Electromagnetic nduction and Waves (Chapters 33-34) The laws of emf induction: Faraday s and Lenz s laws Concepts of classical electromagnetism. Maxwell equations nductance Mutual inductance M Self inductance L. nductors Magnetic field energy Simple inductive circuits RL circuits LC circuits LRC circuits EM waves Maxwell s equations and EM waves Energy and momentum in EM waves: Poynting vector
The other side of the coin We ve seen that an electric current produces a magnetic field. Should we presume that the reverse is valid as well? Can a magnetic field produce an electric current? Yes, magnetic fields can produce electric fields through electromagnetic induction Most of the electric devices that we use as power supplies are electric generators based on induced emf : these generators convert different forms of energy first into mechanical energy and then, by induction, into electric energy Electromagnetic induction was first demonstrated experimentally by Michael Faraday Experimental setup: A primary coil connected to a battery A secondary coil connected to an ammeter Observations: When the switch is closed, the ammeter reads a current and then returns to zero When the switch is opened, the ammeter reads an opposite current and then returns to zero When there is a steady current in the primary circuit, the ammeter reads zero Conclusions: An electrical current is produced by a changing magnetic field The secondary circuit acts as if a source of emf were connected to it for a short time t is customary to say that an induced emf is produced in the secondary circuit by the changing magnetic field. Let s look closer at why this happens
Def: The electric flux of a uniform magnetic field crossing a surface A under an angle θ with respect to the normal to the surface is Acos Gauss s Law Electric Flux n order to quantify the idea of emf induction, we must introduce the magnetic flux since the emf is actually induced by a change in magnetic flux rather than generically by a change in the magnetic field Magnetic flux is defined in a manner similar to that of electrical flux: da A θ f the field varies across the surface, one must integrate the contribution to the flux for every element of surface: θ T m Weber Wb S Ex: The uniform field lines penetrate an area A perpendicular and then parallel to the surface, produce maximum and zero fluxes A 0 A A Anˆ Exercise 1: Gauss s law can be reformulated for magnetic sources. Try to do it yourself: what is the magnetic flux through a closed surface around a magnetic source?
Quiz: 1. Given a solenoid, through which of the shown surfaces is the magnetic flux larger? A 4 A 1 A A 3 A 1 A A 3 A 4. A long, straight wire carrying a current is placed along the axis of a cylindrical surface of radius r and a length L. What is the total magnetic flux through the cylinder? a) L L 0 b) 0 L r c) Zero r
Electromagnetic nduction Faraday s Law and Lenz s Law The involvement of magnetic flux in the electromagnetic induction is described explicitly by Faraday s Law: The instantaneous emf induced in a circuit equals the time rate of change of magnetic flux Φ through the circuit d dt Comments: Since Φ = Acosθ the change in the flux, dφ, can be produced by a change in, A or/and θ f the circuit contains N loops (such as a coil with N turns) N d The negative sign in Faraday s Law is included to indicate the polarity of the induced emf, which is more easily found using: Lenz s Law: The direction of any magnetic induction effect is to oppose the cause of the respective effect Ex: Causes of EM induction can be: varying magnetic fields, currents, emfs, or forces determining a change in flux for instance by varying the area exposed to the magnetic field. nduced effects may be magnetic fields, emfs, currents, forces, electric fields dt
Electromagnetic nduction Applying Lenz s Law Magnetic flux through a loop can be varied by moving a bar magnet in proximity 1. f the bar magnet is moved toward a loop of wire: As the magnet moves, the magnetic flux increases with time The induced current will produce a magnetic field opposing the increasing flux, so the current is in the direction shown This can be seen as a repelling effect onto the incoming bar by the loop seen as a magnet. f the bar magnet is moved away from a loop of wire: As the magnet moves, the magnetic flux decreases with time repels the incoming North The induced current will produce a magnetic field helping the decreasing flux, so the current is in the direction shown This can be seen as a attraction effect onto the bar by the loop seen as a magnet ind ind attracts the departing North
Exercise : Lenz s Law Consider a loop of wire in an external magnetic field ext. y varying ext, the magnetic flux through the loop varies an a current flows through the wire. Let s use Lenz t law to find the direction of the current induced in the loop when a) ext decreases with time, and decreasing ext b) ext increases with time increasing ext
Exercises: 3. Emf induced by moving field: A bar magnet is positioned near a coil of wire as shown in the figure. What is the direction of the induced magnetic field in the coil and the induced current through the resistor when the magnet is moved in each of the following directions. a) to the right b) to the left v b 4. Emf induced by switching field on and off: Find the direction of the current in the resistor in the figure at each of the following times. a) at the instant the switch is closed b) after the switch has been closed for several minutes c) at the instant the switch is opened Problem: 1. Emf induced by reversing field: A wire loop of radius r lies so that an external magnetic field 1 is perpendicular to the loop. The field reverses its direction, and its magnitude changes to in a time Δt. Find the magnitude of the average induced emf in the loop during this time in terms of given quantities.
We ve seen that, in certain conditions, motion can determine an emf we are interested in this phenomenon since it stays behind converting mechanical energy into electric energy To see how it happens, consider a straight conductor of length L moving with constant velocity v perpendicular on a uniform field : the electric carriers in the conductor experience a magnetic force qv along the conductor, as on the figure Notice that the electrons tend to move to the lower end of the conductor, such that a negative charge accumulate at the base Consequently, a positive charge forms at the upper end of the conductor, such that, as a result of this charge separation, an electric field E is produced in the conductor Charges build up at the ends of the conductor until the upward magnetic force (on positive carriers forming a current) is balanced by the downward electric force qe V EL FL q vl ab Motional emf Across a conductor moving in a magnetic field The potential difference between the ends of the conductor is similar with the potential difference between the plates of a charged capacitor: This motional emf is maintained across the conductor as long as there is motion L Ex: The magnetic field of Earth is about 5 10 5 T. Therefore, if a straight 1-m metallic rod is moved perpendicular on the field with a speed of 1 m/s, the emf produced across it ends is about 5 10 5 V q a + + b Fm Fe qv v qe
Motional emf Producing current in a circuit Consider now that the moving bar on the previous slide has a negligible resistance and it slides on rails connected in a circuit to a resistor R, as in the figure As the bar is pulled to the right with a velocity v by an applied force, the free charges move along the length of the bar producing a potential difference and consequently an induced current through R The motional emf induced in the circuit acts like a battery with an emf vl R vl R n general, for any conductor moving with velocity v in a magnetic field we have an alternative expression for Faraday s law: v dl R R + + + v L The charge carriers are pushed upward by the magnetic force vl The integral is around a closed conductor loop force per unit charge acted on the element dl of conductor moving in the external magnetic field
Motional emf Explained using Faraday and Lenz s Laws Alternatively, we can look at the same situation but using Faraday and Lenz s laws: the changing magnetic flux through the loop and the corresponding induced emf in the bar result from the change in area of the loop 1. ncreasing circuit area: The magnetic flux through the loop increases y Lenz s law the induced magnetic field ind must oppose the external magnetic field ext. The direction of the current that will create the induced magnetic field is given by RHR #.. Decreasing circuit area: The magnetic flux through the loop decreases R ind ext F m ind v F applied y Lenz s law the induced magnetic field ind must help the external magnetic field ext. The direction of the current that is reversed compared with the case above. Then, by Faraday s Law, we obtain the same expression for the induced emf: d da dx L Lv dt dt dt R ind ext v ind da dx L
Problems:. Gravity as applied force to induce emf: A metallic rod of mass m slides vertically downward along two rails separated by a distance l connected by a resistor R. The system is immersed in a constant magnetic field oriented into the page. a) Calculate the current flowing through the resistor R when the magnetic force on the rod becomes equal to its weight. b) Calculate the emf induced across the resistor R. c) Use Faraday s Law to compute the speed of the rod when the net force on it is zero. m l R 3. Faraday disk: A thin conducting disk with radius R laying in xy-plane rotates with constant angular velocity ω around z-axis in a uniform magnetic field parallel with z. Find the induced emf between the center and the rim of the disk.
Applications Electric Generators An alternating Current (ac) generator converts mechanical energy to electrical energy by rotating loops of wire in magnetic fields There is a variety of sources that can supply the energy to rotate the loop, including falling water, heat by burning coal or nuclear reactions, etc. asic operation of the generator: as the loop rotates, the magnetic flux through its surface A changes with time, such that an emf is induced For constant angular speed ω = dθ/dt, d d d Acos Asin dt dt dt r θ v A sint sint max Comments: The emf polarity varies sinusoidally (ac signal) ε = ε max when loop is parallel to the field ε = 0 when the loop is perpendicular to the field ω v r v θ = ωt
Concepts of classical electromagnetism nduced electric fields One of the main concepts that you must retain from our discussion about induction is that a changing magnetic flux can produce an electric field (manifested as an emf) even in the absence of a magnetic field in the region where the charges move. Ex: Recall that varying the current in the inner solenoid will induce a current in the outer solenoid even though the field outside the inner solenoid is negligible and the outer solenoid doesn t move An electric field induced by a varying magnetic field flux is called nonelectrostatic field, and it doesn t need the presence of a conductor Moreover, unlike the electrostatic fields (created by charge distributions) they are not conservative For instance, for a stationary loop of induced current or field, the emf is given by Faraday s law: d dt E dl increasing which means that the work done by the field around the loop is not zero for closed paths a property of nonconservative fields Therefore, we cannot associate a potential with this kind of electric field However, a charge in such a field is still acted by a force 0 E E F E E qe or E ind
Concepts of classical electromagnetism Displacement current n conclusion, electromagnetism stems from the symmetry between electricity and magnetism: a changing magnetic flux can produce an electric field and a varying electric field creates a magnetic field. This interplay can be seen by looking at Maxwell s discovery: the displacement current due to a varying electric flux Φ E Consider a parallel plate capacitor: the fact that the ordinary conduction current (apparently) stops on the plates can trouble Ampere s law; for instance, the two surfaces in the figure have the same boundary, but the currents are different ( and 0). To solve the contradiction, Maxwell assumed the existence of a continuation of the current inside the capacitor: the displacement current D given by the change in electric flux: dl 0 D encl 0 0 So, the displacement current has its own measurable magnetic field d dt E encl
Concepts of classical electromagnetism Maxwell equations n 1865, James Clerk Maxwell provided a mathematical theory know today as classical electromagnetism that unified electric and magnetic interactions Maxwell s equations crystallize some of the ideas we ve discussed in class, showing that electric and magnetic fields play symmetric, complementary roles in nature: Thus, for charges and currents in an empty space (no dielectric or magnetic material present) Maxwell s equations are: E da da E dl q encl 0 dl 0 0 0 d dt missing terms: no magnetic monopoles (Gauss s Law) if varies, the resulting E is not conservative d dt E encl (Faraday s Law) (Ampere s Law) electric monopoles: magnetic field is never conservative
Quiz 4. The adjacent magnetic field is decreasing. Which of the fields below is the induced electric field equivalent to an induced current? a) b) c) d) Problem: 4. Magnetic field inside a charging capacitor: A parallel-plate capacitor with circular plates of radius R is being charged by a constant current. Find the magnetic field at an arbitrary distance r from the axis of the capacitor.
y Faraday s law, changing a current in a coil induces an emf in an adjacent coil: this coupling is called mutual inductance Consider two coils with N 1 and N turns. The variation of current in the first coil corresponds to a proportionally varying flux through the second: y Faraday s law N M i N 1 1 d dt mutual induction di 1 M1 dt The mutual-inductance depends on the geometry of the two coils and on the presence of a magnetic material as a core. f the material has linear magnetic properties, the mutual inductance is a constant. The discussion is symmetric in the opposite direction, so we have M M M nductance Mutual inductance N 1 1 1 1 M Henry H i1 i N 1W A
nductance Self inductance Notice that nothing prevents a changing flux to produce an emf in the very coil that produces the actual flux: this phenomenon is called self-inductance: discovered in the 19 th century by Joseph Henry Ex: Consider a current carrying loop of wire f the current increases in a loop, the magnetic flux through the loop surface due to this current also increases: hence, an emf is induced that opposes the change in magnetic flux This opposing emf results in a slowed down increase of the current through the loop Alternatively, if the current decreases, the self-inductance will slow down the rate of decrease The self-induced emf is proportional to the rate of change of the current through the coil: di L dt negative sign indicates that a changing current induces an emf in opposition to that change L is a proportionality constant called the inductance of the coil: Def: f a circuit with N loops carrying a current produces a magnetic flux Φ through each loop surface, the self-inductance is given by L N L Henry (H) S
nductance nductors L characterizes solenoids as elements of circuit called inductors Since the flux is proportional to the current, the inductance of a solenoid does not depend on the current flowing through the coil: it is a characteristic of the device, depending on geometric factors and the magnetic properties of the interior of the coil Ex: Self inductance of a straight solenoid: A straight solenoid with n turns per length, and volume V has inductance given by: A L N N N n 0 A n A L n V 0 0 nductance can be interpreted as a measure of opposition to the rate of change in the current: it determines a potential difference or a back emf across the terminals i A ε back <0 ε back >0 A device with self-inductance (such as a coil) is called an inductor: a circuit element with a certain inductance an additional circuit element besides capacitors and resistors Symbol: L i a b Potential difference: V V V L di dt ab a b back drop if di/dt > 0 raise if di/dt < 0 0 if i = const.
Energy Stored in a Magnetic Field Summary of circuit elements The work done by a battery to produce an increasing current against the back emf of an inductor can be thought of as energy stored L ab in the magnetic field inside the inductor UL 1 L Contrast with the energy dissipated across a current carrying resistor: PR 1 R Or the energy stored in the electric field of a charged capacitor: UC 1 CV A A A U Pdt iv dt L idi d L R C 0nA 0 A Ad V L L di dt The magnetic energy density stored in a straight solenoid inductor is given by u V 1 U L L 0 1 V V n V 1 n 0 0 0 VR V C R Q C 0 this is, in general, the magnetic energy density in vacuum nside a magnetic material such that an iron core inside a solenoid μ 0 is to be replaced with μ : magnetic permeability in the respective material
nductive circuits LR-circuit: principles An inductor can be combined in series with a resistor into a dc-rl circuit to obtain a specific behavior Recall that the resistance R is a measure of opposition to the current while the inductance L measures the opposition to the rate of change of the current. Let s see what s happening in an RL circuit: 1. Close S 1 and open S : the RL series circuit is completed across a battery ε As the current begins to increase, the inductor produces a negative back emf ε L < 0 that opposes the increasing current, so the current doesn t change from 0 to its maximum instantaneously When the current reaches its maximum, the rate of change and the back emf ε L = 0. Open S 1 and close S : the RL series circuit is completed with battery removed Since there is no battery, the current starts to decay, such that the inductor produces a positive back emf ε L > 0 to help the current. f the current becomes zero, the rate of change and the back emf are ε L = 0 Kirchhoff rule applies in both cases (set ε = 0 when current is decaying): i R ε S 1 : i S : S 1 S L ε L <0 ε L >0 di ir L 0 ir L 0 dt
nductive circuits LR-circuit: characteristics 1. The current in the RL circuit in series with a battery increases exponentially to max = ε/r: di 0 Rt L ir L i 1 e dt R The time constant, τ = L/R, for an RL circuit is the time required for the current in the circuit to reach 63.% of its final value A circuit with a large time constant will take a longer time to reach its maximum current i e 1 t max. The current in the RL circuit without a battery decays exponentially from its initial value 0 : di Rt L ir L 0 i 0e dt f the current reached the maximum value before the battery was disconnected, it is given by 0 = max = ε/r i t 0e
nductive circuits LC-circuit: principles An inductor connected across a charged capacitor form an electric oscillator with oscillating current and charge called a dc-lc circuit. Functionality: 1. As the capacitor discharges, current increases from 0 to a maximum value and the potential difference across both elements decreases gradually to 0: the electric energy is stored in the form of magnetic energy. When the current reaches its maximum, the capacitor starts to recharge with an inverse polarity than initially until the current is again zero and the process restarts in the reverse direction electric energy magnetic energy cycle
di q vl vc 0 L 0 dt C The SHO solutions are Charge: Current: nductive circuits RL-circuit: characteristics Kirchhoff rule can be applied to find the equation describing the oscillations of charge and current: 1 LC q Qcost maximum charge initial phase angle given by the charge at t = 0 dq i Q sin t dt dq q 0 We see that the charge on the capacitor satisfies an equation similar with that of a Simple Harmonic Oscillator with angular frequency dt Angular frequency ω = 1/LC Q Q i max i max di dt di dt max max q i di/dt φ = π/ T v v C L q C di L dt L q Qcos t T/ 3T/ φ = 0 T T i C +q q q Qcost t T t T t T i
Exercise 5: LC oscillations compared with a mechanical analog: Recall that in PHY181 we studied the oscillations of a SHO containing a mass m connected to a light spring of force constant k oscillating on a frictionless horizontal surface. a) Do you remember the SHO equation in this case? k m 0 x b) Find the oscillating quantities analogue to electric quantities oscillating in the LC circuit: LC: charge q current i change in current di/dt Spring: Problem: 5. LC oscillator: A power supply with emf ε is used to fully charge up a capacitor C. Then the capacitor is connected to an inductor L. a) What is the frequency and period of the LC circuit? b) Find the maximum charge, the maximum current and the maximum rate of change of current in the circuit. c) Write out the time dependency of the charge, current and rate of change of current considering t = 0 the first time when the capacitor holds only half of its maximum charge. d) Sketch the q vs t graph.
Electromagnetic Waves Maxwell s prediction and Hertz s confirmation Maxwell s electromagnetism modeled the coupling between electric and magnetic fields and predicted that varying magnetic and electric fields may continually create each other and propagate in the form of electromagnetic waves Ex: radio waves, visible light, X-rays etc. Maxwell obtained theoretically the speed of this propagating EM disturbance (speed of light) created by an accelerating charge Ex: A charge oscillating harmonically transmits a wave: the clip shows the wave transmitted transversally along a horizontal line of field Short time after Maxwell s prediction, Heinrich Hertz confirmed the electromagnetic waves experimentally using two LC circuits When in resonance, the circuits exchanged energy, and Hertz hypothesized the energy transfer was in the form of electromagnetic waves So, he confirmed Maxwell s theory by showing the waves existed and had all the properties of light waves such as experiencing reflection, refraction and diffraction
Electromagnetic Waves Produced using an Antenna When a charged particle is accelerated, it must radiate energy, such that, if currents change rapidly in a circuit, some energy is transmitted in the form of EM waves For instance, an alternating voltage applied to an antenna forces the electric charge in the antenna to oscillate and transmit an EM wave Ex: A simple EM-wave emitter Consider two rods connected to a source of alternating current: charges oscillate producing a polarization (a) and thus an electric field E As oscillations continue, the rods become less charged, the E-field near the charges decreases, while the field produced at t = 0 moves away from the rod (b) Then the charge polarization and the corresponding field reverse (c) t = 0 t = T/4 t = T/ t = T Notice that, due to the change of electric field, a magnetic field is generated perpendicular on the electric field E and, as the current changes, the magnetic field spreads out from the antenna in conjunction with the electric field The oscillatory correlated electric and magnetic fields are transmitted in space with speed of light c, and a frequency given by the frequency of the current oscillation in the antenna E E a) b) c) d) E E
Electromagnetic Waves Properties n conclusion, an electromagnetic wave is formed of oscillating electric and magnetic fields perpendicular to each other and perpendicular on the direction of propagation of the wave: so, the electromagnetic waves are transverse waves The electromagnetic wave produced by a charge oscillating harmonically produces at a large distance a sinusoidal wave, ˆ max cos, ˆ cos E x t ye kx t x t z kx t max The directions of the E and vectors are given by a right hand rule (call it #3): where EM-waves travel at the speed of light c which can be written in terms of the electric and magnetic properties of the medium of propagation, ε 0 and μ 0 Also, the theory predicts that the magnitudes of the E and fields are related by: k is the wave number, k = π/λ ω is the angular frequency, ω = π/t E max c max ce L max 0 0 max λ c 1 0 0
Electromagnetic Waves Energy and momentum. Poynting Vector Recalling the energy density carried by the electric and magnetic fields, we obtain the energy carried by an EM-wave in vacuum: 1 1 1 1 u 0E 0 0 0 0E E E 0 0 The elementary amount of energy transported by the wave front A with speed c per unit area per unit time is given by the energy contained in a volume Acdt 1 du 1 udv dx S u c A dt A dt dt 0 E c Def: The energy flow rate is the magnitude of a vector called Poynting vector with the direction given by the direction of wave propagation S 0 E 0 1 E J W S S m s m Therefore, the power imparted by the wave over a certain surface A can be calculated using P S da
Electromagnetic Waves Radiation Pressure Since EM-waves carry energy as they travel through space, this energy can be transferred to objects placed in their path Since the component fields are alternating, the energy carried by EM waves is shared equally by the electric and magnetic fields The average power P per unit area A is called intensity of the wave, given by the average of the Poynting vector over one cycle: Hence, EM-waves transport linear momentum as well as energy. The momentum delivered to a surface is 1. For complete absorption of energy U,. For complete reflection of energy U, Therefore, using Newton s nd law, we see that the force exerted by EM waves perpendicular onto surfaces is given by: F E E c c S max max max dp Adt c dp Adt c max 0 0 0 p p p A complete absorption: F dp c dt A complete reflection: F c
Problem: 6. Radiation pressure: An electromagnetic wave travels in vacuum vertically downward along z-direction with the perpendicular EM-fields in the x and y-directions, as in the figure. The beam strikes perpendicularly a perfectly reflective disk of radius r and mass m. a) Calculate the intensity of the beam knowing that the disk accelerates downward with an acceleration a. b) Show the directions and calculate the maximum amplitudes of the electric and magnetic fields. c) What should be the intensity of the beam required to levitate the disk? x z r y