Title 18. Electromagnetic Induction and Alternating Currents.pdf ✅

Electromagnetic Induction and Alternating CurrentsElectromagnetic Induction 393 18 and Alternating Currents QUICK LOOK EMF is induced in a conductor only if the magnetic flux changes. In S.I. system unit of magnetic flux is weber. If the plane of the coil makes an angle θ with magnetic field, the flux linkedφ θ = ° − BAcos(90 ) Magnetic flux linked with a coil of area A in magnetic field B φ α = = BACos B A. weber Figure: 18.1 Figure: 18.2 Faraday performs an experiment with a magnet and coil; he moved a permanent magnet in and out of a coil or a single loop of wire it induced an Electro Motive Force or emf, in other words a Voltage, and therefore a current was produced. During this experiment, he found how emf is induced in the coil when flux linked with it changes. Lenz’s laws: induced e.m.f. N. ; t φ ε ∆ = − ∆ the minus sign denotes Lenz’s law that it opposes the virtue of which it’s due. Where φ = magnetic flux linked with one turn. Current induced . N i R R t ε φ∆ = = ∆ Charge induced N N( ) Q R R R ∆ ∆ ∆Φ φ φ = = = Where Φ = Nφ is effective flux linkage? EMF induced in a moving rod sin B vl Bvl n ε θ = = Where sin B B n = θ component of magnetic field normal to velocityθ = angle between velocity v and field B. all quantities B, v and l are mutually perpendicular. Figure: 18.3 Figure: 18.4 EMF induced in a rod of length l rotating in a uniform magnetic field 2 2 B r ω ε = (ω = angular velocity) EMF induced between centre end rim of a disc rotating in a uniform magnetic field 2 2 B r ω ε = (r = radius of disc) Figure: 18.5 ω Rotating disc Current direction Copper conductor Uniform magnetic field Rotational direction of disc Z y x I B F ε I υ L × × × × × × × × × × × × × × × × × × × × × × × × × × × × × × × × × × × × × × × × × × × × × × × × × × × × × × × × × × × × × × Induced current Motional emf= vBL Consider a loop of wire moving with velocity v into a stationary magnetic field B out E=qvB down v v FL qvBL emf vBL q = = emf vBL = B out L W + − v B (a) Coil of area A with N Turns Magnetic field away from viewer Induced current (b) Current Force Current Field Thumb Motion Magnetic field lines Coil Coil Normal to plane of the coil θ = 90°, α = 0° α = 0°,θ = 90° α 90° N S
394 Quick Revision NCERT-PHYSICS Figure: 18. 6 EMF induced in a coil rotating in a uniform magnetic field ε ω = NVBA t (sinusoidal) Figure: 18.7 Figure: 18.8 Current in a coil of motor E e i R − = where E = applied e.m.f.; e is the back e.m.f and R = coil resistance. When a magnet falls vertically in a ring placed horizontally, current induced opposes the motion of magnet and acceleration of falling magnet < g. Figure: 18. 9 When a magnet is filling vertically in a ring having a cut and placed horizontally, the e.m.f. is induced in the ring but current is not induced, so acceleration of falling magnet = g. When a magnet falls in a long metallic cylinder, the acceleration of falling magnet decreases and finally magnet attains a constant velocity. If L is self-inductance, then effective flux linked θ = Li Induced e.m.f . di L dt ε = − Work done in establishing the current in a circuit form 0 to i amp 1 2 2 W Li = Self-inductance of a plane circular coil of radius R 2 2 0 0 2 2 N A N R L R R μ μ π = = Self-inductance acts as electrical inertia. Self-inductance of a coil 2 L N ∝ (where N = number of turns in coil) If a ferromagnetic substance is placed inside a coil its self- inductance increases appreciably. Self-inductance of a solenoid of length l, total number of turns N is 2 0N A L l μ = Energy of a coil of self-inductance L is 1 2 2 U Li = (magnetic energy) Magnetic energy in a magnetic field (B) 2 2 m B U μ = × (volume) Joule Magnetic energy density 2 2 m B υ μ = × Joule/m3 Mutual Inductance: Effective flux linked with secondary, Φ =2 1 Mi EMF induced in secondary 1 2 . i M t ε ∆ = − ∆ Figure: 18.10 Mutual inductance of two plane circular coils 2 0 1 2 2 1 2 N N R M R μ π = Mutual inductance of solenoid-circular coil system 2 0 1 2 2 N N R M t μ π = where N1 = number of turns in solenoid of length l. N2 R2 l2 N1 R1 l1 l1 + l1 = l S S N N A < g A = g Charge form 45°-90° Charge form 0°-45° 0° 45° 180° 360° Most rapid Rate of change at Zero amplitudes Least rapid Rate of change at Pack amplitudes φ = × B Aeff Eemf × × × × × × × × B r × × × × × × × × × × × × ω
Electromagnetic Induction and Alternating Currents 395 Combination of Inductance Series: If two coils of self-inductances L1 and L2 having mutual inductance are in series and are far from each other, so that the mutual induction between them is negligible, then net self inductance L L L S = +1 2 Figure: 18.11 When they are situated close to each other, then net inductance 1 2 2 L L L M S = + ± L L L L L 1 2 3 + + + + = ... n eq Parallel : If two coils of self-inductances L1 and L2 having mutual inductance are connected in parallel and are far from each other, then net inductance L is 1 2 1 1 1 L L L P = + ⇒ 1 2 1 2 P L L L L L = + Figure: 18.12 When they are situated close to each other, then MLL MLL LP 21 2 2 21 ±+ − = 1 2 2 1 1 1 1 1 L L L L L eq n = + + − − − + Relation between M, L1 and L2: For two magnetically coupled coils 1 2 M k L L = ; where k − coefficient of coupling or coupling factor which is defined as Magnetic flux linked in secondary ; 0 1 Magnetic flux linked in primary k k = ≤ ≤ Figure: 18.13 In a dynamo or electric generator, mechanical energy is converted into electrical energy. In electric motor (e.g., fan) electrical energy is converted into mechanical energy. In a motor induced e.m.f. is larger at “on” and “off”. In cores of armature coils of dynamo and motor, a soft iron laminated core is used to reduce energy loss caused due to eddy currents. Transformer: One of the main reasons that we use alternating ac voltages and currents in our homes and workplace’s is that it can be easily generated at a convenient voltage, transformed into a much higher voltage and then distributed around the country using a national grid of pylons and cables over very long distances. The reason for transforming the voltage is that a higher distribution voltage implies lower currents and therefore lower losses along the grid. It is a device which raises or lowers the voltage in ac circuits through mutual induction. It consists of two coils wound on the same core. The alternating current passing through the primary creates a continuously changing flux through the core. This changing flux induces an alternating emf in the secondary. Figure: 18.14 Transformer works on ac only and never on dc; The flux per turn of each coil must be same i.e., ; φ φ S P = S P d d dt dt φ φ − = − If Np = number of turns in primary, Ns = number of turns in secondary,Vp = applied (input) voltage to primary, Vs = Voltage across secondary (load voltage or output), p e = induced emf in primary; s e = induced emf in secondary, φ = flux linked with primary as well as secondary, p i = current in primary; s i = current in secondary (or load current) As in an ideal transformer there is no loss of power i.e. P P out in = So V i V i S S P P = and , V e P P ≈ . V e S S ≈ Hence ; s s s p p p p s e N V i k k e N V i = = = = = Transformation ratio (or turn ratio) Laminated sheets Iron core Source Load ~ Input Output P S P Air gap S P S L1 L2 L3 L2 Ln
396 Quick Revision NCERT-PHYSICS Efficiency of transformer ( ) η : Efficiency is defined as the ratio of output power and input power i.e. % 100 100 out S S in P P P V i P V i η = × = × For an ideal transformer out in p P = so η = 100% (But efficiency of practical transformer lies between 70% – 90%) For practical transformer P P P in out losses = + So ( ) 100 100 100 ( ) out out in L in out L in P P P P P P P P η − = × = × = × + Figure: 18.15 Alternating Current Some graphical representation for alternating quantities Figure: 18.16 Table 18.1: Alternating Current and Direct Current Alternating Current (AC) Direct Current (DC) Magnitude and direction both varies with time (Pulsating dc) (Constant dc) Shows heating effect only Shows heating effect, chemical effect and magnetic effect of current. RC Circuit (Growth and Decay) Figure: 18.17 Figure: 18.18 Equation of growth of charge is ( ) / 0 1 t RC q q e − = − Equation of decay of charge, / 0 t RC q q e − = In RC circuit, initial charge on capacitor = 0 and the final charge on capacitor in RC circuit is 0 q CV = In RC circuit, steady state current = 0 and initial charging current in RC circuit is 0 V i R = RL Circuit (Growth and Decay) Figure: 18.19 Figure: 18.20 ( ) Rt L/ V V e L − = ( ) / 1 V Rt L I e R − = − Steady state value τ = L / I 2τ 3τ 4τ 5τ t 37% VL 63% I Imax Transient Time V Switch Battery or D.C. Source VL L R I Inductor Resistor Imax V t RC / I e R − = / 1 t RC Q CV e− = −     Qmax RC Time Charge on capacitor Charging current CV Charging current VC R Charge on capacitor 2RC 3RC 4RC V VC i Switch Resistor, R Capacitor, C Battery or D.C. Source Battery or D.C. Source E or V ~ ω = 2π f (rad/s) f Hz Rectangular + – t i or V + ac super imposed on dc t i or V Sinusoidal + – t i or V Triangular + – t i or V Power Plant Generates Electricity High Voltage Distribution Lines Carry Electricity To Houses Transformer Steps Up Voltage For Transmission Neighborhood Transformer Steps Down Voltage Transformers On Poles Step Down Electricity Before It Enters Houses Voltage Medium Voltage Household Voltage ac Rectifer dc dc Inverter ac