Electric and Magnetic Fields: Concepts, Equations, and Applications - Prof. Murray John Ho, Study notes of Physics

Download Electric and Magnetic Fields: Concepts, Equations, and Applications - Prof. Murray John Ho and more Study notes Physics in PDF only on Docsity! Physics 1120 Review What follows is a rough outline of physics 1120 material, it is by no means complete but is meant to highlight basic ideas and topics. It is certainly not meant to be used as a “crib sheet” for your exam. As usual the equations by themselves are meaningless so please read the text of this document to aid in understanding their meaning. Where vector arrows are omitted for vector quantities it denotes the amplitude only. The reader must determine the direction. 1. Electrostatics A. Electric Fields and Forces The electric field is a vector field which obeys the laws of superposition, i.e. each contributing charge or charge distribution can be considered separately and then the vector sum of all the contributions can be taken to give the total field. Electric field lines are a tool for visualizing a field. The strength of the field is proportional to the density of field lines. Lines originate at positive charges and terminate on negative charges. Electric Field of a point charge: !E = kq r2 r̂ Force on a charged particle due to an electric field: !Fe = q !E Force between two charged particles (combine last two eqns.): !Fe = ±k |q1||q2| r2 r̂ In the last equation the plus or minus sign is chosen so that like charges repel and opposite charges attract. A couple of examples of electric fields for specific charge distributions follow: Electric field for a single infinite sheet of charge: E = " 2#o Electric field of two infinite planes of opposite charge (between the plates): E = " #o This last result can be seen from the superposition of two infinite plate fields, where each plate has an opposite charge (but equal magnitudes). Of course the field outside of two oppositely charged parallel plates is zero (why?). All of the expressions for electric fields that you will see can be derived from Gauss’s Law (see below). Electric flux is defined to be !e = ! !E · d !A where since we are integrating over an area the double integral is implied. Flux is interpreted as the amount of field lines penetrating a surface. If the surface is closed (ie a sphere) then the outward normal is traditionally taken as the positive. For a non-closed surface the choice for the direction of the area vector lies with you. For a uniform electric field and a flat surface the above equation reduces to !e = !E · !A = EA cos ($) (uniform field and flat area only) Here $ is the angle between the area vector and the electric field. To find the electric field for a charge distribution with high symmetry a useful tool is Gauss’s law: !e = " !E · d !A = qenc #o Here qenc is of course the amount of charge enclosed by our (closed) imaginary Gaussian surface. The most common Gaussian surfaces are spheres, cylinders, and boxes. Without high symmetry Gauss’s law is essentially useless. It is only useful in cases where we can choose a surface that is perpendicular to the E 1 field everywhere, and everywhere along that surface the field has the same magnitude. Then we can back the electric field out of the integral, leaving a simple integral over dA which just gives the area of our gaussian surface (know your area formulas). It is then a simple matter to solve for the electric field. Conductors in electrostatic equilibrium have a few properties: • The electric field inside a conductor is zero. (make sure to know why) • On the surface of a conductor the electric field points perpendicular to the surface (or parallel to the normal). This is due to the fact that the entire conductor is an equipotential (see below). B. Electric potential and Electric potential energy Since the electric field is a vector quantity, it is mathematically a pain in the ass to deal with at times. A very useful concept which is both convenient and o"ers a bridge into the world of electronics is electric potential field. As with gravitational potential (with which you are hopefully more familiar) the zero point of the potential can be chosen anywhere we want as long as we are consistent after that choice. Since this means that the absolute value of the potential has no meaning, the only thing that is meaningful are changes in potential. In terms of the electric field, change in electric potential is defined to be #V = ! ! !E · d!l This is a path integral, the dot product serves to pick out components of our path that are along the electric field (through the cosine dependence). Thus if we move perpendicularly to the field we move along an equipotential line. Instead of drawing field lines, we can draw equipotential lines (or surfaces if you are a talented artist) to represent a field. By the preceding argument, field lines are always perpendicular to equipotential surfaces. In the gravity analogy, the gravitational force acts downward, and equipotential lines/surfaces are then planes of constant height. If we are in a uniform field, like in a parallel plate capacitor, and choose a path along the direction of the field, the above integral reduces to #V = !E#l For non-infinite charge distributions we normally take the zero point for the potential to be at infinity. With this convention we can write Electric potential due to a point charge: V = kq r Notice that potential can be either positive or negative depending on the sign of the charge. Since we integrate the field, the electric potential for a point charges falls o" like 1/r in contrast to the electric field that falls o" like 1/r2. Just as an object with mass in a gravitational field can have gravitational potential energy, a object with charge can have electric potential energy (EPE). EPE = qV for a point charge This is the potential energy associated with a charged particle (q) being placed in a electric potential field created by other charge(s). Moving a charged particle in an electric field can change its potential. The work required and the corresponding change in potential are related by the following: #V = ! Workelec q = Workext q = #EPE q The subscripts “elec” and “ext” represent the works done by the field itself and an external agent respectively. The minus sign is important, convince yourself why. Notice that if we move along an equipotential, #EPE is zero and so no work is done by either the field or the external agent. Recall that Work = !F · !d, the dot product of the force and the displacement. Another useful result is the work-kinetic energy theorem: Worknet = #KE 2 When the magnetic flux through a surface changes, an electromotive force (EMF), also known as a potential di"erence, is created that attempts to create a current to maintain the previous flux. Mathematically this is Faraday’s Law # = # # # d!m dt # # # Which says that a changing magnetic flux produces an emf. Don’t confuse the emf symbol # with the universal constant #o. The direction of the resulting current 1 is such that the change in flux is minimized. Faraday’s law is used in all kinds of applications, electric motors, transformers, generators, metal detectors, etc. D. Inductors Inductors are devices that store energy in a magnetic field (similar to the capacitor that stores energy in the electric field). A solenoid (discussed above and in your text) acts as an inductor. Inductance is defined to be L = !m I It turns out that The inductance of a solenoid: Lsolenoid = µoN2A l Notice that the inductance of a solenoid is only a function of its geometry (N-number of turns, A-cross sectional area, l-length). An inductor is essentially a series of many conducting loops, so using Faraday’s law we know that if we try to change the current through an inductor it will fight that change with an induced potential di"erence. This is the potential di"erence across an inductor in exactly this situation: #VL = !L dI dt Note that if we change the current through an inductor very rapidly, for example by closing or opening a switch, the voltage across the inductor can be huge. Inductors have many applications listed above for Faraday’s law as well as many circuit applications. The energy stored in an inductor is given by UL = 1 2 LI2 3. Circuits A. Components and Analysis We’ve already discussed two circuit components, inductors and capacitors. Other basic components include batteries and resistors. A battery’s entire purpose is to maintain a constant potential di!erence between its terminals. Resistors are exactly that, parts of the circuit that resist current flow by dissipating energy. Light bulbs act as resistors, the power that they dissipate is in the form of light and heat. What causes current to flow in a wire or circuit component is an electric field that pushes the charges along. Batteries (as well as changing magnetic flux) produce a potential di"erence which is related to an electric field that exerts forces on the charges. The current density J is related to this electric field strength by J = "E where " is called the conductivity of the material. The inverse of " is ' = 1/" and is called the resistivity. Technically speaking the relation involving J is ohm’s law. You probably call the following ohm’s law: #V = IR 1A current from the potential difference will ensue only if there is a conductor at the location in question. However, the emf is generated regardless of whether there is anything there. 5 which only holds for ohmic materials obeying J = "E . Broken wires do not obey ohm’s law, neither do capacitors. In the above relation causality is di$cult to see. It is the potential di"erence across something that creates the current through it. The resistance of an object (R) is related to its resistivity. For a wire the resistance is given by R = 'L A where A is the cross sectional area and L is the length. So if we double a wire’s length we double its resistance, if we double its radius the resistance goes down by a factor of 4. Commonly in circuits the wires are taken to be ideal, that is they have no resistance so you need not worry about it. This assumption is valid if the resistance of your circuit components is much larger than the wires used which is almost always true. Circuit analysis uses the following rules: • #Vloop = $ i (#V )i = 0 • $ Iin = $ Iout The first says that if you go around a complete loop in a circuit and add up the voltage drops/rises across each component as you cross it you will get zero. You must end where you began. The signs in the equation can be tricky, look at examples in you text etc. The second is the junction rule and simply conveys that all the current into a junction leaves that junction. This says that current is conserved, it is not “used up” by any circuit component. Of course this is not true for potential di"erence which is in a sense “used up” as you progress along the circuit. These two rules let you solve (via a system of linear equations) for the currents in any loop that you choose, which using ohm’s law (V = IR) will give you the potential drop across any circuit component of interest. I mentioned that resistors dissipate power. The amount of power dissipated by a resistor or light bulb with resistance R is P = I#V = I2R = (#V )2 R where #V is the potential drop across the resistor/bulb and I is the current. The numerous expressions are attained from the first using ohm’s law. The expression that is most convenient to use is determined by what you know. There are a few more rules for circuits that are of use, review series and parallel if needed: • The current through any components in series is the same. • The potential di"erence across any components in parallel is the same. B. Component addition We can combine a set of many resistors and replace it by an equivalent resistor. The way they combine di"ers depending on whether they are in parallel or series. RT = R1 + R2 (2 resistors in series) 1 RT = 1 R1 + 1 R2 (2 resistors in parallel) Inductors add just like resistors do. Capacitors are just the opposite: CT = C1 + C2 (2 capacitors in parallel) 1 CT = 1 C1 + 1 C2 (2 capacitors in series) 6 It is also worthy of note that the charge on any two capacitors in series is the same. Also, an uncharged capacitor behaves just like an ideal wire. A fully charged one behaves like an open circuit, where no current can flow. Many complicated circuits can be simplified via these equations. When determining equivalent resistance or capacitance, I suggest you start with a small part of the circuit, draw a new picture at each step, and be careful to only combine components that are strictly in series or parallel. C. Common circuits The RC circuit is a simple circuit that contains a switch, a resistor and a charged capacitor all in series. Zero time is usually defined to be the moment that you close the switch. The RC circuit has some interesting features, the charge on the capacitor changes exponentially with time: Q = Qoe !t/! where & = RC & is called the time constant of the circuit and controls how fast the capacitor will charge or discharge. The current in the circuit behaves similarly: I = Ioe !t/! Notice after a long time (relative the the time constant) the current is essentially zero, this agrees with the fact that a fully charged capacitor acts like an open switch. The RL circuit The current in an LR circuit behaves just like the current in the RC circuit but with a di"erent time constant (how can it be the same, there isn’t any C anymore)... I = Ioe !t/! where & = L R The situation for the LR circuit corresponding to this equation is quite di"erent than the RC circuit, see pg . 1073 in your text. 4. Maxwell’s Equations and solutions All of classical electricity and magnetism can be described using 5 equations. The first is the Lorentz force which combines the forces on a charged particle from electric and magnetic fields: !F = q( !E + !v " !B) (1) The other 4 equations that are needed are the Maxwell’s equations, which combine all the information discussed thusfar... " !E · d !A = Qenc #o Gauss’s Law (2) " !B · d !A = 0 No magnetic monopoles (3) " !E · d!s = ! d!m dt Faraday’s law (4) " !B · d!l = µoIenc + #oµo d!E dt Ampere-Maxwell law (5) These are Maxwell’s equations in integral form, there also exist di"erential forms. The last equation (5) conveys that there are two sources of magnetic fields, changing electric flux and currents. As you have hopefully seen, solutions to Maxwell’s equations are electromagnetic waves that travel at the speed of light! 7