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Appendix A: Fundamentals of Electrostatics

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Appendix A: Fundamentals of Electrostatics Appendix A: Fundamentals of Electrostatics Antonio Arnau and Tomás Sogorb Departamento de Ingeniería Electrónica, Universidad Politécnica de Valencia A.1 Principles on Electrostatics It is usually attributed to Thales of Miletus (600 B.C.) the knowledge of the property of amber (in Greek elektron) of attracting very light bodies, when rubbed. This phenomenon is called electrification and the cause of the phenomenon electricity. The electric manifestations of electrification phenomena on different materials can be opposite and, in these cases, it is said that both materials have acquired qualities of opposite sign. The quality that a material acquires when electrified is called charge, establishing arbitrarily the sign of this quality in relation to the phenomenon that it manifests. In addition, it could be experimentally demonstrated that the electrified bodies interact with each other, and generate repulsive forces when the charges are of the same sign and, of attraction when they are of opposite sign. The fact that the charge is manifested in two opposite ways involves that if a body has as many positive charges as negative, its external electric manifestations are balanced, and we say that it is electrically neutral. Matter is mainly neutral and it is not common to find bodies whose net charge has a considerable value, therefore, the electrical interactions between bodies are in general quite weak. For this reason, for a material to manifest electric properties it is necessary to subject it to some type of action, for example mechanical, as in the case of amber. This external action produces a loss of balance between the positive and negative charges of the rod and allows the electric interaction with other bodies. At this time, it is necessary to introduce the principle of charge conservation in an isolated system. This experimental principle states that in a system in which charge cannot enter or leave, the positive and negative charges can vary with time but its net charge (positive + negative) remains constant. Therefore, when the amber rod is rubbed with a cloth, the charges of the cloth “arbitrarily called negative” pass to the amber rod that now has an excess of negative charge. A redistribution of the charges has taken place between the cloth and the rod but the net charge of the system cloth-rod has not been altered. A.A. Vives (ed.), Piezoelectric Transducers and Applications, doi: 10.1007/978-3-540-77508-9_19, © Springer-Verlag Berlin Heidelberg 2008 498 Antonio Arnau and Tomás Sogorb If the rubbing action on the rod is vigorous, it can be noticed that the electric interactions are stronger. That is to say, the small bodies that are used to show the electric interaction are attracted with more strength. This means that the amber acquires a larger negative charge. However, this charge increase is always in integer multiples of a minimum amount. This minimum amount of electric charge is the negative charge of the electron or the positive charge of the proton. This hypothesis establishes the principle of charge quantization as well as the existence of a natural unit of electric charge. A.2 The Electric Field It is important to notice that it has been necessary to bring the electrified amber rod close to small bodies in order to see the electric interactions. That is, it has been necessary to introduce a test element to check that the electrified amber rod had modified the electric characteristics of its immediate environment in such a way that other bodies, when coming closer to this environment, were subjected to an external force. However, the electrified amber rod modifies the electric characteristics of its environment even if no light bodies, which allow seeing this interference, are placed nearby. For example, we can use an apple to show the gravitational force but this still exists even without the apple. This concept is very important because it indicates that, similarly to the gravitation example, regions of the space are created around the positive and negative charges where they can show electric interactions. If these regions of the space appear due to the presence of punctual charges or superficial or volumetric distributions of static charge, the electric field created is called electrostatic field and the forces on the charges located in this field are determined by Coulomb’s law1. This electric field is of special importance in the study of piezoelectricity. 1 Coulomb’s law provides the force that acts on a charge q’ due to the presence of another punctual and fixed charge q. Its mathematical formulation is: 1 qq' F = 4πε r2 where ε is the permittivity of the medium and r is the distance between charges. Consequently, the force is proportional to the product of the charges and inversely proportional to the square of the distance that separates them. The direction is the straight line that links the charges. The direction is repulsion for charges of the same sign and attraction for charges of opposite sign. Appendix A: Fundamentals of Electrostatics 499 Intensity of the field, or simply field, at a point, is the force that acts on the unit of positive charge placed at this point. Consequently, a charge with value q located at a point, P, of this field would be subjected to a force given by: F(P) = qE(P) (A.1) where E(P) is the intensity of the electric field at point P. Similarly to the force, the field is a vector magnitude with a direction. Generally, electric fields are represented by their so-called lines of force. These lines are a graphic representation of the trajectories that a positive charge would follow, subjected to the influence of the field, in a succession of elementary paths, starting every time from rest. Let us imagine a surface immerse in an electric field. Infinite lines of force will cross this surface. At each point of the surface, there will be a line of force that, at that point, will be characterized by one value of field intensity and one direction. It is called flux Φ of an electric field through this surface: Φ = r r ∫ E o dA (A.2) where dA is a differential element of the surface under study. The product indicated in the previous equation is a scalar product of the two vectors. The previous definition provides an intuitive idea of the electric field as the density of electric flux per area unit. The value of the intensity of electric field at a point can be graphically represented by the number of lines of force that cross the surface unit at this point. Thus, there will be more lines of force in those points where the field intensity is stronger and more lines of force will cross the surface increasing the flux. A.3 The Electrostatic Potential If the potential energy of a punctual charge inside an electrostatic field is defined as a dependent function of point U(P) so that the difference between its values in the initial and final positions is similar to the work acting on the charge, by the force of the field. This potential energy would be formulated as: 2 2 r W = qE dr = U −U 1 ∫ o 1 2 (A.3) 1 and its differential expression would be: 500 Antonio Arnau and Tomás Sogorb r r dW = qE o dr = −dU (A.4) From the previous expression, results: r dU E dr = − = −dV (A.5) o q where dV=dU/q is called potential difference. V is called electrostatic potential, a scalar function dependent on the considered point, and it represents the work carried out to transfer the charge unit from infinite (where it is considered that V(∞)=0) to this point. From the previous expression it is deduced that the field is equal and with opposite sign to the gradient of the potential, that is to say: r ∂V r ∂V r ∂V r E = −grad V = − i − j − k (A.6) ∂x ∂y ∂x r r r where i , j and k are the unitary vectors in the directions of the coordinated axes. A.4 Fundamental Equations of Electrostatics The main problem in Electrostatics is the calculation of the fields produced by different charge distributions. Gauss’ law, which provides the value of the flux of the field through a closed surface, is one of the fundamental vector relationships that allow solving part of the problem. Gauss’ law states that the flux of an electric field through a closed surface is equal to the sum of all the charges enclosed in this surface divided by the permittivity of the vacuum εo. Its mathematical formulation is as follows: r r Q E dA = (A.7) ∫A o ε o where Q represents the total inner charge to surface A. If the distribution of the charge were a volumetric distribution, defined by a charge density ρv, the previous equation would be written: r r 1 E dA = ρ ⋅ dV ∫ o ε ∫ v (A.8) A o V Appendix A: Fundamentals of Electrostatics 501 The integral of the first member of the previous expression can be written as an integral of volume keeping in mind the theorem of divergence2, thus Eq. (A.8) is: r r 1 divE ⋅ dA = ρ ⋅ dV ∫ ε ∫ v (A.9) V o V or its differential expression that is: r ρ div E = v ε (A.10) o Notice that if there is no charge, or there are as many positive charges as negative, inside the closed surface, the field in the inside is constant. Substituting Eq. (A.6) in Eq. (A.10) one reaches Poisson’s Equation, which is the fundamental equation of electrostatics: r ∂2V ∂2V ∂2V ρ (x, y, z) div E = div gradV = + + = − v (A.12) ∂ 2 ∂ 2 ∂ 2 ε x y z o Notice that if the volumetric density of net charge is null, or the positive and negative charges are balanced, inside a volume, the potential inside this volume keeps a linear relationship with the distance.
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