Explanation Given: The charge density ρ v of parallel plate capacitor is ρ o sin ( π z 2 d ) . Concept used: The Laplacian operator ∇ 2 is obtained from divergence of a gradient for potential V and is known as Poisson’s equation. ∇ 2 V = − ρ v ε The Laplace equation for cylindrical coordinates is shown below. ∇ 2 V cylindrical = 1 ρ ∂ ∂ ρ ( ρ ∂ V ∂ ρ ) + 1 ρ 2 ( ∂ 2 V ∂ ϕ 2 ) + ∂ 2 V ∂ z 2 Calculation: The electric potential V is only a function of z . Substitute ρ o sin ( π z 2 d ) for ρ v in the Poisson equation. ∇ 2 V = − 1 ε ⋅ ρ o sin ( π z 2 d ) = − ρ o ε sin ( π z 2 d ) Equate Laplace equation and Poisson’s equation. ∂ 2 V ∂ z 2 = − ρ o ε sin ( π z 2 d ) Integrate the above equation. ∫ ∂ 2 V ∂ z 2 d z = − ρ o ε ∫ sin ( π z 2 d ) d z ∂ V ∂ z = − ρ o ε ( − cos ( π z 2 d ) ) 1 ( π 2 d ) + C = 2 d ρ o π ε cos ( π z 2 d ) + C Integrate the above equation. ∫ ∂ V ∂ z d z = ∫ ( 2 d ρ o π ε cos ( π z 2 d ) + C ) d z V ( z ) = 2 d ρ o π ε sin ( π z 2 d ) 1 π 2 d + C z + D = 2 d ρ o π ε ⋅ 2 d π sin ( π z 2 d ) + C z + D = 4 d 2 ρ o π 2 ε sin ( π z 2 d ) + C z + D From the boundary conditions, the constants C and D are obtained and these boundary conditions are V ( 0 ) = 0 and V ( d ) = V d . Substitute 0 for V ( z ) and 0 for z in the expression of V . 0 = 4 d 2 ρ o π 2 ε sin ( 0 ) + 0 + D D = 0 Substitute V d for V ( z ) , 0 for D and d for z in the expression of V . V d = 4 d 2 ρ o π 2 ε sin ( π d 2 d ) + C d + 0 C d = V d − 4 d 2 ρ o π 2 ε C = V d − 4 d 2 ρ o π 2 ε d = V d d − 4 d ρ o π 2 ε So, the expression for potential V can be written as, V ( z ) = 4 d 2 ρ o π 2 ε sin ( π z 2 d ) + ( V d d − 4 d ρ o π 2 ε ) z + 0 = 4 d 2 ρ o π 2 ε sin ( π z 2 d ) + ( V d d − 4 d ρ o π 2 ε ) z Therefore, the expression for potential V is 4 d 2 ρ o π 2 ε sin ( π z 2 d ) + ( V d d − 4 d ρ o π 2 ε ) z

Fundamentals of Electromagnetics with Engineering Applications

1st Edition

ISBN: 9780470105757

Author: Stuart M. Wentworth

Publisher: Wiley, John & Sons, Incorporated

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Chapter 2, Problem 2.69P

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The new expression for potential V and electric field E . Also, the surface charge density ρs on each plate.

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Chapter 2, Problem 2.70P

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Fundamentals of Electromagnetics with Engineering Applications

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Ch. 2 - Prob. 2.11P Ch. 2 - Prob. 2.12P Ch. 2 - Prob. 2.13P Ch. 2 - A 20.0â€“cmâ€“long section of copper pipe has a...Ch. 2 - A line charge with charge density 2.00nC/m exists...Ch. 2 - You are given two zâ€“directed line charges of...Ch. 2 - Suppose you have a segment of line charge of...Ch. 2 - A segment of line charge L=10.nC/m exists on the...Ch. 2 - In free space, there is a point charge Q=8.0nC at...Ch. 2 - Prob. 2.20P Ch. 2 - Sketch the following surfaces and find the total...Ch. 2 - Consider a circular disk in the xâ€“y plane of...Ch. 2 - Suppose a ribbon of charge with density S exists...Ch. 2 - Sketch the following volumes and find the total...Ch. 2 - You have a cylinder of 4.00â€“in diameter and...Ch. 2 - Consider a rectangular volume with...Ch. 2 - Prob. 2.27P Ch. 2 - Prob. 2.28P Ch. 2 - Given D=2a+sinazC/m2, find the electric flux...Ch. 2 - Suppose the electric flux density is given by...Ch. 2 - Prob. 2.31P Ch. 2 - A cylindrical pipe with a 1.00â€“cm wall thickness...Ch. 2 - Prob. 2.34P Ch. 2 - Prob. 2.35P Ch. 2 - A thickâ€“walled spherical shell, with inner...Ch. 2 - Prob. 2.37P Ch. 2 - Determine the charge density at the point...Ch. 2 - Given D=3ax+2xyay+8x2y3azC/m2, (a) determine the...Ch. 2 - Suppose D=6cosaC/m2. (a) Determine the charge...Ch. 2 - Suppose D=r2sinar+sincosaC/m2. (a) Determine the...Ch. 2 - Prob. 2.42P Ch. 2 - A surface is defined by the function 2x+4y21nz=12....Ch. 2 - For the following potential distributions, use the...Ch. 2 - A 100nC point charge is located at the origin. (a)...Ch. 2 - Prob. 2.46P Ch. 2 - Prob. 2.47P Ch. 2 - Prob. 2.48P Ch. 2 - Suppose a 6.0â€“mâ€“diameter ring with charge...Ch. 2 - Prob. 2.50P Ch. 2 - Prob. 2.51P Ch. 2 - The typical length of each piece of jumper wire on...Ch. 2 - A 150â€“m length of AWGâ€“22 (0.644 mm diameter)...Ch. 2 - Determine an expression for the power dissipated...Ch. 2 - Find the resistance per unit length of a stainless...Ch. 2 - A nickel wire of diameter 5.0 mm is surrounded by...Ch. 2 - Prob. 2.57P Ch. 2 - A 20nC point charge at the origin is embedded in...Ch. 2 - Suppose the force is very carefully measured...Ch. 2 - The potential field in a material with r=10.2 is...Ch. 2 - In a mineral oil dielectric, with breakdown...Ch. 2 - Prob. 2.62P Ch. 2 - For z0,r1=9.0 and for z0,r2=4.0. If E1 makes a 300...Ch. 2 - Prob. 2.64P Ch. 2 - Consider a dielectricâ€“dielectric chargeâ€“free...Ch. 2 - A 1.0â€“cmâ€“diameter conductor is sheathed with a...Ch. 2 - Prob. 2.67P Ch. 2 - For a coaxial cable of inner conductor radius a...Ch. 2 - Prob. 2.69P Ch. 2 - Prob. 2.70P Ch. 2 - A parallelâ€“plate capacitor with a 1.0m2 surface...Ch. 2 - Prob. 2.72P Ch. 2 - Prob. 2.73P Ch. 2 - Given E=5xyax+3zaZV/m, find the electrostatic...Ch. 2 - Suppose a coaxial capacitor with inner radius 1.0...

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