Physics (2) Dr. Yazid Delenda

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1 PHYS 104 Physics (2) Dr. Yazid Delenda Department of Physics, Faculty of Sciences and Arts at Yanbu, Taibah University - Yanbu Branch, KSA. yazid.delenda@yahoo.com Lecture notes available at Contents 1 Electric fields 2 2 Gauss s law 5 3 Electric potentials 7 4 Capacitance and dielectrics 10 5 Current and resistance 11 6 Direct-current circuits 12 7 Magnetic fields 14 1

2 8 Sources of magnetic fields 16 9 Faraday s law Inductance Alternating-current circuits Electromagnetic waves 25 1 Electric fields Electric Charge: The fundamental entity in electrostatics is the electric charge. There are two types of charge: positive and negative. Charges of the same sign repel each other and those with opposite signs attract each other. Charge is a conserved quantity; the net charge in an isolated system is constant. Furthermore charge is quantised; it only exists in discrete packets that are some integral multiple of the electron (or proton) charge, which equals e = C. Structure of matter: Matter consists of neutrons, protons and electrons. The positive protons and electrically neutral neutrons in a nucleus are held by nuclear forces. The negative electrons surround it at a (average) distance much greater than the nucleus size. Electric interactions are responsible for the structure of atoms. Conductors are materials that permit electric charge to move easily within them. Most metals are in fact good conductors of electricity, e.g. Copper. Insulators permit charges to move much less readily. Most non-metals are insulators, e.g. Wood. Induced charge: When a charge is brought near a neutral conducting system, a separation of charges in the conductor occurs. This phenomenon is known as polarisation. If then we connect the conductor to another one, charges move to the new conductor in a way that opposite charges always attract each other. Finally one can disconnect the two conductors to get oppositely charged systems (charging by induction) see fig 1. Coulomb s law: The electric force on a point charge q due to another point charge Q, whose position vectors are r and R respectively, is: F = qq 4πǫ 0 r R r R 3, where ǫ 0 is the primitivity of free space, ǫ 0 = C 2 N 1 m 2. In a simpler form we may write this force as: F = qq ˆd 4πǫ 0 d 2, 2

3 Figure 1: Charging two spheres by induction where ˆd is a unit vector of the vector d which is the position vector of the point charge q with respect to Q ( d = r R). Thus the magnitude of this force is: F = k qq d 2, with k = 1/(4πǫ 0 ) = N.m 2 /C 2, and d the distance between the point charges. The principle of superposition of forces states that when two or more charges, each exerts a force on another charge q, the total force on the charge q is the vector sum of the forces exerted by the individual charges. Note that in electrostatic studies, charges have no net motion. Electric field is a vector quantity that represents the force per unit charge exerted on a test charge at any point, provided that the test charge is small enough that it does not disturb the charges that cause this field. Thus the electric field at position r produced by a single point charge Q at position R, is given by: or E = Q 4πǫ 0 r R r R 3, E = Q 4πǫ 0 ˆd d 2, such that the force exerted on a test charge q placed at r is given by F = q E. 3

4 The electric field in this case has the magnitude: and has units N/C. E = k Q d 2 The electric field points away from a positive charge and points towards a negative one. Principle of superposition for electric fields: The electric field at a given point in space due to any combination of point charges is the vector sum of the fields caused by the individual charges. To compute the electric field at a given point in space due to a continuous distribution of charges we divide the distribution into infinitesimal elements and calculate the field caused by each element, and then carry out the vector sum (or component sum), usually by integration. Charge distributions are described by linear charge density λ, surface charge density σ or volume charge density ρ. Electric field lines: Field lines provide a graphical representation of the electric fields. At any point on a field line, the tangent to the field line gives the direction of the electric field at that point. Points where field lines are closer together E is larger, and where field lines are far apart E is smaller. See for example figs 2 and 3. Figure 2: Field lines from a positive, and from a negative charge Electric dipole is a pair of electric charges equal in magnitude (q) but opposite in sign, separated by a distance d (as in fig 3). Note: The electric field produced by a dipole falls like 1/r 3 when r d, where r is the distance to the dipole. Electric dipole moment is defined to be p = qd in magnitude and has direction from the negative charge to the positive charge. The torque experienced by a dipole in a uniform external electric field E is given by: τ = p E 4

5 Figure 3: Field lines from a dipole and thus the work done by the external electric field if the dipole rotates by and angle dφ, is dw = τ. dφ = pesinφdφ, where φ is the angle between p and E. Thus we write (after integration) the potential energy: U = p. E 2 Gauss s law Electric flux is a measure of the flow of electric field through a given closed surface. It is equal to the product of an element area da and the perpendicular component of E, integrated over a closed surface: Φ E = EcosφdA = E da = E.dA, where φ is the angle between E and ˆn, the vector perpendicular to the surface at the element surface-area da, and d A = daˆn. Note ˆn points outwards of the closed surface. Gauss s law: The total electric flux through a closed surface equals the total net charge enclosed by the surface divided by the permittivity of free space ǫ 0. Note Φ E = Q enc ǫ 0 When excess charge is placed on a conductor (at rest), it resides on the surface, and the electric field equals zero inside the conductor ( E = 0). 5

6 Applications of Gauss s law: The following table lists electric fields caused by several symmetric charge distributions. In the table q, Q, λ, σ refer to the magnitudes of the quantities. Charge distribution Region Electric field Single point charge q distance r from q E = 1 4πǫ 0 q r 2 Charge q on surface of a conducting sphere of radius R outside sphere, r > R E = 1 4πǫ 0 q r 2 inside sphere, r < R E = 0 Infinite wire of linear charge density λ distance r from wire E = 1 λ 2πǫ 0 r Infinite conducting cylinder of radius R and linear charge density λ Outside cylinder, a distance r > R from axis Inside cylinder, a distance r < R from axis E = 1 λ 2πǫ 0 r E = 0 Solid insulating sphere of radius R and charge Q uniformly distributed throughout the volume outside sphere, r > R E = 1 4πǫ 0 Q r 2 inside sphere, r < R E = 1 4πǫ 0 Q R 3r Infinite conducting plate of surface charge density σ Any point E = σ 2ǫ 0 Two oppositely charged conducting plates with surface charge densities σ and σ Any point between plates E = σ ǫ 0 For the last case, E = 0 outside the region between the plates. Note: The electric field caused by charges on the surface of a conductor at equilibrium in the region just outside its surface is perpendicular to its surface and has magnitude σ/ǫ 0, where σ is the surface density at that point. 6

7 3 Electric potentials Electric potential energy The electric force caused by any collection of charges is a conservative force. In analogy to the gravitational potential energy it is easy to show that the electric potential energy for a charged particle q 0 due to another charge q a distance r away (that is work done by the electric force on the particle q 0 when it moves from r to ) is given by: U = 1 4πǫ 0 qq 0 r, whichhasunitsofjoules, andisnegativeifchargesrepel(i.e. qq 0 < 0)andpositiveotherwise. Here the electric potential energy reference (point where electric potential energy reference is zero) is defined at. A collection of charges q i produce an electric potential energy for particle q 0 : U = q 0 q i, 4πǫ 0 r i where r i is the distance of the i th particle from q 0. However these charges do themselves have electric potential energy. The total potential energy for such a system is U = 1 q i q j, 4πǫ 0 r ij where the sum runs over all pairs of particles (i,j), with i j. A particle q 0 in a uniform electric field E has electric potential energy: i,j i U = q 0 Ed where d is a distance to the surface of reference of electric potential energy. Note that for a positive charge the electric potential energy increases in the opposite direction of the electric field. Electric potential (a.k.a. voltage) is defined by the electric potential energy per unit test charge q 0, produced by a charge q, and thus it has units of J/C, or as is commonly known V (Volt). Thus for a single charged particle q, the electric potential produced a distance r from it is: V = U q 0 = 1 4πǫ 0 q r, and that produced by a collection of charges q i is: V = 1 q i, 4πǫ 0 r i where r i are the distances from the point charges to the point where V is measured. Just as electric potentia energy, the electric potential (or voltage) is always measured relative to a reference, which in the above cases is chosen to be at. 7 i

8 For a continuous charge distribution the electric potential (voltage) is: V = 1 dq 4πǫ 0 r. The potential difference between two points a and b (the potential of a with respect to b), is in general given by: a V ab = V a V b = E. dl b which can easily be seen by considering the work-energy theorem or the definition of the potential energy. Thus a uniform electric field has electric potential associated to it: e.g. between the plates of a capacitor. V = Ed, Potential gradient The gradient of the electric potential equals the negative of the electric field: E = V which means the voltage increases in the opposite direction of the electric field. For a radial electric field (one independent of θ and φ) we have: E r = dv dr which suggests that V/m is a suitable unit of measurement of the electric field. Equipotential surfaces are surfaces in space where the potential energy is the same at every point. The electric field is always perpendicular to equipotential surfaces since E = V (In the same way F = U). For example the surface of a conducting material is an equipotential surface when all charges are at rest (equilibrium) and thus E is tangent to the surface at any point in the surface. Cathode ray tube is a device that accelerates electrons and ejects them into a uniform electric field. Electrons experience a motion similar to projectile motion. Electrons are scattered off a heated element with zero kinetic energy and are accelerated through a potential difference V 1. Then they enter a uniform electric field E perpendicular to the initial velocity of the electrons v 0, which have acquired kinetic energy E k = ev 1. The motion of electrons along the xaxis obeys the equation: 2eV1 x = v 0 t = t. m e 8

9 y V 1 d θ E x y l R Electron generator Figure 4: Cathode ray tube Along the yaxis the position of the electrons is described by: y = 1 2 at2 = ev 2 2m e d t2, where we used the fact that the kinetic energy is 1 2 m ev 2 = E k = ev 1 and the constant acceleration of the electrons is equal to a = F/m e = ee/m e = ev 2 /(m e d), with V 1 the potential difference between the electron generator and cathode, while V 2 is the potential difference between the parallel plates. The time of leaving the uniform electric field is t = l/v 0 = l me 2eV 1, and here the velocity of the electron has the components: 2eV1 v x = v 0 =, m e The angle of deflection satisfies: v y = at = al/v 0 = ev 2 m e d l me 2eV 1. tanθ = v y v x = ev 2 m e d l m e 2eV 1 = V 2 V 1 l 2d, which gives us the position of the electrons on the screen: y = V 2 V 1 lr 2d. 9

10 4 Capacitance and dielectrics Capacitors and capacitance: A capacitor is any pair of conductors separated by an insulating material. When the capacitor is charged there are charges of equal magnitude Q and opposite signs on the two conductors, and the potential difference V ab of the positively charged plate with respect to the negatively charged one is proportional to Q (see later). The capacitance C of a capacitor is defined by: C = Q V ab. A parallel-plates capacitor is made of two parallel conducting plates each with area A, separated by a distance d (d A). If these plates are separated by vacuum, we know that the electric field between the plates is: E = σ ǫ 0 = Q Aǫ 0, and the potential difference is Thus the capacitance is: V = E.d = d Aǫ 0 Q. C = Q V = A d ǫ 0, which has units of farads (F), 1F= 1C/V= 1C 2 /(N.m). Capacitors in series and parallel: If capacitors C i are attached in series, then they all have the same charge Q, which implies that the equivalent capacitance satisfies: 1 = 1, C eq C i and if they are attached in parallel, they all have the same voltage V, which implies that the equivalent capacitance satisfies: C eq = C i. The energy stored in a charged capacitor: Consider the force acting on one plate due to the otherplate. Infacteachindividual chargedparticleinagivenplatedoesnotfeelanet electric field due to the charged particles in the same plate (it is in equilibrium by symmetry). It only feels the net electric field due to the charged particles in the other plate which is σ/(2ǫ 0 ) = E/2, where E is the actual electric field between the two plates. Hence we deduce that the net force on one plate due to the other is F = QE/2, which is constant, meaning that the electric potential energy is simply U = QEd/2. Thus: U = 1 2 QV = 1 2 CV 2 = Q2 2C. 10

11 This energy can be thought of as residing in the electric field between the plates of the capacitor. Then the energy density (energy per unit volume) can simply be expressed as: u = U Ad = 1 2 ǫ 0E 2. Dielectrics: When the space between the conductors of a capacitor is filled with a dielectric material, and for a constant charge on the plates of the capacitor, the total effective electric field within the capacitor is reduced by a factor K due to the polarisation of the dielectric material, E eff = KE 0, where E 0 is the electric field in the absence of the dielectric material. The effective electric potential decreases by the same factor V = KV 0, and hence the effective capacitance increases by the same factor K. For a parallel-plates capacitor the new capacitance is: C = C 0 K = ǫ 0 A K d ǫa d where ǫ = ǫ 0 /K is the permittivity of the dielectric material (ǫ > ǫ 0 ). The energy density stored in a capacitor with a dielectric material is the same as above with the replacement ǫ 0 ǫ. 5 Current and resistance Current is defined by the rate of flow of charge through a given area I = dq dt, It has SI units of Ampere, 1A = 1C/s. For a cylindrical wire of cross-sectional area A and a number density of charge carriers n, with charges q moving at drift velocity v d, the (average) current flowing through the area A is: I = n q v d A where we used the fact that the amount of charge is dq = nqdv, with dv = A.dx = A.v d dt. Current density is the current per unit area. From before we cast this into: j = nq v d Conventionally the direction of the flow of current is defined as the direction of motion of positive charges (or equivalently of the external field that causes the current), even when the actual charge carriers are negative or of both signs. Therefore voltage increases in the opposite direction of current flow, and current heads towards the negative pole. Resistivity: The resistivity ρ of a material is defined by the ratio of the magnitudes of electric field and current density, ρ = E/j. The SI unit of resistivity is ohm-meter (Ωm). Good conductors have a small resistivity while good insulators have a large resistivity. Ohm s law states that the resistivity of a material is a constant independent of the electric field. Resistivity depends on temperature. 11

12 Ohm s law: For materials that obey Ohm s law, the potential difference V across a particular sample of material is proportional to the current I that flows through the material V = RI, where R is the resistance of the sample. In terms of resistivity ρ, length l and cross-section area A of the material, we have R = ρl A The SI unit of measurement of resistance is the ohm, 1Ω =1 V/A. Power A circuit element with a potential difference V ab and a current I puts energy into a circuit if the current direction is in the same direction as the electric potential gradient (or opposite to the electric field) in the device, and takes energy from the circuit if the current apposes the potential gradient (or in the same direction as the electric field). The rate of flow of energy (power) is given by: P = IV ab. A resistor always takes energy from a circuit, and the power consumption of a resistor is: P = IV = RI 2 = V 2 /R. 6 Direct-current circuits Circuits and electromotive force: A complete circuit is a conducting loop that provides a continuous current-carrying path. A circuit must contain a source of electromotive force (emf) denoted ε, which has units of volt (V). An ideal emf source provides a constant voltage independent of the current through it, however real sources have some internal resistance r and so the terminal voltage across it (and thus across the circuit) is given by: V ab = εir Resistors in series: When several resistors R 1, R 2, R 3,...are attached in series, the equivalent resistance R eq is the sum of the individual resistances: R eq = R 1 R 2 R 3 The same current flows through all the resistors in a series connection. Resistors in parallel: When several resistors R 1, R 2, R 3,...are attached in parallel, the equivalent resistance R eq is given by: 1 R eq = 1 R 1 1 R 2 1 R 3 All resistors in a parallel connection have the same potential difference between their terminals. 12

13 Kirchhoff s rule for junctions is based on the principle of conservation of charges, and it states that The algebraic sum of currents into any junction must be zero Kirchhoff s rule for loops is based on the conservative nature of the electric force and on the conservation of energy. It states that The algebraic sum of potential differences around any closed loop must be zero. Time constant of R-C circuit When a capacitor C is charged by a battery in series with a resistor R, the current flowing in the circuit as a function of time is: I = I 0 exp(t/rc) and the charge in the capacitor as a function of time is give by: and the voltage across the capacitor is: Q = Q f (1exp(t/RC)), V = Q C = V 0(1exp(t/RC)) where I 0 is the initial current and Q f is the terminal charge of the capacitor, Q f = V 0 C, I 0 = V 0 /R, with V 0 the terminal voltage across the capacitor or equivalently the emf of the circuit. When a capacitor is discharged we have: Q = Q 0 exp(t/rc) I = I 0 exp(t/rc) V = V 0 exp(t/rc) The constant τ = RC has dimensions of time and represents the time in which the capacitor is discharged to 1/e 37% of its initial charge. 13

14 7 Magnetic fields Magnetic field and magnetic forces Magnetic interactions are fundamentally interactions between moving charged particles. These interactions are caused by the vector magnetic field, denoted B. A charged particle q moving with velocity v in a magnetic field B will experience a force F, given by: F = q v B The direction of this magnetic force is perpendicular both to the velocity of the particle and to the magnetic field. The magnitude of this force is F = q vbsinθ where θ is the smaller angle between v and B. The SI unit of measurement of magnetic field B is the Tesla (T), where 1 T = 1 N/(A.m). Other units include the Gauss: 1G = 10 4 T. When a charged particle moves in a magnetic field, the work done by the magnetic force on the particle is zero because the displacement is always perpendicular to the direction of the force. The magnetic field can alter the direction of the particle s velocity vector, but it cannot change its speed. Magnetic field lines and magnetic flux Magnetic fields can also be represented graphically in terms of magnetic field lines. At each point the magnetic field B is tangent to the magnetic field line at that point. Points where field lines are closer together the field magnitude is larger. The magnetic flux through an area is defined by Φ B = B da = BcosφdA = The SI unit for magnetic flux is the Weber (Wb). 1Wb=1Tm 2. Gauss s law for magnetism states that: B.d A The magnetic flux through any closed surface is always zero As a result, magnetic field lines always form closed loops (curling field lines) and thus there are no point sources for magnetic fields (monopoles). 14

15 Applications of the motion of a charged particle in a uniform magnetic field The magnetic force is always perpendicular to the velocity vector. A particle moving under the action of a magnetic field alone always moves with constant speed. In a uniform magnetic field a particle with initial velocity perpendicular to the field moves in a circle with radius given by R = mv q B and angular frequency ω = v R = q B m f = q B 2πm Velocity selector An electron beam of velocity v passes through a uniform electric field E and a uniform magnetic field B such that v, E and B are perpendicular to each other pairwise. Thus only electrons with speed E/B will pass through the fields without deflecting. e/m experiment Electrons of mass m and charge e are accelerated through potential V and gain velocity squared v 2 = 2eV/m. The potential V is tuned so that electrons pass through a velocity selector without deflecting, i.e. such that v = E/B. Thus: e m = 1 2V Mass spectrometers We use a velocity selector to select the speed of ions (v = E/B ), which enter a uniform magnetic field B at right angle (hence circular motion). Radius of path is R = mv/( q B). Thus the mass of the ions can be separated according to path radius: E 2 B 2 m = R q BB. E Magnetic force on current-carrying conductor The force on a straight segment l of a conductor carrying current I (in the direction of l) in a uniform magnetic field B is given by: F = I l B For an infinitesimal segment this reads: d F = Id l B Force and torque on a current loop A current loop with area A and current I in a uniform magnetic field B experiences no net force, but does experience a torque of magnitude τ = IBAsinφ, with φ the angle between B and the 15

16 area vector for the loop A (in a way that I and A respect the right-hand rule, with A in direction of thumb). In terms of the magnetic dipole moment the torque is µ I A τ = µ B and analogously to the electric dipole in an external electric field, the potential energy of a current loop of magnetic dipole moment µ in an external magnetic field B is U = µ. B = µbcosφ Note that the magnetic dipole moment depends only on the current and the area, and is independent of the shape of the loop. Magnetic dipole in a non-uniform magnetic field: In a non-uniform magnetic field (example that of a magnet), a current loop will experience a net force repulsive from the north if µ points to it. We can explain why two similar poles repel by considering one of them as being put in a non-uniform magnetic field of the other, with the directions of µ for both magnets apposing each other. The reason why unmagnetised iron magnetizes is that µ tends to align with the magnetic field on the atomic scale. DC motor A magnetic field exerts a torque on a current in the rotor. Motion of the rotor through the magnetic field causes an induced emf called back emf. Hall effect: is a potential difference measured perpendicular to the direction of current in a conductor, when the conductor is placed in a magnetic field ( B v d ). Electrons are attracted to one side of the conductor leaving positive ions in the opposite side. This causes an electric field perpendicular to the direction of the current. The Hall voltage is determined by the requirement that the associated electric field must just balance the magnetic force on a moving charge. Hall effect measurements can be used to determine the density n of charge carriers and their sign from the relation nq = J xb y E z 8 Sources of magnetic fields The magnetic field B created by a moving charge q with velocity v, as measured at position r with respect to the source point (location of q), is: B = µ 0 q v ˆr, 4π r 2 where ˆr is a unit vector of r, µ 0 is a universal constant known as the permeability of free space: µ 0 = 4π 10 7 NA 2. 16

17 The principle of superposition for magnetic fields: The total magnetic field produced by several moving charges is the vector sum of the fields produced by the individual charges. The law of Biot and Savart: the magnetic field db created by an element d l of a conductor carrying current I (in the same direction as d l), as measured at position r with respect to this element, is: db = µ 0 Id l ˆr. 4π r 2 Thus the field created by a finite current-carrying conductor is the line integral of this expression over the length of the conductor. For example, the magnetic field a distance r form a long, straight conductor carrying current I has magnitude B = µ 0I 2πr. The magnetic field lines are circular coaxial with the wire, with the direction given by the right hand rule (current is with direction of thumb). Also, the magnetic field produced by a circular conducting loop of radius a, carrying current I, measured at a distance x form the centre of the loop along its axis has magnitude: B = µ 0 Ia 2 2(x 2 a 2 ) 3/2 Thus for N loops this expression is multiplied by N. At the centre of the loop, where x = 0, we have: B = µ 0IN 2a The direction is also given by the right-hand rule ( B with thumb) The interaction force per unit length between two long parallel conductors with currents I and I separated by a distance r has magnitude: F/L = µ 0II 2πr. Currents in parallel wires attract if they are in the same direction and repel if they are in opposite directions. Ampère s law The circulation of the magnetic field around any closed loop (the line-integral of the magnetic field dotted with the line element d l) equals µ 0 times the net current passing through the area enclosed by the path: B.d l = µ 0 I enc The positive sense of the current is defined by the right hand rule. 17

18 Thus we may draw a table similar to that for electric fields: Current distribution Region Magnetic field Long (infinite) straight conductor of current I distance r from conductor B = µ 0I 2πr Circular loop of radius a carrying current I (for N loops multiply by N) On axis of loop a distance x from centre At centre of loop B = B = µ 0I 2a µ 0 Ia 2 2(x 2 a 2 ) 3/2 Long cylindrical conductor of radius R carrying current I Outside cylinder, a distance r > R from axis Inside cylinder, a distance r < R from axis B = µ 0I 2πr B = µ 0I 2π r R 2 Long, closely wound solenoid with n turns per unit length and current I Inside solenoid, near its centre B = µ 0 ni Just outside solenoid B 0 Tightly wound toroidal solenoid (toroid) with N turns and current I Within the space enclosed by the windings, distance r from symmetry axis Outside the space enclosed by windings B = µ 0NI 2πr B 0 Field lines and current enter a magnetic coil from the same side Displacement current ConsideraninfinitewireconductingacurrentI. Ampère slaw: B.d l = µ 0 I enc. Here I enc is the current that passes through any surface that is bound by the loop over which B.d l is integrated, which can be deformed so as to cover one surface of a capacitor at infinity (a long distance away from point of measurement of field), then there will be no net current (since charge eventually gets accumulated in the capacitor), meaning that I = 0, which would also mean B = 0. However B = µ 0 I/(2πr), is unaffected by a capacitor at 18

19 . To resolve this we must add another current, known as displacement current, whose magnitude is determined so as to let B as before. Mathematically we can rewrite Ampère s law (modified) as: B.d l = µ 0 (I enc i D ) where i D = ǫ 0 dφ E dt is the displacement current, and Φ E is the flux of the electric field out of the surface bound by the integration loop. Note the inevitable connection of the magnetic field and electric field! 9 Faraday s law Faraday s law states that the induced emf in a closed loop equals the negative of the time rate of the flux of the magnetic field through the loop: ε = dφ B dt, where Φ B = B.d s is the flux of magnetic field out of the surface of the loop. This relation is valid whether the flux change is caused by changing the magnetic field, motion of the loop, or both. Lenz s law states that the induced emf always tends to appose or cancel out the change that caused it. Motional electromotive force: If a conductor with length L moves with speed v in a uniform magnetic field of magnitude B, and if the length and velocity are both perpendicular to the field, the induced emf is: ε = vbl More generally, when part or all of a closed loop moves in a magnetic field B, the induced emf is: ε = ( v B).d l Induced electric field: When an emf is induced by changing the magnetic flux through a stationary closed path, there is an induced electric field E of non-electrostatic origin, such that: E.d l = dφ B dt The field is non-conservative and cannot be associated with a potential. Eddy currents are induced in the volume of a bulk piece of conducting material, such as a metal, when it is in a changing magnetic field or moved through a field. 19

20 Maxwell s equations are the combination of Gauss s law for electric field, and for magnetic field, Ampere s law and faraday s law. They are: E.d s = Q enc S ǫ 0 B.d s = 0 S E.d l = dφ B dt B.d l = µ 0 I ǫ 0 µ 0 dφ E dt where the first law is Gauss s law for electric fields, the second is Gauss s law for magnetic fields, the third is Faraday s law, and the last is Ampère s law. (1) 10 Inductance Mutual inductance When a change in current I 1 in one circuit causes a change in magnetic flux in a second circuit, an emf ε 2 is induced in the second circuit; and vise versa: ε 2 = M di 1 dt ε 1 = M di 2 dt TheconstantM iscalledthemutualimpedancebetween thetwocoilsanddependsonthegeometry of the two coils and the material between them. If the circuits are coils of wire with N 1 and N 2 turns, respectively, the mutual impedance can be expressed in terms of the average flux Φ B2 through each turn of coil 2 that is caused by the current I 1 in coil 1 or vise versa M = N 2Φ B2 I 1 = N 1Φ B1 I 2 The SI unit of measurement of the mutual impedance is the Henry (H) 1H= 1Ω.s = 1 Wb/A = 1 V.s/A. Self-inductance A change in current I in any circuit induces an emf ε in that circuit called self-induced emf: ε = L di dt 20

21 The constant L, is called the inductance or self-inductance, depends on the geometry of the circuit and the material surrounding it. The inductance of a coil of N turns is related to the average flux Φ B through each turn caused by the current I in the coil: L = NΦ B I The potential difference across an inductor depends on the current flowing through it: V ab = V a V b = L di dt where the current I is directed from a to b. The potential gradient apposes the direction of the current if the current rate is positive (energy is put in the coil), and the potential gradient is in the same direction as the current (energy is put in the circuit) if the rate of the current is negative. Magnetic-field energy An inductor with inductance L carrying current I has energy: U = 1 2 LI2 This is the energy associated with the magnetic field of the conductor. If the field is in vacuum, the magnetic energy density u (energy per unit volume) is u = B2 2µ 0 If not in vacuum, we replace the permeability in vacuum µ 0 with the corresponding permeability of the material µ. R L circuit in DC In an RL circuit containing a resistor R and an inductor L, and a source of emf ε, the current flowing through the resistance as a function of time is: I = I 0 (1exp(t/τ)) where τ = L R is the characteristic time of thecircuit. I 0 is theterminal current throughthe resistor I 0 = ε/r. As time approaches infinity the rate of change of current goes to zero and so the potential difference across the inductor is essentially zero (which means the circuit reduces to a simple resistance). Now consider a steady current flowing through the circuit (after a long time). If the emf is suddenly dropped to zero the current in the circuit will decay: I = I 0 exp(t/τ) 21

22 C L circuit in DC Consider a charged capacitor C with initial charge Q connected with an inductor L in a closed loop. Then all quantities in the circuit (Voltage across each component, current in inductor, and charge in capacitor) will oscillate with angular frequency 1 ω = CL Such a circuit is analogous to a mechanical harmonic oscillator, with inductance L analogous to mass m, and the reciprocal of the capacitance 1/C analogous to the constant k of the restoring force, the charge q analogous to the displacement x and current I analogous to the velocity v. R L C circuit in DC An L-R-C circuit contains an inductor L, a resistor R and a capacitor C. The circuit undergoes damped oscillations for sufficiently small resistance R. The frequency ω of the damped oscillations is 1 ω = LC R2 4L 2 As the value of R increases damping increases. For the value of R = 4L/C the oscillations are critically damped and for values of resistance greater than this value overdamping occurs and the circuit simply decays. 11 Alternating-current circuits An alternator or AC source produces and emf that varies sinusoidally with time. A sinusoidal voltage or current can be represented by a phasor, a vector that rotates counterclockwise with constant angular speed ω equal to the angular frequency of the sinusoidal quantity. Its projection along the x-axis at any time represents the instantaneous value of the quantity. For a sinusoidal current the average root-mean-square (rms) current I rms and voltage V rms are related to the maximum values of current I 0 and voltage V 0 by: If the current I in an AC circuit is and the voltage across the circuit is I rms = I 0 2, V rms = V 0 2 I = I 0 cosωt V = V 0 cos(ωtφ) then φ is called a phase angle, and is the angle of the voltage phasor relative to the current phasor. 22

23 Resistor From Ohm s law V = IR we immediately deduce that the current and voltage across a resistor are in phase, and the maximum voltage across the resistor V 0 = I 0 R. Note: V = V ab = I ab R, where V ab = V a V b and I ab is current flow from a to b. If I is positive (meaning current flowing from a to b) then V ab > 0 (meaning the potential of a is greater than the potential of b), and vice versa. Inductor The voltage across an inductor is V ab = LdI ab /dt. Thus if I ab = I 0 cos(ωt) then V ab = L di dt = IωLsinωt = IωLcos(ωtπ/2) which means that the voltage across an inductor leads the current by π/2 phase, andthat the maximum voltage is V 0 = I 0 ωl. Thus the inductive reactance of an inductor is X L = ωl. Capacitor The voltage across a capacitor is V ab = q/c. where q is the charge in the plate in the side a (being positive or negative). The current in the circuit in the direction a to b is given by I ab = dq dt = CdV ab dt Thus: dv = 1 C Idt V = 1 C I 0 cosωtdt = 1 ωc I 0sinωt = 1 ωc I 0cos(ωtπ/2) which means that the voltage across a capacitor lags the current by π/2 phase, and that the maximum voltage is V 0 = I 0 /ωc. Thus the capacitive reactance is X C = 1/ωC. LRC circuit in series We use phase diagrams to specify the impedance of the circuits. The total voltage phasor is the vector sum of the L and R and C vector phasors (Kirchhoff). This way it turns out that the impedance Z of the circuit is Z = R 2 (ωl1/ωc) 2 and the phase angle between the total voltage across the LRC circuit and the current satisfies the relation tanφ = ωl1/ωc R such that V = V 0 cos(ωtφ) and V 0 = ZI 0 (recall I = I 0 cosωt). Note that the rms voltage is related to the rms current by V rms = ZI rms Power The instantaneous power is given by P = IV. When the power is negative it means that the circuit is returning energy to the power source. The average power in the circuit is P av = V rms I rms cosφ 23

24 where cosφ is called the power factor of the circuit, being 1 for a pure resistor and 0 for a pure inductor and/or capacitor. Generally this factor is cos φ = R/Z. Resonance in AC circuit When the current becomes maximum (for any given voltage) and the impedance becomes minimum the circuit is said to be in resonance. This happens when ωl = 1 ωc in which case the impedance equals the resistance Z = R, and here the angular frequency of oscillation of circuit is ω = 1/ LC. At resonance the voltage and current are in phase. 24