Explanation Given data: The conductor extends between 3 m and 7 m . The current is 15 A . The point P is ( 2 , ϕ , 0 ) . Redraw the given figure is as shown in Figure 1. Calculation: The differential length vector is given as, d l = z ^ d z The cross product is written as, d l × R ^ = d z ( z ^ × R ^ ) = ϕ ^ sin θ 1 d z The value of cos θ 1 is calculated as, cos ( θ 1 ) = z − z 1 ( z − z 1 ) 2 + r 2 The value of cos θ 2 is calculated as, cos ( θ 2 ) = z − z 2 ( z − z 2 ) 2 + r 2 The value of sin θ 1 is calculated as, sin ( θ 1 ) = r z 2 + r 2 From Biot-Savart’s law, the magnetic field due to the current carrying wire is given as, H = I 4 π ∫ z 1 = 3 z 2 = 7 d l × R ^ R 2 Substitute ϕ ^ sin θ 1 d z for d l × R ^ , r z 2 + r 2 for sin ( θ 1 ) in the above formula. H = I 4 π ∫ z 1 = 3 z 2 = 7 ϕ ^ sin θ 1 R 2 d z = I 4 π ∫ z 1 = 3 z 2 = 7 ϕ ^ r z 2 + r 2 R 2 d z Substitute ( z 2 + r 2 ) for R 2 in the above expression. H = I 4 π ∫ z 1 = 3 z 2 = 7 ϕ ^ r z 2 + r 2 ( z 2 + r 2 ) d z = I 4 π ∫ 3 7 ϕ ^ r ( z 2 + r 2 ) 3 2 d z (1) The value of d z is calculated as, z = r tan θ Differentiate the above expression as, d z = r sec 2 θ d θ So, the limits for the integration will be θ 1 to θ 2 . Substitute r sec 2 θ d θ for d z and r tan θ for z in equation (1). H = I 4 π ∫ θ 1 θ 2 ϕ ^ r ( z 2 + r 2 ) 3 2 r sec 2 θ d θ = I 4 π ∫ θ 1 θ 2 ϕ ^ r ( ( r tan θ ) 2 + r 2 ) 3 2 r sec 2 θ = I 4 π ∫ θ 1 θ 2 ϕ ^ r ( r 2 tan 2 θ + r 2 ) 3 2 r sec 2 θ = I 4 π ∫ θ 1 θ 2 ϕ ^ r ( r 2 sec 2 θ ) 3 2 r sec 2 θ Further simplify the above expression

Fundamentals of Applied Electromagnetics (7th Edition)

7th Edition

ISBN: 9780133356816

Author: Fawwaz T. Ulaby, Umberto Ravaioli

Publisher: PEARSON

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Chapter 5, Problem 8P

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The expression for the magnetic field H at an arbitrary point P due to the linear conductor.

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Fundamentals of Applied Electromagnetics (7th Edition)

Ch. 5.1 - What are the major differences between the...Ch. 5.1 - Prob. 2CQ Ch. 5.1 - How is the direction of the magnetic moment of a...Ch. 5.1 - If one of two wires of equal length is formed into...Ch. 5.1 - An electron moving in the positive x direction...Ch. 5.1 - A proton moving with a speed of 2 106 m/s through...Ch. 5.1 - A charged particle with velocity u is moving in a...Ch. 5.1 - A horizontal wire with a mass per unit length of...Ch. 5.1 - A square coil of 100 turns and 0.5 m long sides is...Ch. 5.2 - Two infinitely long parallel wires carry currents...

Ch. 5.2 - Devise a right-hand rule for the direction of the...Ch. 5.2 - What is a magnetic dipole? Describe its magnetic...Ch. 5.2 - Prob. 6E Ch. 5.2 - A wire carrying a current of 4 A is formed into a...Ch. 5.2 - Prob. 8E Ch. 5.3 - What are the fundamental differences between...Ch. 5.3 - Prob. 9CQ Ch. 5.3 - Compare the utility of applying the BiotSavart law...Ch. 5.3 - Prob. 11CQ Ch. 5.3 - A current I flows in the inner conductor of a long...Ch. 5.3 - The metal niobium becomes a superconductor with...Ch. 5.5 - What are the three types of magnetic materials and...Ch. 5.5 - What causes magnetic hysteresis in ferromagnetic...Ch. 5.5 - Prob. 14CQ Ch. 5.5 - The magnetic vector M is the vector sum of the...Ch. 5.6 - With reference to Fig. 5-24, determine the single...Ch. 5.7 - Prob. 15CQ Ch. 5.7 - What is the difference between self-inductance and...Ch. 5.7 - Prob. 17CQ Ch. 5.7 - Use Eq. (5.89) to obtain an expression for B at a...Ch. 5 - An electron with a speed of 8 106 m/s is...Ch. 5 - When a particle with charge q and mass m is...Ch. 5 - The circuit shown in Fig. P5.3 uses two identical...Ch. 5 - The rectangular loop shown in Fig. P5.4 consists...Ch. 5 - In a cylindrical coordinate system, a 2 m long...Ch. 5 - Prob. 6P Ch. 5 - Prob. 7P Ch. 5 - Prob. 8P Ch. 5 - The loop shown in Fig. P5.9 consists of radial...Ch. 5 - An infinitely long, thin conducting sheet defined...Ch. 5 - An infinitely long wire carrying a 25 A current in...Ch. 5 - Prob. 12P Ch. 5 - Prob. 13P Ch. 5 - Prob. 14P Ch. 5 - A circular loop of radius a carrying current I1 is...Ch. 5 - Prob. 16P Ch. 5 - Prob. 17P Ch. 5 - Prob. 18P Ch. 5 - Three long, parallel wires are arranged as shown...Ch. 5 - A square loop placed as shown in Fig. P5.20 has 2...Ch. 5 - Prob. 21P Ch. 5 - Prob. 22P Ch. 5 - Repeat Problem 5.22 for a current density J=zJ0er.Ch. 5 - In a certain conducting region, the magnetic field...Ch. 5 - Prob. 25P Ch. 5 - Prob. 26P Ch. 5 - Prob. 27P Ch. 5 - A uniform current density given by J=zj0 (A/m2)...Ch. 5 - A thin current element extending between z = L/2...Ch. 5 - In the model of the hydrogen atom proposed by Bohr...Ch. 5 - Iron contains 8.5 1028 atoms/m3. At saturation,...Ch. 5 - The xy plane separates two magnetic media with...Ch. 5 - Given that a current sheet with surface current...Ch. 5 - In Fig. P5.34, the plane defined by x y = 1...Ch. 5 - The plane boundary defined by z = 0 separates air...Ch. 5 - Prob. 36P Ch. 5 - Prob. 37P Ch. 5 - A solenoid with a length of 20 cm and a radius of...Ch. 5 - Prob. 39P Ch. 5 - The rectangular loop shown in Fig. P5.40 is...Ch. 5 - Determine the mutual inductance between the...

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