EXPERIMENTAL FACTS: INTRODUCTION. Introduction. Experimental facts. Coulomb s Law. Repulsive force between two positive charges.

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1 Chapter 1 ELECTROSTATICS Introduction Experimental facts Coulomb s Law Units Principle of Superposition Figure 1 Repulsive force between two positive charges. EXPERIMENTAL FACTS: Force Field Concept of Electric Field Representation of an E field Summary Appendix 1: Newton s Law of Gravitation INTRODUCTION The existence of electric charge both as static charge and electric current, caused by moving electric charges, are manifest in many ways. Lightning is one dramatic example of the build up of static electric charge in clouds, while Aurora Borealis, the deflection of electron beam in you computer display, and the flow of electric current and power in conductors all are examples of moving charges. The study of electricity is best achieved by considering first electrostatics, that is, electric fields and forces for static charges. This then will lead to a discussion of electric current flow, or dynamic electricity. The next stage will be the discussion of magnetism produced by electric currents. Finally electricity and magnetism will be unified into one common subject called electromagnetism. The laws of electromagnetism can be condensed into four basic laws of nature, the Maxwell equations. These lead to the prediction of electromagnetic waves, such as light or radio waves. Electromagnetism, as epitomised by Maxwell s equations, led to the discovery of Einstein s theory of relativity, the understanding of optics, as well as the binding of atoms, molecules, and matter. As with all of physics, the laws of electrostatics must be based on experimental facts, which are discussed next. Experimental studies of electrostatics by physicists led to the following facts. 1)Objects can carry a property called charge, q, that is manifest by the electrostatic forces between charges. )There are two flavours of charge. BenjaminFranklin called them positive and negative, that is, he defined the sign of the charge. For example, charge is generated by rubbing rubber on fur leaving a charge Franklin definedasnegativeonrubberandpositiveonthefur. Similarly, rubbing silk on glass leaves the silk negatively charged and glass positively charged according to Franklin. 3)Like charges repel, unlike charges attract, just like the attraction in the animal kingdom. The pith balls demonstrate this fact. 4)Charge is quantized, that is, all electrons and protons have charge that is identical in magnitude but that the charge on the proton is positive and the charge on the electron is negative, using the sign convention chosen by Benjamin Franklin. The magnitude of the charge on the proton and electron is = Coulombs, a unit that will be defined soon. The normal atom comprises an equal number of protons and electrons and is found to be electrically neutral to This illustrates the exactly equal magnitudes and opposite signs of the charges on the electron and proton. Since there are electrons per gram-molecule of hydrogen, the effectsofthisquanti- zation are only manifest in microscopic behavior, such as the forces between atoms. 5)Charge is conserved. Thisisoneoftheimportant conservation laws of nature. Although it is observed that charges can be created or destroyed, the net charge is always conserved. For example, when an electron and positron annihilate into two gamma rays, the equal and opposite charges cancel leaving a neutrally charged vacuum. 6)The electrostatic force between point charges, q 1 and q is accurately proportional to 1. where r is the separation distance between the charges. The exponent on the distance dependence has been measured to be.000 to an accuracy of one part in 10 9 over a 1

2 Figure Force vector diagram range of distances from to )The electrostatic force always lies on a line separating the two point charges. Note that two points define a line. If the force did not point along this line then we would have to assume asymmetry of space. COULOMBS LAW These experimental facts lead to Coulomb s Law which says that the force F10 exerted by q 1 on q 0 pointing from q 1 to q 0 is: F 10 = cr 10 (1.1) where cr 10 = r is the unit vector pointing from q 1 to q 0. and k is the Coulomb constant. By Newton s third law, the force F01 exerted on q 1 by q 0 is equal but opposite to F 10 Thus we have that F 01 = cr 01 = F10 (1.) Note that the force is repulsive, that is positive, between like charges, whereas the force is attractive, that is negative, for unlike charges. UNITS The SI unit of charge, q, is the Coulomb. TheCoulomb is definedintermsoftheampere which is a current of 1 Coulomb per second. In this system of units, since r is in meters and the force in Newtons, we find that the constant k = Nm /C. For esoteric reasons, that will become obvious later, physicists write this constant as =14 0 which defines 0 in terms of this Coulomb s constant k. That is; the permittivity of free space 0 = You should have noticed the similarity of Coulomb s law of electrostatics and Newton s Law of Gravitation discussed in appendix 1. In electrostatics, charge plays the role of mass. However, there are several important differences. There are two flavours of charge and only one flavour of mass. The gravitational force is attractive whereas the electric force between same-sign charges is repulsive. The constant k for electrostaticsis10 36 times larger than the gravitational constant G. Note that the electrostatic force between two one- Coulomb charges, separated by 1 meter, is , i.e. about 10 6 tons. An appreciation of the strength of the electrostatic force is illustrated by considering two people sitting 1 meter apart. They do not feel the minute gravitational attraction, that is why we only perceive gravitational effects when very large masses, like the earth, are involved. By contrast, the two people would experience an electrostatic force between them of 10 1 tons if there was an imbalance of only 1% in the number of electrons and protons in each of them. It is because of the exact cancellation of the proton and electron charges in atoms that we do not explode with enormous energy due to electrostatic forces. As already mentioned, it is the electrostatic force that binds atoms, molecules, and solids and liquids. Even the energy from a nuclear reactor comes primarily from electrostatic forces. The uranium or plutonium nuclei deform into dumb bell shapes and it is the enormous electrostatic replusion between these two pieces of charged nuclear matter that liberates most of the energy manifest in nuclear fission reactors or nuclear weapons. In summary, for the SI system of units, Coulomb s Law is often written as F 10 = cr 10 (1.3) where charge q is in Coulombs, r in meters, and = Nm /C. PRINCIPLE OF SUPERPOSITION. Thediscussionsofarhasfocussedonforcesbetween two point charges. The Principle of Superposition is an important experimental fact that is important for calculation of electrostatic forces between ensembles of many point charges as is the usual situation. Consider three charges, 0 1 and. The net electrostatic force on 0 can be shown experimentally to be the vector sum of the forces on 0 due to each of the other charges. That is, the net force F 0 on 0,is; F 0 = F10 + F0 (1.4)

3 known, then it is possible to solve Newton s Laws of motion for a particle in this force field just like solving for the motion of a space craft in the gravitational force field of the earth. CONCEPT OF ELECTRIC FIELD Figure 3 Force diagram for superposition of two charges acting on q 0 This may seem obvious but it is not generally true. It is a statement that the force between any pair of charges is unchanged if additional charges are introduced. That is, the electrostatic force depends only on the location and strengths of each of the charges. This is called a local field force. It is obeyed by the electrostatic and gravitational forces but not the nuclear force, for which the force between any two nucleons is changed when additional nucleons are introduced. FORCE FIELD It was stated that the electric force field has the following properties: 1) The vector force depends only on the position and time. ) The net force is the vector superposition of the separate forces. 3) The force is proportional to the probe charge. Considerthecaseof =,then F 0 = F 1 + F 1 1 = cr cr 0 " F1 # F = 0 + (1.7) 0 0 As a consequence it is possible to factor out the probe charge q 0 by writing the above equation as; F0 F1 F = + (1.8) This leads us to the concept of electric field, analogous to gravitational field. The definition of the electric field, E,isthat; The fact that the electric force on a probe charge q 0 depends only on the location and the strengths of all the charges, leads to the concept of a force field. Using Coulomb s law: E F (1.9) F 0 = cr 0 (1.5) 0 and the Principle of Superposition, allows the net force on q 0 due to q 1 and q to be written as; F 0 = X 0 0 " 1 cr 0 = X 0 cr 0 # (1.6) Note the charge q 0 can be factored out of the bracket. Thus we can see that the force on q 0 depends only onthetheprobechargeq 0 and its location relative to asetoffixed other charges. The force on the probe charge, q 0 for a system of fixed other charges, can be calculated for any location of q 0. This introduces the concept of a force field F 0 ( 0 ) which can be evaluated anywhere in space, at time t, just like we calculate the gravitational force on a given mass at a given time anywhere in space. Once the force field is That is, electric field is the force on a probe charge, perunitpositiveprobecharge. That is, it is the force, in Newtons, that a probe charge of +1 Coulomb would experience at that location. The superposition of the force field due to two charges, given above, can be written as: E = E 1 + E (1.10) Where E 1 and E are the electric fields due to charges q 1 and q respectively and E is the net vector field. That is, the electric field also obeys the Principle of Superposition. Note that the electric field has the following properties: 1) It is completely specified at every spatial location ) It is a vector field 3) It is continuous and differentiable. Onecanreversetheabovedefinition and express the force field in terms of the electric field; 3

4 F = E (1.11) Note that the electric field has the magnitude and direction of the force experienced by a positive unit charge, that is, it points away from positive charges and towards negative charges. The SI unit of E field is Newtons/Coulomb. Theimportanceoftheelectricfield concept is that one can separate the calculation of the electric field and the net forces on any probe charge. This is like separating the calculation of the gravitational field at all points around the earth and the calculation of the forces on some body in the earths gravitational field. This factorization of electrical forces makes solving of equations of motion much simpler than repeatedly calculating the electrostatic force field at each position of the charge. This factorization occurs because of both the Principle of Superposition and the fact that the probe charge factors out due to the simplicity of Coulomb s Law. It is fortunate that we do not have to treat each atomic charge individually, superposition can be used to add the electric field from each individual atomic charge to get some net electric field which we can use without treating each of the 10 4 individual atomic charges individually. This is the same as summing the gravitational field over all atoms in an object, that is, superposition applies to the gravitational field which is the gravitational force per unit mass.. One may question the abstract concept of electric field that exists even when a probe charge is absent. You would not step out of an airplane at 10,000 without a parachute since you believe that a gravitational field exists around the earth. Similarly it is believed that an electric field can exist even in the absence of a probe charge. The electric field is real, for example, the transmission of electromagnetic waves through outer space shows us that the electric and magnetic fields are real concepts even in the high vacuum of outer space, that is why you see the light from distant stars. If you do not believe in the reality of the electric field, then go sailing on a lake with a metal masted boat during a lightning storm, it will be just a matter of time before you will be convinced. It is useful to combine the following; 1) concept of electric field ) superposition 3) Coulomb s law which leads to the following equation for the total electric field, at the point p, due to an ensemble of n point charges q : E = 1 X 4 cr (1.1) where the unit vector c points from i to p. This can be used to evaluate the electric field due to a distribution of point charges as will be discussed next lecture. Figure 4 charges. E fieldatapointpforndiscretepoint Superposition of electric forces implies superposition of electric fields. That is, the total electric field produced by n point charges, as given above, can be written as E 0 = X E 0 (1.13) where E 0 is the electic field due to a single charge Once the electric field is known,then at any position it is possible to compute the electrostatic force on any charge q. Knowing this force, and Newton s laws of motion, then it is possible to compute the resultant motion of charged objects. REPRESENTATION OF THE E FIELD Vector fields, such as the electric field, magnetic field and gravitational field have both a magnitude and direction. This is in contrast to a scalar field like temperature, gravitational potential, which only have a magnitude. It is convenient to express a vector field in terms of perpendicular components. For example, in cartesian coordinates the components are; E = b i + b j + b k (1.14) Note that the electric field is only a function of position, E( ) and not the probe charge. Of course the instantaneous location and strength of all the other charges is required to calculate E as will be discussed next lecture. Faraday found it convenient to visualize the abstract electric field in terms of lines of force. Occasionally it is convenient to represent the electric field by drawing field lines that indicate the direction of the electric field. The convention is to draw the direction of the field lines in the direction of the electrostatic force that a positive charge would experience at that point. That is, the electric field lines radiate from a positive charge and end at a negative charge. The number of lines per unit area, also called areal density, represents the magnitude of the electric field. This 4

5 SUMMARY The main points in this lecture are the following. The force between two point charges is given by Coulomb s law: Coulomb s Law: Figure 5 Cartesian components of electric field F 10 = cr 10 The Principle of Superposition states that the electrostatic forces add vectorially Principle of Superposition F 0 = F10 + F0 Because the electrostatic force depends only on position and time, obeys superposition, and is proportional to the probe charge, it is possible to factor out the probe charge from the force which allows defining the concept of the electric field E Electric Field E = F Figure 6 dipole. The electric field distribution for an electric The combination of Coulomb s Law, superposition and the concept of electric field gives that the electric field at a point P due to a system of charges is given by latter statement can be seen by considering a single positive point charge. The number of lines radiating from the charge is constant. However, since the surface area of a sphere equals 4 then the number of lines per unit area for any concentric sphere must fall off as 1 just like the radial dependence of the E field due to a point charge. This concept will be discussed further next lecture. The distribution of electric field can be demonstrated by observing the polarization of floating grass seeds in an electric field. The polarization cause the individual grass seeds to act like little magnets in a magnetic field and align themselves along the electric field lines. Fig. 6 shows the electric field distribution for an electric dipole, which has physically-separated identical-magnitude positive and negative charges. System of point charges: E = 1 4 X cr Superposition of electric forces implies superposition of electric fields. That is, the total electric field produced by n charges is given by; E 0 = X E 0 Once the electric fieldisknown,thenatanyposition it is possible to compute the electrostatic force on any charge q. Knowing this force and Newton s laws of motion, then it is possible to compute the resultant motion of charged objects. Evaluating electric fields is fundamental to electromagnetism. As will be discussed later, even light is a manifestation of oscillating electric and magnetic fields. The next two lectures will discuss calculation of electric fields including development of Gauss Law which supercedes Coulomb s law. 5

6 APPENDIX 1: Newton s Law of Gravitation It is useful to compare Newton s Law of Gravitation and Coulomb s Law because they obey the same mathematical relations and gravitation is more familiar to you. Newton realized that astronomical observations and Kepler s Laws of planetary motion all could be understood if he assumed that a gravitational force exists between any two objects, either earth-bound or heavenly bodies. Let 0 and 1 be two point masses separated by a distance 10,pointingfrom 1 to 0. In 1686 Newton postulated that the force exerted on 0 by 1, 10 is given by: 10 = c 10 (1.15) where c 10 = is a unit vector pointing from 1 to 0 and G is the Universal Gravitational Constant G = 6 67x Note that force points towards 1 that is, 10 is negative which corresponds to an attractive force. The gravitational force is minute between kg-sized bodies. For example, two 1 kg bodies 1 meter apart have a gravitational attraction of From Newton s third law of motion, we have that the force on 1 due to 0, 01, is equal and opposite to 10 i.e., 01 = 10 (1.16) The Principle of Superposition also applies to the gravitational force, that is, the gravitational forces acting on a given body due to several other bodies add vectorially like normal vectors. Thus one can calculate the gravitational attraction between macroscopic bodies by taking the vector sum over the gravitational attraction between all atoms involved. The concept of a gravitational field can be introduced because the ratio 0 0 is independent of the probe mass m 0, because of the form of Newton s Law of Gravitation. That is, the gravitational field, at some position, is the force in Newtons, that a 1 kg mass would experience at that position. 10 = 0 10 (1.17) From superposition we have that the gravitational fields due to an assemble of masses is the vector sum of the gravitational field due to each mass. That is, 0 = Σ 0 (1.18) The reality of the gravitational field is unlikely to be challenged. You believe that there is a gravitational field acting even if it is not manifest by observing a body fall. In P113 you solved problems of motion of bodies under gravitational attraction near the surface of the earth. It can be shown that the gravitational force due toasphereofmass, at or above the surface, is the same as if it is a point mass. Thus the gravitational field due to a spherical mass equals = b (1.19) where b is the radial unit vector. Note that combining the definition of the gravitational field and Newton s Laws of motion, gives that the force on mass 0, 0 ³ 0 = 0 = 0 b = 0 (1.0) That is, the gravitational field equals the gravitational acceleration at that location. At the earth s surface, this acceleration is a constant called That is; =981 (1.1) The negativce sign is because the force is attractive and thus the acceleration is in the negative direction, c 10, that is, towards the centre of the earth. Reading assignment: Giancoli, Chapter 1.1 to 1.8 6