Complement to Physics 259

Transcription

1 Complement to Physics 259 P. Marzlin 1 1 Institute for Quantum Information Science, University of Calgary, 2500 University Drive NW, Calgary, Alberta T2N 1N4, Canada I. INTRODUCTORY REMARKS The purpose of this document is to highlight the most important concepts introduced in Physics 259 and to provide additional information about mathematical techniques which are used but not properly introduced in the book. You do not need to memorize all equations, it is much more important to understand the underlying principles. In the final exam you will be provided with an equation sheet which will be the same as in Download the final exam of 2004 to see which equations you do not need to memorize. II. CHAPTER 21: CHARGES AND ELECTRIC FORCE The Coulomb force created by a charge q 1 (unit: 1 C) at position u 1, acting on charge q 2 at position u 2 is given by F = q 1q 2 1 ˆr (1) 4πε 0 r2 ˆr = r r (2) r = u 2 u 1 = distance vector (3) r = r = rx 2 + ry 2 + rz 2 (4) The distance vector always points from the source (here u 1 ) to the point where we observe the force (here u 2 ). This is guaranteed if one defines it as the difference between the position vectors of the two points, with the position vector of the source being subtracted from that of the point of observation. The distance vector is a very important concept that appears frequently for both electric and magnetic fields. Make sure to understand it very well. The Coulomb force created by q 2 acting on q 1 is simply F. This corresponds to r r = u 1 u 2. The Electric field (unit: N/C) created by charge q 1 observed at position u 2 is given by E = F q 2 (5) = q 1 1 ˆr 4πε 0 r2 (6) E points always away from positive and towards negative charges. The electric fields of several charges can be found by superposition of the fields created by the individual charges. Definition for a continuous charge distribution (charged line, surface, or volume): E = dq 1 ˆr(q) (7) 4πε 0 r(q) 2 How such integrals can generally be calculated is discussed in the appendix. Electric field lines indicate the direction of E at each point in space. The closer electric field lines are together, the stronger the electric field in that area. Electric field lines are always perpendicular to equipotential surfaces and indicate the direction in which the potential changes maximally. Since this direction is unique, field lines can never cross. III. CHAPTER 22: ELECTRIC FIELDS AND GAUSS S LAW The Electric flux is defined as Φ = E da (8) da = n da, where n is the normal vector of a small surface element 1 of size da. For a closed surface, da always points outwards. E da = E n da. This means that we integrate over the normal component E n = E n of the electric field on the surface. In most cases in Phys259, this normal component is constant. In this case the integration simplifies to Φ = E n da = E n where A is the total area of the surface. 1 See appendix for details on surface integrals. da = E n A (9)

2 2 Gauss s law states that for a closed surface Φ = E da = Q (10) ε 0 where Q is the total charge in the enclosed volume. Gauss s law allows you to infer the enclosed charge if you know the electric field. It generally does not allow to infer the magnitude and direction of E at any position. However, if one has a symmetry Gauss s law is a formidable tool to derive the electric field. Charged conductors are special materials. In a conductor many electrons can move almost freely. As a consequence they are redistributed by applied electric fields until their own electric field cancels the applied field: inside a conductor the electric field is zero and the electric potential is constant. For the same reason, the electric field is perpendicular to the surface of a conductor. If there is a cavity inside a conductor, the electric field is adapted so that the field in the conductor vanishes: if there is no charge inside the cavity, the electric field is zero in the cavity. if there is a charge Q inside the cavity, it induces charges on the boundary of the cavity (of total charge Q) so that the electric field does not penetrate into the conductor. Generally, conductors allow charges to move (almost) freely, but they block electric fields. Insulators prohibit charges to move, but electric fields can penetrate insulators. IV. CHAPTER 23: ELECTRIC POTENTIAL The electric potential difference (unit: 1V = 1 J/C) between two points is defined as V ab = b a dl E. (11) Here, the integration is along an arbitrary path between the two points 2. The path doesn t matter (as long as it connects the two points a and b) because of the conservative nature of Coulomb s force. V ab generally is the antiderivative of the electric field. Since the latter is 2 See appendix for details on path integrals. only determined up to a constant, the electric potential itself is also determined up to a constant. In most cases, this constant is chosen so that V 0 for r. The potential energy of a charge Q at point u is given by U( u) = QV ( u). If a charged particle of mass M moves through an electric field, its total energy is conserved: 1 2 M v2 + U( u) = const (12) where v is the velocity of the particle when it is at position u. Special cases of the potential: Point charge Q at a distance r: V = Q 1 4πε 0 r (13) Collection of point charges Q i at positions u i, observed at a position u: V ( u) = i with r i = u u i. Continuous charge distribution V ( u) = 1 dq 4πε 0 r Q i 1 (14) 4πε 0 r i (15) where r is the distance between the charge dq and the point u. The potential is continuous, i.e., it has no jumps. Equipotential surfaces are points at which the electric potential is equal. Inside a conductor the potential is constant, i.e., the interior of a conductor represents an equipotential surface. The electric field is the negative of the gradient of the electric potential: E = V = E x î + E y ĵ + E z ˆk (16) = V (x, y, z) x (17) = V (x, y, z) y (18) = V (x, y, z) z (19) E x E y E z The gradient gives the direction in which the change of the potential is largest. Its magnitude tells us how steep the potential is. The electric field always points

3 3 in the downhill direction of the potential (direction of the largest change of V ). In any direction n that is perpendicular to the gradient the potential does not change at all: we are then moving on a equipotential surface. This is also the reason why the electric field lines are perpendicular to the equipotential surfaces. V. CHAPTER 24: CAPACITORS AND DIELECTRICS The Electric field of an infinite charged sheet: if the sheet is made of a non-conductor, lies in the y z-plane at x = 0 and has surface charge density σ, the electric field is given by E = σ 2ε 0 sgn(x) î (20) Its magnitude is constant; for a positive charge (σ > 0) it always points away from the sheet, for σ < 0 it always points towards the sheet. For several parallel sheets, the total electric field at a given point is the superposition of the electric field of each sheet. Keep in mind that the direction of field created by the individual sheets only depends on whether you are on the left or on the right side of that sheet ( sgn(x) in the equation above). For a conducting charged plate the electric field is zero inside the conductor. The field is mirrored to the outside, so that it is twice as strong: E = { σ ε 0 î for x 0 0 for x < 0 (21) Between two conducting, oppositely charged plates the field is E = σ ε 0 î between the plates (22) It may look odd that it is the same as the field produced by a single conducting plate. The effect of the oppositely charged second plate is like that of an induced charge which guarantees that the electric field does not penetrate into the second plate. The potential between the plates grows linearly, V (x) = σ ε 0 x + const (23) A capacitor is simply a pair of oppositely charged, separated conductors. It serves as a storage device for electric energy. The capacity (unit: 1 F = 1 C/V) is defined as C = Q V (25) where Q is the charge on one of the conductors and V the potential difference. The capacity of two plates, each of area A, separated by a distance d is C = ε 0 A d (26) In capacitor circuits the total capacity C is calculated as C = C 1 + C 2 for two parallel capacitors (27) 1 C = 1 C C 2 for two capacitors in series (28) The wires which connect the capacitors can be considered as equipotential surfaces (if their resistance is neglected). No current can flow over a capacitor because there is no conducting connection between the two plates. However, if a potential difference is applied the plates get charged. The energy in a capacitor is stored in the electric field between the plates. It is given by U = 1 2 CV 2 = 1 2C Q2 = 1 QV (29) 2 The energy density (= energy /unit volume, unit: J/m 3 ) is given by u = 1 2 ε 0 E 2 (30) If a dielectric material is placed into a capacitor its capacity is changed according to C = KC 0 (31) where K 0 is the dielectric constant of the material. Alternatively, one can replace ε 0 by ε = Kε 0, all equations (including the energy and energy density) then have the same form as in a vacuum. The total potential difference between the plates is V = Ed (24) VI. CHAPTER 25: ELECTRIC CURRENTS AND RESISTORS where E is the magnitude of the electric field and d the distance between the plates. When a potential difference is applied to a conducting wire, the associated electric field accelerates the

4 4 free electrons inside. Because of collisions with the stationary atoms the electrons then move with an average velocity, the drift velocity v d. We then can define the rate at which the charges move through a unit area, the current density Not all circuits with resistors can be analyzed using these equations. An example where a resistor is neither in parallel nor in series with the other parts of a circuit is the following: J = nqv d = I A (32) where A is the area of the wire and the current I (unit: 1 A = 1 C/s) is the rate at which the charges flow through this area. A current can only flow in closed loops (which may include the ground) and has the same value everywhere in a wire as long as no junctions are crossed. It flows in the downhill direction of the electric potential. An electromotive force (emf) such as a battery provides a force that moves the charges uphill again. The plus-terminal of an emf corresponds to the higher potential. In many media, the current density is proportional to the applied electric field. The (inverse of the) proportionality factor is called resistivity (unit: Ω m) ρ, J = E ρ. (33) Since I = AJ and V = EL, where L is the length of the wire and V the potential difference between point a and b, we find the relation V ab = RI (34) with the resistance (unit: 1 Ω = 1 V/A) R = ρ L A. (35) A resistor is simply a piece of material with a given resistance. The potential difference V ab = RI represents the voltage drop over the resistor, which is caused by the loss of energy in the resistor due to the collisions. The power P (= work per unit time) which is delivered to or extracted from a circuit element is given by VII. P = V ab I = I 2 R = V 2 ab R CHAPTER 26: KIRCHHOFF S RULES AND R-C CIRCUITS (36) For two (or more) resistors in series or parallel one can calculate their combined resistance by R = R 1 + R 2 for two resistors in series (37) 1 R = for two capacitors in parallel(38) R 1 R 2 In this case one sometimes can exploit symmetries. For instance, if all the resistors in the example have the same resistance then the potential difference between the two ends of the central resistor is zero (since the voltage drop over the other resistors is the same for the upper and the lower branch). Thus, no current flows through the resistor and it does not contribute to the total resistance of the circuit. More generally one can analyze any circuit using Kirchhoff s rules: I = 0 (junction rule) (39) V = 0 (loop rule) (40) (41) Kirchhoff s rules form a set of linear equations that is sufficient to determine all quantities of interest (e.g., currents and emfs) for a given circuit. The junction rule states that the sum of all currents flow towards or away from a junction should be zero. To use it you first have to introduce a sign convention by drawing arrows for all currents in all branches of the circuit. I would start at the plus terminal of an emf and let the current flow away from it. The directions are arbitrary: if you get a negative result for a current, it simply flows in the direction opposite of the arrow you have introduced. For instance, the junction rule for the first junction in the figure is I 1 I 2 I 3 = 0 (42) since currents with arrows pointing towards a junction are counted positive and arrows with arrows pointing away from a junction are counted negative. The loop rule states that the sum of all potential changes over a closed loop must be zero (since the potential is uniquely defined for any point in the circuit). In applying it one has to be careful about the sign of the potential difference. If you go from the minus to the plus terminal of an emf E you go uphill, so one adds

5 5 a term +E in the loop rule. Since a resistor R corresponds to a loss of energy we add a term of RI if we go downhill, i.e., in the direction of the current s arrow. If we follow the loop in the opposite direction of the arrow, we add a term of +RI. Again it does not matter if the actual current is not flowing in the direction of the arrow since this will be taken into account in the final solution of the linear system of equations. The arrows only represent a convention which direction of the current flow is considered to be positive. For instance, in loop 1 of the figure the loop rule reads I 3 R 1 + I 4 R 3 + I 2 R 2 = 0. (43) If a capacitor is part of the circuit we talk about an R-C circuit. A positive charge will be accumulated at that terminal where the potential is higher. If you pass the capacitor in the direction of a current arrow then the change in potential should be Q/C. Since a capacitor breaks a circuit no current can flow through it. However, when the capacitor is not fully charged a current flows towards it and delivers the charges to charge it. This non-stationary process can also be described by Kirchhoff s rules which then provide a differential equation. For a simple R-C circuit with one resistor, one capacitor, and one emf the charging process is described by Q(t) = CE(1 e t/(rc) ) (44) I(t) = dq dt = E R e t/(rc) (45) RC is the time at which the current drops to the fraction 1/e of the initial current (when the capacitor is uncharged). VIII. CHAPTER 27: MAGNETIC FIELD AND MAGNETIC FORCES A magnetic field B (unit: 1 T = 1 N/(A m)) causes a force F = Q v B (46) to act on a particle of charge Q and velocity v. This force is nonzero only for moving charged particles and is perpendicular to both the magnetic field and the velocity of the particle. It therefore does not change the magnitude of the velocity. the cross product v B has a magnitude of vb sin α where α is the angle between v and B. The direction of the cross product can be found using the left-hand rule: If your thumb corresponds to v and your middle finger to B then your index points in the direction of the cross product. In free space a particle in a homogeneous magnetic field moves on a circle of radius R = mv/(qb). In a conductor this is different. Since the atoms are tightly bound to their place in the wire, an applied magnetic field can only affect the electrons and moves them to a different position. This leads to a charge separation which in turn gives rise to an electric field. In equilibrium the total force F = Q v B + Q E (47) vanishes so that the induced electric field is perpendicular to both the current and the magnetic field and has a magnitude of E = vb. This is the Hall effect. Magnetic field lines show the direction of the magnetic field. The thinner they are distributed, the weaker is the magnetic field in the corresponding region. Magnetic fields point out of the North pole of a magnet and into the South pole. For the Earth, the geographic North pole is close to a magnetic South pole so that the North pole of any compass points towards the magnetic South pole (geographic North) of the Earth. The reason is that unlike poles attract each other and like poles repel each other. The magnetic flux (unit: 1 Wb = 1 N m/a) through a surface is an important quantity for induction processes and given by Φ B = B da (48) Gauss s law for magnetic fields states that the total flux through a closed surface is always zero, B da = 0. (49) The magnetic force on a conducting wire can be derived from that on the current-carrying particles. For a straight wire of length L pointing in the direction of a unit vector ˆn one defines a length vector l = Lˆn. The force on the wire is then given by F = I l B. (50) The force on a curved wire can be calculated by dividing the wire into small straight segments, df = I dl B. (51) The total force on the wire is then given by the line integral F = I dl B. (52)

6 6 The torque τ on a closed wire loop in a plane can be expressed by its magnetic dipole moment µ, τ = µ B (53) µ = IAˆn (54) where A is the area enclosed by the loop and ˆn the normal vector of this area. The corresponding potential energy is µ B. where I encl is the total current enclosed by the path. Ampere s law is particularly useful when symmetries can be exploited. For instance, for a long solenoid the path could be chosen to start at the centre of the solenoid and go radially outward (see Fig ). A circular path would in this case not help since it would not enclose any current: the closed path should always be chosen such that the enclosed currents are aligned with the normal vector of the enclosed area. IX. CHAPTER 28: SOURCES OF MAGNETIC FIELDS X. CHAPTER 29: ELECTROMAGNETIC INDUCTION Magnetic fields are generally created by moving charges. The magnetic field of a moving point charge is given by B = µ 0 Q v ˆr 4π r 2 (55) where the notation is the same as in Eq. (1). This means that the corresponding field lines are circles whose normal vector points in the direction of the velocity v of the point charge. The magnitude of the field drops like 1/r 2 with increasing distance r from the point charge. The law of Biot and Savart, B = µ 0 Idl ˆr 4π r 2, (56) allows to compute the magnetic field of a line current. The magnetic field for some important special cases is B = µ 0 ni (solenoid with n loops/unit length)(57) B = µ 0I (long straight wire) (58) 2πr B = µ 0NI 2a B = µ 0NI 2πr (N circular loops of radius a) (59) (toroidal solenoid) (60) The direction of the magnetic field can be found using the right-hand rule: If your right thumb points in the direction of a straight current segment then the fingers coincide with the magnetic field lines of the corresponding magnetic field. Apere s law allows to calculate the magnetic field created by a current distribution. The line integral of the magnetic field over any closed path fulfills B dl = µ 0 I encl (61) Induction is the phenomenon that a change of the magnetic field flowing through a current loop, or a change in the geometry of the loop, induce an electromotive force E in the loop. Mathematically it is described by Faraday s law, E = dφ B dt, (62) where Φ B is the magnetic flux through the area enclosed by the loop. If there are N windings of the loop E has to multiplied by N. The direction of the induced emf can be determined by Lenz s law: the direction of any induction effect is such as to oppose the cause of the effect. With this rule in mind it is a good idea to review Fig For instance, if the magnetic field flowing through a conducting coil is increased the induced emf E causes a current to flow in the coil. This current in turn produces a magnetic field that (partially) cancels the original change in the magnetic field. Since a current flows from the plus terminal to the minus terminal of an emf this determines the direction of E. An induction effect can also simply be achieved by moving a conductor through a magnetic field. the motional electromotive force can be expressed as E = ( v B) dl. (63) This is equivalent to Eq. (62) if the magnetic field is time independent. Note that the integral is taken over the closed loop and that the resulting expression is zero if the magnetic field and/or the velocity do not change on the loop. If the loop moves with a constant velocity and enters into an area with a constant orthogonal magnetic field this equation simplifies to E = BdA/dt where A(t) is the area of that part of the loop which is already inside the magnetic field.

7 7 XI. CHAPTER 30: INDUCTANCE A change in the current of one loop causes a change in the magnetic field created by this current and can induce an emf in another loop. This phenomenon is called mutual inductivity. Since the magnetic field is proportional to the current I one finds E 1 = M di 2 dt E 2 = M di 1 dt M = N 2Φ B2 I 1 (64) (65) = N 1Φ B1 I 2 (66) where M is the mutual inductance (unit: 1 H = 1 J/A 2 ) and φ B2 the magnetic flux through loop 2 caused by the magnetic field created by a current in loop 1 (see, for instance, example 30.1). A current in a single loop can also induce an emf in the same loop. The corresponding self inductance L and the induced emf fulfill E = L di dt L = NΦ B I (67). (68) As a circuit element, an inductor causes a potential drop of LdI/dt. This means, if we apply Kirchhoff s loop rule and follow the loop in the direction of the current arrow, we add a term LdI/dt. The sign is the same as that for a resistor, but the inductor only leads to a potential difference if the current is changing. An inductor allows to store an amount of energy of U = 1 2 LI2 (69) in the magnetic field of the inductor. The corresponding energy density is given by u = 1 2µ 0 B 2. (70) A second application of an inductor is to soften the change of a current. In a simple R-L circuit with one battery of strength E 0, a resistor R and an inductor L where the current is initially zero one finds for the change in the current I(t) = E 0 R (1 e (R/L)t ). (71) XII. OTHER REMARKS Try to understand the examples in the book. In Physics, learning by doing is essential. Try to solve as many problems as possible. On the Phys259 web page you may find recommended problems for each chapter (click on Text Probs). You may need some elementary concepts of mechanics. In particular, for a particle that is subject to a constant acceleration a for a time t, velocity of accelerated object: v = at (72) corresponding traveled distance: s = 1 2 at2 (73) XIII. kinetic energy of an object: T = 1 2 mv2 (74) wrk agnst force over distnce s: W = F s (75) corresponding power: P = F d s dt (76) = F v (77) APPENDIX: INTEGRATION OVER LINES, SURFACES, AND VOLUMES A. Line integrals For the definition of the electric potential and for Ampere s law we use integrals of the form E dl (78) P where P denotes some path in space. A proper definition of dl can be given if the path is described by a parametrized curve l(φ), where φ is a real parameter. In Phys 259 you will only encounter the case of a straight or a circular path. The case of a straight path is easier to discuss without using integrals. We will therefore present the general formalism using the example of a circular path of radius R in the x-y-plane. This path can be described by l(φ) = R cos(φ) î + R sin(φ)ĵ. (79) A semi-circle then would start at φ = 0, i.e., l = Rî, and end at φ = π, i.e., l = Rî. The line integral can then be expressed as an ordinary integral over the variable φ: the integration element dl is simply given by dl dl =. (80)

8 8 This vector is nothing but the difference vector between two nearby points on the curve: dl dl = (81) 1 ( l(φ + ) l(φ)) (82) = l(φ + ) l(φ) (83) dl The vector is tangent to the path P. Its length, dl = dl, (84) corresponds to the length of the line segment dl. The dl unit vector /(dl) is the tangent vector to the curve. In our example we have dl = R sin(φ)î + R cos(φ)ĵ (85) dl = ( R sin(φ)î + R cos(φ)ĵ) (86) dl = ( R sin(φ)) 2 + (R cos(φ)) 2 (87) = R. (88) The integral is now very easy to perform. Suppose we want to integrate the field E = E 0 î over the semicircle. We only need to insert all expressions to obtain P E dl = = = π 0 π 0 π 0 E dl (89) E 0 î ( R sin(φ)î + R cos(φ)ĵ) (90) E 0 ( R sin(φ)) (91) π = RE 0 sin(φ) (92) 0 = RE 0 [cos(φ)] π 0 (93) = 2RE 0 (94) We encountered another type of integral when we had to calculate the electric field produced by a line charge density λ. Since the latter is not a vector the integration is not over dl but over dl. For instance, the total charge on a line could be calculated as Q = λdl (95) where dl can generally be expressed as given above. For constant λ the integral dl simply gives the P length of the curve. P B. Surface integrals Surface integrals can be calculated in a similar way as line integrals. While in Phys 259 you probably will not need the approach presented here it may nevertheless give you a better idea of how they are generally calculated. Let s consider an integral like that of the electric or magnetic flux, B da (96) where da is the area of a small surface element times the normal vector on the surface. To calculate these quantities we again start with a parametrization A(θ, φ) of the surface. We need two parameters because a surface is two-dimensional. An example is the surface of a sphere of radius R which can be described by A(θ, φ) = R{cos(φ) sin(θ)î+sin(φ) sin(θ)ĵ+cos(θ)ˆk} (97) For each value of the parameters this describes a point on the sphere. For instance, the North pole is given by A(0, 0) = Rˆk. We can describe any point on the sphere by values φ [0, 2π] and θ [0, π]. For a fixed value of θ, the vector da θ = d A dθ (98) dθ describes how much A is changed if we change the angle θ by a small amount dθ. Likewise, da φ = d A (99) describes how much A is changed if we change the angle φ by a small amount. These two vectors form a small parallelogram whose area corresponds to the surface element da. If the angle between the two vectors is α the area is da = da θ da φ sin(α) (100) = da θ da φ (101) since the magnitude of the cross product is proportional to sin(α). In addition, the cross product is orthogonal to the surface since da φ and da θ are tangent. We therefore find that the surface element is given by da = d A dθ d A dθ. (102)

9 9 In our example we have da = R{cos(φ) cos(θ)î + sin(φ) cos(θ)ĵ sin(θ)ˆk} (103) dθ da = R{ sin(φ) sin(θ)î + cos(φ) sin(θ)ĵ} (104) da = R 2 sin(θ){cos(φ) sin(θ)î (105) + sin(φ) sin(θ)ĵ + cos(θ)ˆk} dθ (106) da = R 2 sin(θ) dθ (107) It may be helpful to remark that if we integrate of a surface charge density it is da and not da that has to be used. Again one uses a parametrization of each point inside the volume, A(R, θ, φ), this time with three parameters. An example is given in Eq. (97) above. We can again introduce tangent vectors associated with a small change of each of the three parameters. The volume element can then be expressed as da dv = dr ( da dθ d A ) dr dθ (108) dθ C. Volume integrals This concept is hardly used in the book and you will not need it in Phys 259. I therefore will be very brief. In the example one finds dv = R 2 sin(θ) dr dθ.