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  1. 1. Ways to induce emf or current: Since Φ = BAcosθ, An AC supply of Vrms = 240 V provides the same power as a emf can be induced (1) by changing A to change Φ DC supply of constant V = 240 V. × × × × when the area of the Vp V pp V Ip loop inside B increases, Ip = , I pp = , I rms = rms , I rms = R R R 2 Physics Unit 4 Summary Sheets (09–12) × × × × the induced current I is as 2 Free download and print from I shown. Vp 1 2 Vp I p ©Copyright 2009 P = Pav = = I pR = or – × × × × 2R 2 2 + Magnetic field B (Unit: tesla (T); or weber per square metre V2 2 (wbm-2) ): × × × × P = Pav = rms = I rms R = Vrms I rms . + when a conductor moves R S In a power station the generator always rotates at the same × × × × across the magnetic field rate. If the power consumption by homes and offices is lines from a to b (change higher (lower), more (less) energy is required to maintain N S × × I× × in area enclosed by dotted the same rotation rate, 50 Hz in Australia. – lines), induced emf is Transformer: An electrical device that is used to change the × a × × b × shown as + –. voltage of an AC power supply without changing the power Aust to be delivered. N (2) by changing B to change Φ Working of a transformer: Alternating current at the when magnet moves closer primary (input) coil produces an alternating B inside the soft to the loop, magnetic field iron core. The secondary (output) coil is linked to the increases and the induced primary through the core, a changing B in the core results in N current I is as shown. a changing Φ in the secondary coil. According to Faraday’s I Law an emf is induced in the secondary coil (output). I Step-up transformer N S > N P ; step-down N S < N P . (3) by changing θ (either by rotating the loop or the magnet) to For an ideal (100% efficiency) transformer, PS = PP , × × × • • • change Φ × × × • • • NS V I VS I S = VP I P , = S = P . × × × I • • • N P VP IS I S N S N Power loss Ploss and voltage drop Vdrop occur when I I electricity is transmitted over a long distance by Use right-hand grip rule to find direction of B. transmission lines. Magnetic force on current-carrying wire: Direction of induced current and polarity of output terminals of F a generator are determined by Lenz’s Law: an induced current VA Transmission lines current I, resistance R VB < VA I in a conducting loop flows in a direction such that the magnetic × × × field of the induced current opposes the change in magnetic flux Vdrop = V A − VB = IR ; that produces it. The terminal that the induced current flows to × × × F = BIL is +, the other –. 2 Vdrop × × × ∆Φ Ploss = I 2 R = Vdrop I = . B Faraday’s Law: ξ av = n where n is the number of R ∆t Direction of F L (Right-hand slap rule) loops in the coil through which the flux passes. Ploss ∝ I 2 , power loss is greatly reduced by lowering I in 1 the lines; this can be achieved by increasing the voltage for ξ av ∝ n , ξ av ∝ ∆Φ , ξ av ∝ . ∆t transmission in order to deliver the same power, ∴step-up Magnetic force on current-carrying coil of n loops: AC generator (alternators): Alternating emf induced by a transformer is used at the power station end. At the Zero force rotating conducting coil (loop area A) in a magnetic field B is consumer end, step-down transformer is used to reduce made accessible with slip-rings connected to the terminals of voltage to 240 V. × × × × Load curve is a graph showing the demand of electric the coil, and the external circuit is connected to the rings via conducting brushes. power over a time period. Area under the curve represents L × × × × ξ the total consumption of electrical energy during the period. E = P∆t . θ Kilowatt-hour (kwh) is a unit of electrical energy. × × × × 1 kwh = 3.6 MJ r T = 360o Force into page Zero net torque If the slip-rings are replaced by a split-ring commutator, the Force out of page on coil device is a DC generator. Thermal oscillations of electrons in atoms give off Net clockwise torque electromagnetic radiation (visible light). In an incandescent on coil (view from above) ξ light bulb, the atoms in the filament are excited by heating, Refer to above diagrams, the two forces (each F = nBIL in T = 180o θ and they give off their excess energy as wave trains (with wide spectrum of wavelengths) of light, each lasts about and out of the page) exert a turning effect called torque 10-8s. The emitted light is the sum of such wave trains that (τ = rF) on the coil causing it to speed up its rotation in the The amplitude of the AC emf is called the peak voltage Vp . bear a random phase relation to each other and they are first quarter turn. In the second quarter turn the torque is in incoherent. Two light globes produce incoherent light, the opposite direction causing the coil to slow down. In a Vp ∝ n , Vp ∝ B , Vp ∝ A , Vp ∝ f . hence no interference pattern. Thomas Young simple DC motor, the direction of the coil current is DC motor has the same construction, and can be used demonstrated the wave nature of light with his double-slit reversed with a split-ring commutator every 180o turn experiment to obtain an interference pattern. He used when the net torque is zero. Hence the torque on the coil as a DC generator by turning the coil mechanically. Alternating emf can also be induced by rotating a permanent sunlight through a narrow slit as the light source and then remains in the same direction, allowing the coil in the motor through the double slits. Lights through the double slits are to keep turning in the same direction. magnet (or electromagnet) beside a coil. External circuit is connected directly to the terminals of the coil and no slip rings coherent because they are split from the same wave trains are required. from the single slit, ∴ there is an interference pattern. Laser Magnetic flux Φ = BAcosθ. Unit: weber (wb) is a very coherent source because it is monochromatic AC power supply delivered to homes and offices are generated by rotating an electromagnet between two connected coils at (single wavelength) and wave trains are emitted A, area of region enclosed by loop simultaneously, hence a very clear interference pattern. B f = 50Hz (T = 1 = 0.02 s = 20 ms). P Path difference f θ 0V ξ S1 pd = S2P – S1P Normal to region Bright & dark d fringes appear At t = 0 S2 on screen Electromagnetic induction is the generation of electricity S N L by changing magnetic flux. The generated current is called ξ (V) induced current I; the generated voltage is called induced Vp Vpp Constructive interference (bright): pd = 0λ, 1λ, 2λ, .... emf ξ. Destructive interference (dark): pd = 0.5λ, 1.5λ, 2.5λ, .... ∆Φ ξ 0 10 20 t (ms) Spacing between fringes increases when wavelength λ Magnitude of ξ av = , I= where ∆t time ∆t R Peak voltage Vp = 340 V, peak-to-peak voltage Vpp = 680 V, increases, screen distance L increases and/or slits separation taken for the change, R resistance of the loop. Vp d decreases. λred > λ green > λblue > λviolet . f = c . root-mean-square voltage Vrms = = 240 V. λ 2
  2. 2. Diffraction of light also demonstrates the wave These spectra are evidence for quantised atomic energy levels. Standing waves in closed resonant tube of length L: nature of light. Electrons move around a nucleus with discrete energies. When Overtones Harmonics λ f = v/λ Gap Diffraction pattern has an electron jumps from high to low energy level, it loses energy Fundamental first 4L/1 1(v/4L) width a bright central region in discrete amount equal to the difference between the two levels and results in emission of a photon of the same energy. First third 4L/3 3(v/4L) w between 2 dark fringes Second fifth 4L/5 5(v/4L) hf = EH – EL. de Broglie used the idea of standing matter waves to explain the quantised energy levels of an atom. The For closed tubes only odd harmonics exist. Obstacle Diffraction pattern has only matter waves that persist are those for which the Dynamic microphone: Sound moves the cone and the width a dark shadow of the circumference of the orbit is an integral multiple of λ. attached coil of wire in a magnetic field to and fro. obstacle with a bright Travelling sound wave through air is longitudinal because the Electromagnetic induction produces an emf (signal) at the w fringe at the centre air molecules oscillate parallel to the direction of propagation of terminals of the coil. Ribbon (or velocity) microphone: Air caused by constructive the sound wave. A sequence of high (compression) and low movement due to sound waves moves the metallic ribbon in interference of light (rarefaction) air pressure is generated and it propagates a magnetic field. Electromagnetic induction generates emf around the obstacle outwards from the source carrying the sound energy with it. between the ends of the ribbon. Condenser microphone: The back plate and the front metallic membrane form a Extent of diffraction ∝ λ . Significant effect when λ ≈ 1 . ∆p (at a particular time) ∆p (at a particular point) capacitor (charged with a battery). Sound waves cause the w w More diffraction when λ is longer and/or w is smaller. membrane to vibrate and change the spacing between the x t plate and the membrane. This causes the output voltage Photoelectric effect demonstrates particle-like nature of λ T (signal) to change. In electret-condenser microphone a light. Light Accel./ f = 1 , speed of sound v = λ = fλ . permanently charged electret material is used for the T T membrane, thus eliminating the need of a charging battery. Setup V retard. Crystal microphone: uses a thin strip of piezoelectric v( solid ) > v( water ) > v(hotair ) > v (coolair) . v is voltage crystal attached to a diaphragm that is sent into vibration by constant when sound travels in the same medium, sound waves, causing the crystal to deform and produce a I λ f voltage (signal). ∴ λ ∝ 1 and 2 = 1 . f is constant when sound travels A dynamic loudspeaker has the same basic construction as f λ1 f 2 I (Constant freq.) I (Constant int.) a dynamic microphone. The input signal changes the current H v2 λ2 in the coil and results in a varying magnetic force on the coil from a medium into another, ∴v ∝ λ and = . L intensity H L frequency v1 λ1 that is attached to the cone. Vo V V Enclosure formed by baffles: to prevent the sound from the Sound intensity I measures the amount of energy (J) arriving at back of the speaker cone cancelling the sound from the front retard accel retard accel a square metre of surface in a second. It is defined as because of destructive interference due to phase difference. Max EK E P Directional spread of sound waves (diffraction): Sound Metal 1 Metal 2 Same gradient for I= = , E energy received, A area exposed, ∆t time diffracts when it passes by the edge of a barrier. Refer to A∆t A both metals = h exposed. Unit: Js-1m-2 or Wm-2. For a small sound source in the diffraction of light. Extent of diffraction ∝ λ . w is the f (Planck’s constant) open, the sound energy spreads outwards spherically, w φ2 width of obstacle or opening. High pitch (high f, short λ) φ2 work function of metal 2 P 1 I b ra2 sound diffracts less than low pitch. I= , P is the power of source, ∴ I ∝ , = . Threshold frequencies fo1 and fo2 4πr 2 r 2 I a rb2 A loudspeaker is omni-directional, (i.e. it radiates sound φ = hfo , max EK = qVo When the distance r from the source is doubled, intensity I is a energy spherically in all directions) when λ > 4 , w is the Failure of the wave model to explain the photoelectric w quarter of the original value. effect According to the wave model, light is a continuous diameter of speaker cone. The higher the frequency the less wave and the intensity is related to its amplitude, which I omni-directional it becomes. Sound intensity level L = 10 × log10 in dB, measures the energy of the wave. Therefore an electron can 10−12 Frequency response of human ear: is most sensitive to absorb any amount of light energy, depending on the time sound of frequency 4000Hz, e.g. of the three sounds, 100Hz, ∆L 4 kHz and 10 kHz, at the same dB level at the ear, the 4kHz interval it is exposed to the light wave. The wave model If L −12 If failed to explain why (1) maximum kinetic energy remained ∆L = 10 × log10 . I = 10 10 , = 10 10 . will sound the loudest to the listener. To make 100Hz and the same when the intensity was changed; (2) maximum Ii Ii 10kHz the same loudness as 4 kHz, increase their dB level. kinetic energy changed with the frequency of light used; If Loudness is measured in phons. The loudness of a sound is (3)there was a threshold frequency for each metal used. When I is doubled, i.e. = 2 , ∆L = +3 dB. compared with the loudness of 1 kHz sound. The loudness Einstein’s interpretation of photoelectric effect-the Ii of a x dB 1 kHz sound is x phons. Sounds at different photon model: A beam of light is a stream of particles If frequencies, which are as loud as the x dB 1 kHz sound have 1 a loudness of x phons. The following graph shows a curve of called photons. Light of a single frequency f consists of When r is doubled, = , ∆L = − 6 dB. photons of the same energy E = hf = hc/λ . Ii 4 equal loudness (30 phons) for different frequencies. There are more photons in a more intense beam, hence Threshold of hearing 10-12 Wm-2 0 dB L(dB) higher current. When photons strike a metal, some will be Normal conversation 10-6 60 80 Above curve, absorbed by the electrons in the metal. To have louder than 30 phons 30-phon curve photoelectrons emitted, the energy of each photon must be Car alarm 1 m away 10-2 100 high enough for the electrons to overcome the bonding Threshold of pain 1 120 30 Softer softer below curve energy (i.e. the work function φ). As the photons penetrate Jet engine 30 m away 102 140 into the metal they collide with other electrons before they 0 20 1k 15 k f (Hz) After reflection, f, λ and v remain the same. When the forward are absorbed. Each collision lowers the photon frequency Frequency response curve of a microphone: is a graph of and the reflected travelling waves superpose each other, a (energy) slightly, the Compton effect. ∴ electrons at the output intensity level versus frequency for a constant input. standing wave (a sequence of loud and soft sound at fixed surface escape with higher (max) kinetic energy than those Zero dB is assigned to 1 kHz sound as the reference level. positions quarter of a wavelength apart) is formed between the inside metal, max EK = hf – φ for surface electrons. L(dB) source and the wall. Pressure antinodes (max fluctuation in air The Compton effect and photon momentum: The particle 10 pressure) give loud sound and pressure nodes (min fluctuation) nature of light was further supported by the Compton effect. 0 give soft sound. Photon momentum p = E = h . Wall – 10 c λ Loud- loud soft The two models (wave and particle) of light appear to be 50 150 1 k 4k 6k f (Hz) Speaker L S L S L S L inconsistent with each other but both have been shown to be This graph indicates that the microphone responds equally ∆p well to frequencies between 150 and 4 kHz, more sensitive valid depending on the circumstances. This dual nature of light is known as wave-particle duality. to over 4 kHz, less sensitive to below 150 Hz. x Wave nature of matter: de Broglie proposed that a moving Frequency response of multi-speaker system: material particle also has wave-particle duality. Wavelength λ L(dB) woofer tweeter of particle is related to its momentum (like a photon). Every object has its own natural frequencies of vibration. If an Fairly flat between energy source at one of these frequencies interacts with the h h h 20 and 20 kHz de Broglie λ= = = . These equations are object, the latter will be set into vibration, i.e. a standing wave p mv 2mEk is formed. The object is in resonance. The natural frequencies of vibration are called resonant frequencies. 0 20 20 k f (Hz) valid when λ in m, m in kg, v in ms-1, EK in J, h in Js. A single loudspeaker on its own (e.g. the woofer or tweeter) The diffraction of electrons from the surface of a metal Standing waves in a stretched string of length L: tends to ‘colour’ the sound it produces, i.e. some crystal confirmed the wave nature of matter. Overtones Harmonics λ f = v/λ frequencies are louder than others due to resonance. An An electron with the same λ as a photon has the same Fundamental first 2L/1 1(v/2L) ideal loudspeaker system would need to have the same momentum as the photon, p = h/λ. First second 2L/2 2(v/2L) loudness at all frequencies, i.e. a fairly flat response curve. When a gas or metal vapour is heated, the gas or vapour Second third 2L/3 3(v/2L) Some loudspeaker enclosures have tubes (called ports) put glows and emits a characteristic diffraction pattern (obtained in them. Size and depth of ports can be changed to absorb Note: v is the speed of travelling wave in the string. with a diffraction grating) called an emission spectrum. sound of particular frequencies to produce a flat response. When sunlight passes through a gas/vapour, some dark lines Standing waves in open resonant tube of length L: The appear in its spectrum called absorption spectrum, caused vibration of the air column in the tube forms a standing wave. by the absorption of certain wavelengths of sunlight by the Has the same pattern of harmonics as strings but v is the speed atoms or molecules in the gas/vapour. of travelling sound wave in the tube.