described by 0 R 2, 0 ...Ch. 3 - A vector field is given in cylindrical coordinates...Ch. 3 - At a given point in space, vectors A and B are...Ch. 3 - Given vectors...Ch. 3 - Prob. 31P Ch. 3 - Prob. 32P Ch. 3 - Transform the vector A=Rsin2cos+cos2sin into...Ch. 3 - Transform the following vectors into cylindrical...Ch. 3 - Transform the following vectors into spherical...Ch. 3 - Find the gradient of the following scalar...Ch. 3 - For each of the following scalar fields, obtain an...Ch. 3 - The gradient of a scalar function T is given by...Ch. 3 - Prob. 39P Ch. 3 - For the scalar function V = xy2 z2, determine its...Ch. 3 - Evaluate the line integral of E=xxyy along the...Ch. 3 - Prob. 42P Ch. 3 - Prob. 43P Ch. 3 - Each of the following vector fields is displayed...Ch. 3 - Prob. 45P Ch. 3 - For the vector field E=xxzyyz2zxy, verify the...Ch. 3 - For the vector field E=r10erz3z, verify the...Ch. 3 - A vector field D=rr3 exists in the region between...Ch. 3 - For the vector field D=R3R2, evaluate both sides...Ch. 3 - For the vector field E=xxyy(x2+2y2), calculate (a)...Ch. 3 - Repeat Problem 3.50 for the contour shown in Fig....Ch. 3 - Verify Stokess theorem for the vector field...Ch. 3 - Prob. 53P Ch. 3 - Prob. 54P Ch. 3 - Verify Stokess theorem for the vector field B = (r...Ch. 3 - Prob. 56P Ch. 3 - Prob. 57P Ch. 3 - Find the Laplacian of the following scalar...

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