ECE 106 Winter 2013 Course Notes

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University of Waterloo
Electrical and Computer Engineering
ECE 106
Michael Balogh

ECE 106 - Electricity and Magnetism Kevin Carruthers Winter 2013 Structure of an Atom An atom has a nucleus comprised of protons (p+) and neutrons (n). It is surrounded by an ▯19 electron (e-) cloud. Each has a charge of ▯1:6 ▯ 1C (note that a single Coloumb would be fucking huge.). Any atom with an equal amount of electrons and protons is electrically neutral. If one or more electrons is removed, the atom is positively charged. Extra electron(s) make the atom negatively charged. In a conductor, the electrons’ bond to the nucleus is very weak. This means the electrons are free to move, and thus we say the conductor has a low internal resistance. By bringing a charged object nearby, we can induce a charge into such a medium. In an insulator, the electrons are not free to move; however, the atoms can be slightly "reshaped" internally. An insulator has a high internal resistance. We can also induce charge in inductors: this is known as polarizability. Electromagnetic Force Electromagnetic force is the second strongest of the four fundamental forces, weaker only than the strong force, which has extremely short range. The electromagnetic force has in▯nite range, as it is inversely proportional to the distance between interacting charges; it may grow extremely weak, but will never lose all e▯ect. It can also both attract and repel, depending on the relative di▯erences (from zero) of interacting charges. Electric Charge Electric charge is the fundamental unit upon which electromagnetism acts, much like mass/energy is for gravity. Charge is discrete in nature, which means all values of charge are some inte- 1 ger multiple of the charge of an electron: electromagnetism’s fundamental unit. Note that charge can thus be negative or positive, and that opposite charges exert an attractive force. Conversely, like charges repel. Charge is conserved - it can neither be created nor destroyed, and thus in any isolated (closed) system, the total charge can never change. Electric Field The electric ▯eld E is de▯ned as the force created by a charge, divided by the charge of the interacting particle kq~ E = 2d d (I’ll de▯ne k soon). Note that positive charges will create a ▯eld pointing away from them, ~ and vice versa, as evidenced by the d. In the future, I will not show this direction vector unless necessary for clarity, but it is present in virtually all calculations. For any charge q in an electric ▯eld, we can calculate the electric force acting on it by F = qE This gives us the general equation for electric force F = kq0 1 d2 Note the similarity to the equation for gravitational force. Vector Diagrams of the Electric Field When sketching ▯elds, the direction of arrows shows the direction of the ▯eld, and the density of arrows shows the relative strength of the ▯eld (qualitatively). Since ▯elds are vectors, we can then add multiple ▯elds through vectorial addition (and/or superposition). Coloumb’s Law The force acting between two charges is equal to q q k 0 1 d 2 where k is Coloumb’s constant and k = 1 = 8:99▯10 . " 0 8:854▯10 ▯12 is the permitivity 4▯0 of free space, which does indicate that the value of "0(and thus k) will change depending on the medium through which the ▯eld travels. 2 Dipoles A dipole is a area where two equal but opposite charge are in close proximity (separated by d, where d is very small). When a dipole exists, the approximate electric ▯eld is 1 E / rn where n < 2. We can also do vector addition with the following d d r1+ 2 = r2+ 2 = r thus giving us the simpli▯cation for the electric ▯eld of a dipole system E = kqd r2 Dipole Moment The dipole moment is written as ~ = q ▯ d which is the charge multiplied by its displacement from the other dipole. Note that this simpli▯es our earlier dipole equation to kp~ E = r2 Charge Distribution Charge Density The charge density rho is equal to Q ▯ = V This gives k▯ dE = r2 dV and Z E = k▯ ▯ dV r2 Note that this is a vectorial integration, as r is changing in direction and size. This can only be easily done in symmetrical cases. 3 For a charge distributed evenly over a disk of radius r ▯ = Q ▯r2 and dQ = ▯2▯r dr The disk would create an electric ▯eld such that k dQ dE = 2 2 3 4▯"(r + z ) 2 and Z r d▯ 2▯r z E = 2 2 2 ▯ dr 0 4▯"(r + z ) which is ▯ ▯ 2▯▯z 1 1 ▯z E = ▯ p = p 4▯" 2 r + z2 4" r + z 2 Superposition Superposition is basically a simple concept: in electromagnetism, forces add. In a given system with N charges, the force extered on one of them is XN kq0 1 F = r2 i=1 Given charge density, we can integrate this sum to ▯nd N X kq▯▯V F = 2 i=1 r given N chunks of volume ▯V and charge ▯▯V . Technically, this formula is an approxima- tion, but we can use integrals to determine the exact answer with Z F = kq▯ ▯ dV r2 V Uniformly Distributed Charge For any point on an object with uniformly distributed charge, we can give it a width of dy thus giving us dE = k dQ r1 k▯ dy = r + y 2 4 So we have a general equation of the form Z L E = ▯dE 0 where L is the length of the object. Note that this is a vectorial integral: as such, the direction of the ▯eld will always be perpendicular to the object (ie symmetrical along the horizontal axis). Because this integral is symmetrical, we can integrate only along the axis, thus giving us Z 2 E = L▯dE ▯2 Z L 2 k▯r = 3 ▯ dy ▯2 r1 Z L 2 k▯r = 3 ▯ dy ▯L (r + y )2 2 k▯L = q 2 L2 r r + 4 1 Thus as L ! 1, E / r For charges uniformly distriuted on a ring, we have k▯ dL dE = 0 2 (r ) or for any one direction k▯ dL z dE = 0 2 0 2 (r ) (r ) where z is the direction along it’s z-axis, so k▯z Z E = ▯dL (r )2 Z where ▯dL is the circumferance. Thus gives us the general form of this equation ▯rz E z 3 2"(r + z ) 2 5 Electric Fields for In▯nite Plains A plane can be written as a disk with an in▯nite radius. Then 1 p ! 0 r + z2 so E = ▯ 2" Note that this has no relation to z, and is constant with distance. Electric Flux Electric ux is the ow of some vectorial quantity (ie charge) through a given area. We can also think of it as (qualitatively) the number of ▯eld lines crossing a given area. It is given as the overlap between the amount of ow and the given area, thus giving us d▯E= E dA where dA is the area of an open object. This gives us Z ▯ = E ▯ dA E or as a non-vectorial implementation ▯E= EAcos▯ where ▯ is the angle between the perpendicular vector to the object and the direction of the ▯eld lines. By convention, we consider ux entering an object to be negative and ux leaving an object to be positive. This gives us an essential property: that for any closed surface we have ▯E= 0, since as much ux enters the object as it does leave. Gauss’ Law If we have a closed surface with an electric ▯eld pointing in only one direction (example: a sphere of radius r with a point charge q enclosed in it’s center), this gives us Z Z kq kq r 1 ▯ E E ▯ dA = ▯ d▯r = = q r2 r "0 "0 6 If the surface is not spherical, we have the same equation: we can create a spherical region around the charge, segregate this area, and perform superposition to ▯nd our answer (k4▯q+ 0 = k4▯q). Note that for more than one charge this gives us N 1 X ▯ E qi "0 i=1 This is Gauss’ Law: that the electrix ux through a closed surface S is equal to the total charge contained inside S (divided by the permitivity of free space). Mathematically, we have qwithin ▯ E "0 Note that we may refer to any closed surface as a gaussian surface, since this law holds true in all such cases. Using Gauss’ Law Gauss’ Law is indescribable useful in situations involving symmetry. The simplest case is in that of spherical symmetry. Example: suppose we have a spherical distribution of charge with density ▯(r) (ie the density is a function of the radius). Enclose this distribution within a spherical surface S of radius r. Gauss’ Law gives q ▯ E within "0 Since the ▯eld is spherically symmetric, E must point radially and be proportional to r. It can only depend on r, which means it is constant over any spherical surface. Thus we have the integral 1 2 ▯ E r E(r) "0 The charge within S is given by integrating the charge density over the volume of S Z q = ▯(r) ▯ dV V so Z ▯(r) ▯ dV E(r) = V = q(r) "0r2 "0r2 Note that outside of the sphere there is no charge (▯ is constant), so we have E(d) = ▯d 3" 0 3 ▯r = 3" d 2 0 7 Or in other words: the electric ▯eld grows linearly with d inside the sphere, but falls o▯ inversely proportional to the square distance outside of the sphere (ie when d > r). Thus the external ▯eld is exactly that of a point charge 3 q = ▯r 3"0 Energy Work Work is a measure of the force exerted to move a charge from one location to another. It is given by Z W = F ▯ dr where F is the opposite of the force exerted upon that charge. This gives Z W(r !0r ) 1 F ▯ dr Z r0 kq0 1 = ▯ r2 ▯ dr r1 kq0 1 kq 0 1 = ▯ r0 r1 An important fact to note is that the path travelled does not make a di▯erence to the amount of work done; the only things which matter are the start and end positions. Field Energy Field energy can be equated with pressure: it is the measure of force caused by an object’s electric ▯eld per it’s area. It is calculated by multiplying the electric ▯eld of a "hole" within the object by the charge density of the disk formed by creating this hole. This gives us P = (E obj▯ E disk▯ We can use this quantity to help us calculate work done with dW = F dr = PA dr In essence, this gives us "We had to put dW = (energy density) dV amount of work into the system to create that ▯eld." 8 Potential Di▯erence Potential di▯erence is the measure of work which would have to be done to move a unit of charge from one location to another per unit of charge. It is given by Z b ▯ba = ▯ E ▯ ds a which can be derived by solving for the amount of work, then dividing out the charge (since work done is proportional to the charge). It can also be repesented by a V . Potential The electric potential of an object is its potential di▯erence with respect to some ▯xed point. Though this point is generally in▯nity, that is not always the case. When it is we have Z r ▯(r) = ▯ E ▯ ds 1 We can also rede▯ne potential di▯erence in terms of potential (yes, this will be as painfully obvious as it sounds). Given a and b, the potential di▯erence between them is equal to ▯ba= ▯(b) ▯ ▯(a) If we suppose our ▯eld is created from a point charge q at the origin we have Z rkq kq ▯(r) = ▯ ▯ dr = 1 r2 r which gives us a formula we can use to generalize this. Given superposition, we have the potential at some point P from N contributing charges as X ▯(P) = kq i ri i=1 this can then be further generalized to account for continuous distributions with Z k▯ ▯(P) = ▯ dV V r Obviously, these equations are only useful if we can set the reference point to in▯nity. This, in turn, is only possible if the charge distribution is of a ▯nite size. If the distribution is in▯nitely large, we will ▯nd the equation to diverge as the reference point approaches in▯nity. In such a case, simply pick a di▯erent reference point. 9 Charged Conductors Since we can also express work as W = ▯q(▯(b) ▯ ▯(a)) we can see that no work is required to move a charge between two points at the same potential. We also know that the electric potential is constant everywhere on the surface of a charged conductor in equilibrium, and the the electric potential is constant everywhere inside a conductor and is equal to the value at its surface. Any surface on which all points are at the same potential is called an equipotential surface. The potential di▯erence of any two points on said surface is zero, no work is required to move charge at a constant speed on the surface, and the electric ▯eld is always perpendicular to this surface. Electrostatic Equilibrium A conductor is in electrostatic equilibrium when no net motion of charge occurs within it. If this is true, then we also know the following ▯ The electric ▯eld is zero everywhere inside the conductor. ▯ Any excess charge on an isolated conductor resides entirely on its surface. ▯ The electric ▯eld outside a charged conductor is perpendicular to its surface. ▯ For any non-spherical (ie misshapen) conductor, charge accumulates on its "sharpest" points. Electric Current Current is de▯ned as the rate at which charge ows through a surface. Mathematically, we have q I = t which is measured in amperes A = C . s The direction of current is de▯ned to be the direction at which a positive charge would ow (note that this is opposite the direction of electron movement!). We refer to moving charge through a conductor as a mobile charge carrier. Electrons ow in the opposite direction of an electric ▯eld, as given by the "opposites at- tract" rule-of-thumb. As a charged p
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