Elements of Electromagnetics
Elements of Electromagnetics
7th Edition
ISBN: 9780190698669
Author: Sadiku
Publisher: Oxford University Press
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Chapter 3, Problem 18P

(a)

To determine

The gradient of given scalar field and their value at specified point.

(a)

Expert Solution
Check Mark

Explanation of Solution

Given:

The scalar field (V(x,y,z)) is 10xyz2x2z.

The specified point (P) is (1,4,3).

Calculation:

Calculate the gradient of scalar field using the relation.

  gradV(x,y,z)=V(x,y,z)xax+V(x,y,z)yay+V(x,y,z)zaz

  gradV(x,y,z)=(10xyz2x2z)xax+(10xyz2x2z)yay+(10xyz2x2z)zaz=(10yz2(2x)z)ax+(10xz)ay+(10xy2x2)az=(10yz4xz)ax+10xzay+(10xy2x2)az

Calculate the gradient of scalar field at specified point using the relation.

  gradV(x,y,z)=(10yz4xz)ax+10xzay+(10xy2x2)az

  gradV(x,y,z)={(10(4)(3)4(1)(3))ax+10(1)(3)ay+(10(1)(4)2(1)2)az=(120+12)ax30ay+(402)az=132ax30ay42az

Thus, the gradient of scalar field is (10yz4xz)ax+10xzay+(10xy2x2)az_ and their value at the point (1,4,3) is 132ax30ay42az_.

(b)

To determine

The gradient of given scalar field and their value at specified point.

(b)

Expert Solution
Check Mark

Explanation of Solution

Given:

The scalar field (U(ρ,ϕ,z)) is 2ρsinϕ+ρz.

The specified point (Q) is (2,90°,1).

Calculation:

Calculate the gradient of scalar field using the relation.

  gradU(ρ,ϕ,z)=U(ρ,ϕ,z)ρaρ+U(ρ,ϕ,z)ϕaϕ+U(ρ,ϕ,z)zaz

  gradU(ρ,ϕ,z)=(2ρsinϕ+ρz)ρaρ+(2ρsinϕ+ρz)ϕaϕ+(2ρsinϕ+ρz)zaz=(2sinϕ+z)aρ+(2ρcosϕ)aϕ+(ρ)az=(2sinϕ+z)aρ+2ρcosϕaϕ+ρaz

Calculate the gradient of scalar field at specified point using the relation.

  gradU(ρ,ϕ,z)=(2sinϕ+z)aρ+2ρcosϕaϕ+ρaz

  gradU(ρ,ϕ,z)=(2sin(90°)+(1))aρ+2(2)cos(90°)aϕ+(2)az=(2(1)1)aρ+(4)(0)aϕ+2az=aρ+2az

Thus, the gradient of scalar field is (2sinϕ+z)aρ+2ρcosϕaϕ+ρaz_ and their value at (2,90°,1) is aρ+2az_.

(c)

To determine

The gradient of given scalar field and their value at specified point.

(c)

Expert Solution
Check Mark

Explanation of Solution

Given:

The scalar field (W(r,θ,ϕ)) is 4rsinθcosϕ.

The specified point (R) is (1,π/6,π/2).

Calculation:

Calculate the gradient of scalar field using the relation.

  gradW(r,θ,ϕ)=W(r,θ,ϕ)rar+W(r,θ,ϕ)θaθ+W(r,θ,ϕ)ϕaϕ

  gradW(r,θ,ϕ)=(4rsinθcosϕ)rar+(4rsinθcosϕ)θaθ+(4rsinθcosϕ)ϕaϕ={((4r2)sinθcosϕ)ar+(4r(cosθ)cosϕ)aθ+(4rsinθ(sinϕ))aϕ=4r2sinθcosϕar+4rcosθcosϕaθ4rsinθsinϕaϕ

Calculate the gradient of scalar field at specified point using the relation.

  gradW(r,θ,ϕ)=4r2sinθcosϕar+4rcosθcosϕaθ4rsinθsinϕaϕ

  gradW(r,θ,ϕ)={412sin(π6)cos(π2)ar+41cos(π6)cos(π2)aθ41sin(π6)sin(π2)aϕ=4(0.5)(0)ar+4(0.866)(0)aθ4(0.5)(1)aϕ=0+02aϕ=2aϕ

Thus, the gradient of scalar field is 4r2sinθcosϕar+4rcosθcosϕaθ4rsinθsinϕaϕ_ and their value at (1,π/6,π/2) is 2aϕ_.

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Elements of Electromagnetics

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