Electric Potential in a Uniform Electric Field

Dielectrics in Electric Fields Gorur G. Raju University of Windsor- Windsor, Ontario, Canada MARCEL MARCEL DEKKER, INC. DEKKER NEW YORK BASEL Library of Congress Cataloging-in-Publication Data A catalog record for this book is available from the Library of Congress ISBN: 0-8247-0864-4 This book is printed on acid-free paper Headquarters Marcel Dekker, Inc 270 Madison Avenue, New York, NY 10016 tel 212-696-9000, fax 212-685-4540 Eastern Hemisphere Distribution Marcel Dekker AG Hutgasse4, Postfach 812, CH-4001 Basel, Switzerland tel 41-61-260-6300, fax 41-61-260-6333 World Wide Web http //www dekker com The publisher offers discounts on this book when ordered in bulk quantities For more information, write to Special Sales/Professional Marketing at the headquarters address above Copyright © 2003 by Marcel Dekker, Inc. All Rights Reserved. Neither this book nor any part may be reproduced or transmitted in any form or by any means, electronic or mechanical, including photocopying, microfilming, and recording, or by any information storage and retrieval system, without permission in writing from the publisher Current printing (last digit) 10 987654321 PRINTED IN THE UNITED STATES OF AMERICA TM Copyright n 2003 by Marcel Dekker, Inc. All Rights Reserved. POWER ENGINEERING Series Editor H. Lee Willis ABB Inc. Raleigh, North Carolina 1. Power Distribution Planning Reference Book, H. Lee Willis 2. Transmission Network Protection: Theory and Practice, Y. G. Paithan- kar 3. Electrical Insulation in Power Systems, N. H. Malik, A. A. AI-Arainy, and M. I. Qureshi 4. Electrical Power Equipment Maintenance and Testing, Paul Gill 5. Protective Relaying: Principles and Applications, Second Edition, J. Lewis Blackburn 6. Understanding Electric Utilities and De-Regulation, Lorrin Philipson and H. Lee Willis 7. Electrical Power Cable Engineering, William A. Thue 8. Electric Systems, Dynamics, and Stability with Artificial Intelligence Applications, James A. Momoh and Mohamed E. EI-Hawary 9. Insulation Coordination for Power Systems, Andrew R. Hileman 10. Distributed Power Generation: Planning and Evaluation, H. Lee Willis and Walter G. Scott 11. Electric Power System Applications of Optimization, James A. Momoh 12. Aging Power Delivery Infrastructures, H. Lee Willis, Gregory V. Welch, and Randall R. Schrieber 13. Restructured Electrical Power Systems: Operation, Trading, and Vola- tility, Mohammad Shahidehpour and Muwaffaq Alomoush 14. Electric Power Distribution Reliability, Richard E. Brown 15. Computer-Aided Power System Analysis, Ramasamy Natarajan 16. Power System Analysis: Short-Circuit Load Flow and Harmonics, J. C. Das 17. Power Transformers: Principles and Applications, John J. Winders, Jr. 18. Spatial Electric Load Forecasting: Second Edition, Revised and Ex- panded, H. Lee Willis 19. Dielectrics in Electric Fields, GorurG. Raju 20. Protection Devices and Systems for High-Voltage Applications, Vladimir Gurevich ADDITIONAL VOLUMES IN PREPARATION TM Copyright n 2003 by Marcel Dekker, Inc. All Rights Reserved. TO MY PARENTS. MY WIFE, PADMINI, AND OUR SON, ANAND WHO GA VE ME ALL I VALUE. SOME DEBTS ARE NEVER REPAID IN FULL MEASURE. TM Copyright n 2003 by Marcel Dekker, Inc. All Rights Reserved. TM Copyright n 2003 by Marcel Dekker, Inc. All Rights Reserved. SERIES INTRODUCTION Power engineering is the oldest and most traditional of the various areas within electrical engineering, yet no other facet of modern technology is currently undergoing a more dramatic revolution in both technology and industry structure. This addition to Marcel Dekker's Power Engineering Series addresses a fundamental element Electric Potential in a Uniform Electric Field Electric Potential in a Uniform Electric Field Bởi: OpenStaxCollege In the previous section, we explored the relationship between voltage and energy In this section, we will explore the relationship between voltage and electric field For example, a uniform electric field E is produced by placing a potential difference (or voltage) ΔV across two parallel metal plates, labeled A and B (See [link].) Examining this will tell us what voltage is needed to produce a certain electric field strength; it will also reveal a more fundamental relationship between electric potential and electric field From a physicist’s point of view, either ΔV or E can be used to describe any charge distribution ΔV is most closely tied to energy, whereas E is most closely related to force ΔV is a scalar quantity and has no direction, while E is a vector quantity, having both magnitude and direction (Note that the magnitude of the electric field strength, a scalar quantity, is represented by E below.) The relationship between ΔV and E is revealed by calculating the work done by the force in moving a charge from point A to point B But, as noted in Electric Potential Energy: Potential Difference, this is complex for arbitrary charge distributions, requiring calculus We therefore look at a uniform electric field as an interesting special case 1/8 Electric Potential in a Uniform Electric Field The relationship between V and E for parallel conducting plates is E = V / d (Note that ΔV = VAB in magnitude For a charge that is moved from plate A at higher potential to plate B at lower potential, a minus sign needs to be included as follows: –Δ V = VA – VB = VAB See the text for details.) The work done by the electric field in [link] to move a positive charge q from A, the positive plate, higher potential, to B, the negative plate, lower potential, is W = –ΔPE = – qΔV The potential difference between points A and B is –Δ V = – (VB – VA) = VA – VB = VAB Entering this into the expression for work yields W = qVAB Work is W = Fd cos θ; here cos θ = 1, since the path is parallel to the field, and so W = Fd Since F = qE, we see that W = qEd Substituting this expression for work into the previous equation gives qEd = qVAB The charge cancels, and so the voltage between points A and B is seen to be 2/8 Electric Potential in a Uniform Electric Field VAB = Ed E= VAB d } (uniform E - field only), where d is the distance from A to B, or the distance between the plates in [link] Note that the above equation implies the units for electric field are volts per meter We already know the units for electric field are newtons per coulomb; thus the following relation among units is valid: N / C = V / m Voltage between Points A and B VAB = Ed E= VAB d } (uniform E - field only), where d is the distance from A to B, or the distance between the plates What Is the Highest Voltage Possible between Two Plates? Dry air will support a maximum electric field strength of about 3.0×106 V/m Above that value, the field creates enough ionization in the air to make the air a conductor This allows a discharge or spark that reduces the field What, then, is the maximum voltage between two parallel conducting plates separated by 2.5 cm of dry air? Strategy We are given the maximum electric field E between the plates and the distance d between them The equation VAB = Ed can thus be used to calculate the maximum voltage Solution The potential difference or voltage between the plates is VAB = Ed Entering the given values for E and d gives VAB = (3.0×106 V/m)(0.025 m) = 7.5×104 V or 3/8 Electric Potential in a Uniform Electric Field VAB = 75 kV (The answer is quoted to only two digits, since the maximum field strength is approximate.) Discussion One of the implications of this result is that it takes about 75 kV to make a spark jump across a 2.5 cm (1 in.) gap, or 150 kV for a cm spark This limits the voltages that can exist between conductors, perhaps on a power transmission line A smaller voltage will cause a spark if there are points on the surface, since points create greater fields than smooth surfaces Humid air breaks down at a lower field strength, meaning that a smaller voltage will make a spark jump through humid air The largest voltages can be built up, say with static electricity, on dry days A spark chamber is used to trace the paths of high-energy particles Ionization created by the particles as they pass through the gas between the plates allows a spark to jump The sparks are perpendicular to the plates, following electric field lines between them The potential difference between adjacent plates is not high enough to cause sparks without the ionization produced by particles from accelerator experiments (or cosmic rays) (credit: Daderot, Wikimedia Commons) Field and Force inside an Electron Gun (a) An electron gun has parallel plates separated by 4.00 cm and gives electrons 25.0 keV of energy What is the electric field strength ...The rich and the poor are two locked caskets of which each contains the key to the other. Karen Blixen (Danish Writer) 1 INTRODUCTORY CONCEPTS I n this Chapter we recapitulate some basic concepts that are used in several chapters that follow. Theorems on electrostatics are included as an introduction to the study of the influence of electric fields on dielectric materials. The solution of Laplace's equation to find the electric field within and without dielectric combinations yield expressions which help to develop the various dielectric theories discussed in subsequent chapters. The band theory of solids is discussed briefly to assist in understanding the electronic structure of dielectrics and a fundamental knowledge of this topic is essential to understand the conduction and breakdown in dielectrics. The energy distribution of charged particles is one of the most basic aspects that are required for a proper understanding of structure of the condensed phase and electrical discharges in gases. Certain theorems are merely mentioned without a rigorous proof and the student should consult a book on electrostatics to supplement the reading. 1.1 A DIPOLE A pair of equal and opposite charges situated close enough compared with the distance to an observer is called an electric dipole. The quantity » = Qd (1.1) where d is the distance between the two charges is called the electric dipole moment, u. is a vector quantity the direction of which is taken from the negative to the positive •jr. charge and has the unit of C m. A unit of dipole moment is 1 Debye = 3.33 xlO" C m. TM Copyright n 2003 by Marcel Dekker, Inc. All Rights Reserved. 1.2 THE POTENTIAL DUE TO A DIPOLE Let two point charges of equal magnitude and opposite polarity, +Q and -Q be situated d meters apart. It is required to calculate the electric potential at point P, which is situated at a distance of R from the midpoint of the axis of the dipole. Let R + and R . be the distance of the point from the positive and negative charge respectively (fig. 1.1). Let R make an angle 6 with the axis of the dipole. R Fig. 1.1 Potential at a far away point P due to a dipole. The potential at P is equal to Q R_ (1.2) Starting from this equation the potential due to the dipole is , QdcosQ (1.3) TM Copyright n 2003 by Marcel Dekker, Inc. All Rights Reserved. Three other forms of equation (1.3) are often useful. They are (1.4) (1.5) (1.6) The potential due to a dipole decreases more rapidly than that due to a single charge as the distance is increased. Hence equation (1.3) should not be used when R « d. To determine its accuracy relative to eq. (1.2) consider a point along the axis of the dipole at a distance of R=d from the positive charge. Since 6 = 0 in this case, (f> = Qd/4ns 0 (1.5d) =Q/9ns 0 d according to (1.3). If we use equation (1.2) instead, the potential is Q/8ns 0 d, an error of about 12%. The electric field due to a dipole in spherical coordinates with two variables (r, 0 ) is given as: 17 r n _!_ n l-—*r-—* 9 (iy) Partial differentiation of equation (1.3) leads to Equation (1.7) may be written more concisely as: TM Copyright n 2003 by Marcel Dekker, Inc. All Rights Reserved. (1.10) Substituting for § from equation (1.5) and changing the variable to r from R we get 1 1 47TGQ r r We may now make the substitution r r 3r ^ r Equation (1.12) now becomes 3//vT (1.11) (1.12) (1.13) Fig. 1.2 The two components of the electric field due to a dipole with moment TM Copyright n 2003 by Marcel Dekker, Inc. All Rights Reserved. The electric field at P has two components. The first term in NANO EXPRESS Open Access Gold colloidal nanoparticle electrodeposition on a silicon surface in a uniform electric field D Buttard 1,2* , F Oelher 1 and T David 1 Abstract The electrodeposition of gold colloidal nanoparticles on a silicon wafer in a uniform electric field is investigated using scanning electron microscopy and homemade electrochemical cells. Dense and uniform distributions of particles are obtained with no ag gregation. The evolution of surface particle density is analyzed in relation to several parameters: applied voltage, electric field, exchanged charge. Electrical, chemical, and electrohydrodynamical parameters are taken into account in describing the electromigration proce ss. 1. Introduction The emerging fields of nanoscience and nanoengineering are helping us to better understand and control the funda- mental building blocks in the physics of materials [1,2]. The manipulation of nano-objects is also essential and requires expertise in several domains (mechanics, electro- chemistry, optics ) [3-5]. The traditional top-down approach is by far the most widespread within the micro- electronics industry, but it relies on a complex lithography technique that results in very high production costs. Alter- native approaches are theref ore bei ng investiga ted with a view to achieving a spontaneous self-assembly of nano- components. Among these approaches, the so- called bot- tom-up method is attracting increasing attention. Based on this method, the self-organization of gold nanoparticles on a planar surface is providing new solutions for electrical or catalytic systems [6,7]. However, the deposition of parti- cles on a substrat e [8,9] must confor m to several criteria such as irreversibility of the deposition process [10], stabi- lity, and high density. Deposition of gold coll oidal nano- particles can be achieved with different methods. For instance, the electrophoretic deposition method (EPD) [11,12] uses a uniform external electric field to drive the suspended particles from the solution toward the substrate surface. The advantage of the EPD method is that it requires no special surface passivation on the colloidal particles and it can be controlled conveniently by the applied field [13 ,14]. The deposition proce ss, however, is complex [15] and many questions remain unanswered, despite the extensive use of EPD. In this article, we describe the uniform electric field- assisted deposition of gold colloidal nanoparticles from an aqueous solution onto a planar silicon surface. The adsorption of nanoparticles onto silicon is described and the surface density obtained is investi gated in function of the usual experimental param eters: applied voltage, elec- tric field, and initial nanoparticle density existing in the solution. 2. Material and methods Gold colloidal nanoparticles from the British Bio Cell Company were deposited on standard p-type silicon wafers, <111>-oriented, with a low electrical resistivity ( r <0.01Ωcm) to ensure a good ohmic contact in the electroch emical cell. Prior to particle deposition, the sili- con wafers were deoxidized using vapor hydrofluoric acid (HF) at room temperature above a liquid HF solution with 49 vol.%. Thanks to this process, the silicon surface of the wafer is free of the native silicon oxide that usually covers a silicon surface. The colloidal solution is an aqueous-sta- bilized dispersion of gold nanoparticles (particle purity 99.9%) with Hindawi Publishing Corporation Boundary Value Problems Volume 2009, Article ID 949124, 27 pages doi:10.1155/2009/949124 Research Article Electroelastic Wave Scattering in a Cracked Dielectric Polymer under a Uniform Electric Field Yasuhide Shindo and Fumio Narita Department of Materials Processing, Graduate School of Engineering, Tohoku University, Aoba-yama 6-6-02, Sendai 980-8579, Japan Correspondence should be addressed to Yasuhide Shindo, shindo@material.tohoku.ac.jp Received 25 April 2009; Revised 2 May 2009; Accepted 18 May 2009 Recommended by Juan J. Nieto We investigatethe scattering of plane harmonic compression and shear waves by a Griffith crack in an infinite isotropic dielectric polymer. The dielectric polymer is permeated by a uniform electric field normal to the crack face, and the incoming wave is applied in an arbitrary direction. By the use of Fourier transforms, we reduce the problem to that of solving two simultaneous dual integral equations. The solution of the dual integral equations is then expressed in terms of a pair of coupled Fredholm integral equations of the second kind having the kernel that is a finite integral. The dynamic stress intensity factor and energy release rate for mode I and mode II are computed for different wave frequencies and angles of incidence, and the influence of the electric field on the normalized values is displayed graphically. Copyright q 2009 Y. Shindo and F. Narita. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited. 1. Introduction Elastic dielectrics such as insulating materials have been reported to have poor mechanical properties. Mechanical failure of insulators is also a well-known phenomenon. Therefore, understanding the fracture behavior of the elastic dielectrics will provide useful information to the insulation designers. Toupin 1 considered the isotropic elastic dielectric material and obtained the form of the constitutive relations for the stress and electric fields. Kurlandzka 2 investigated a crack problem of an elastic dielectric material subjected to an electrostatic field. Pak and Herrmann 3, 4 also derived a material force in the form of a path-independent integral for the elastic dielectric medium, which is related to the energy release rate. Recently, Shindo and Narita 5 considered the planar problem for an infinite dielectric polymer containing a crack under a uniform electric field, and discussed the stress intensity factor and energy release rate under mode I and mode II loadings. This paper investigates the scattering of in-plane compressional P and shear SV waves by a Griffith crack in an infinite dielectric polymer permeated by a uniform electric 2 Boundary Value Problems field. The electric field is normal to the crack surface. Fourier transforms are used to reduce the problem to the solution of two simultaneous dual integral equations. The solution of the integral equations is then expressed in terms of a pair of coupled Fredholm integral equations of the second kind. In literature, there are two derivations of dual integral equations. One is the one mentioned in this paper. The other one is for the dual boundary element methods BEM6, 7. Numerical calculations are carried out for the dynamic stress intensity factor and energy release rate under mode I and mode II, and the results are shown graphically to demonstrate the effect of the Kanakamedala et al. Radiation Oncology 2010, 5:38 http://www.ro-journal.com/content/5/1/38 Open Access CASE REPORT © 2010 Kanakamedala et al; licensee BioMed Central Ltd. This is an Open Access article distributed under the terms of the Creative Com- mons Attribution License (http://creativecommons.org/licenses/by/2.0), which permits unrestricted use, distribution, and reproduc- tion in any medium, provided the original work is properly cited. Case report Lack of Cetuximab induced skin toxicity in a previously irradiated field: case report and review of the literature Madhava R Kanakamedala*, Satyaseelan Packianathan and Srinivasan Vijayakumar Abstract Introduction: Mutation, amplification or dysregulation of the EGFR family leads to uncontrolled division and predisposes to cancer. Inhibiting the EGFR represents a form of targeted cancer therapy. Case report: We report the case of 79 year old gentlemen with a history of skin cancer involving the left ear who had radiation and surgical excision. He had presented with recurrent lymph node in the left upper neck. We treated him with radiation therapy concurrently with Cetuximab. He developed a skin rash over the face and neck area two weeks after starting Cetuximab, which however spared the previously irradiated area. Conclusion: The etiology underlying the sparing of the previously irradiated skin maybe due to either decrease in the population of EGFR expressing cells or decrease in the EGFR expression. We raised the question that "Is it justifiable to use EGFR inhibitors for patients having recurrence in the previously irradiated field?" We may need further research to answer this question which may guide the physicians in choosing appropriate drug in this scenario. Introduction The ErbB or epidermal growth factor family is a family of four structurally related, EGFR/ErbB1/HER1, ErbB2/neu/ HER2, ErbB3/HER3, and ErbB4/HER4. ErbB receptors are comprised of an extracellular region or ectodomain, a single transmembrane spanning region, and a cytoplas- mic tyrosine kinase domain [1]. Epidermal growth factor receptors (EGFR), upon activation by their respective ligands, undergo a transformation from the inactive monomeric form into an active homo or hetero-dimer. This process stimulates its intrinsic intracellular protein- tyrosine kinase activity [2]. Mutation, amplification, or dysregulation of the EGFR family leads to uncontrolled division and predisposes the individual to cancer development [3]. EGFR over-expres- sion has also been correlated with disease progression, poorer prognosis, and reduced sensitivity to chemother- apy [4]. Inhibiting the EGFR - by directly blocking the extracellular EGFR receptor domain with monoclonal antibodies or by inhibiting the intra-cytoplasmic ATP binding site with tyrosine kinase inhibitors (TKI's) - rep- resents an accepted form of targeted cancer therapy[5]. Data from a large, randomized, phase III study of patients with locally advanced squamous cell carcinoma (SCC) of the head and neck suggests that blockade of the EGFR pathway may improve the efficacy of radiation therapy and improve survival [6]. In this study, EGFR blockade was achieved with the monoclonal antibody Cetuximab (Erbitux). There was no significant difference in the rate of mucositis seen in either treatment arm, but there was a higher incidence of grade 3/4 skin reactions when the combined high dose radiation/Cetuximab was employed. Nonetheless, the addition of Cetuximab was associated with a significant improvement in overall sur- vival (median 54 v 28 months; p = 0.02) compared to radi- ation alone. EGFR inhibition, whether with antibodies or TKI, causes a cutaneous rash in almost 70% of patients receiv- ing such therapy; generally it involves the face, neck, and upper chest. The severity of rash has been correlated to progression-free survival in cetuximab and erlotinib * Correspondence: mkanakamedala@ci.umsmed.edu 1 Department of Radiation ... potential The electric field is said to be the gradient (as in grade or slope) of the electric potential For continually changing potentials, ΔV and Δs become infinitesimals and differential calculus... smooth surfaces Humid air breaks down at a lower field strength, meaning that a smaller voltage will make a spark jump through humid air The largest voltages can be built up, say with static electricity,... difference and electric field strength are related Give an example What is the strength of the electric field in a region where the electric potential is constant? Will a negative charge, initially at