(a) Explanation Given data: The vector field is given as, B = r ^ r cos ϕ + ϕ ^ sin ϕ (1) Redraw the diagram as shown in Figure 1. Calculation: The line integral around the semicircular contour is given as, ∮ C B ⋅ d l = ∮ L 1 B ⋅ d l + ∮ L 2 B ⋅ d l + ∮ L 3 B ⋅ d l + ∮ L 4 B ⋅ d l (2) Here, ∮ C B ⋅ d l is the line integral around the given contour, The integral is written as, ∮ L 1 B ⋅ d l = ∫ ( r ^ r cos ϕ + ϕ ^ sin ϕ ) ⋅ ( r ^ d r + ϕ ^ r d ϕ + z ^ d z ) = ∫ r = 1 2 r cos ϕ | ϕ = 0 , z = 0 d r + ∫ ϕ = 0 0 ( r sin ϕ ) | z = 0 d ϕ = ∫ r = 1 2 r d r + ∫ ϕ = 0 0 r sin ϕ d ϕ = [ r 2 2 ] 1 2 + r [ − cos ϕ ] 0 0 Further simplify the above expression. ∮ L 1 B ⋅ d l = [ 4 2 − 1 2 ] = 3 2 The integral is written as, ∮ L 2 B ⋅ d l = ∫ ( r ^ r cos ϕ + ϕ ^ sin ϕ ) ⋅ ( r ^ d r + ϕ ^ r d ϕ + z ^ d z ) = ∫ r = 2 2 r cos ϕ | z = 0 d r + ∫ ϕ = 0 π 2 ( r sin ϕ ) | r = 2 , z = 0 d ϕ = 0 + ∫ ϕ = 0 π 2 2 sin ϕ d ϕ = 2 [ − cos ϕ ] 0 π 2 Further simplify the above expression (b) To determine The value of ∫ S ( ∇ × B ) ⋅ d s .

Fundamentals of Applied Electromagnetics (7th Edition)

7th Edition

ISBN: 9780133356816

Author: Fawwaz T. Ulaby, Umberto Ravaioli

Publisher: PEARSON

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Spherical Coordinate System

The spherical coordinate system is a three-dimensional coordinate system for space where the position of a point is specified by three numbers- the radial distance of that point from a fixed source, its polar angle measured from a certain zenith, and the…

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Chapter 3, Problem 53P

(a)

To determine

The value of ∮CB⋅dl around the given contour.

(b)

To determine

The value of ∫S(∇×B)⋅ds.

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Chapter 3, Problem 52P

Chapter 3, Problem 54P

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Chapter 3 Solutions

Fundamentals of Applied Electromagnetics (7th Edition)

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The value of ∮ C B ⋅ d l around the given contour.

Solution Summary: The author calculates the line integral around the semicircular contour by redrawing the diagram as shown in Figure 1.

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The value of ∮CB⋅dl around the given contour.

The value of ∫S(∇×B)⋅ds.

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Chapter 3 Solutions