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15 - Electromagnetic Induction

Electromagnetic Induction

  • Orested discovered magnetic field due to current carrying coil.
  • Faraday discovered electromagnetic induction (production of induced EMF and induced current by time rate of change of magnetic flux)
  • The induced current depends upon the speed with which conductor moves and upon the resistance of loop.
  • The induced current can be increased by:
    • using stronger magnetic field.
    • moving the loop faster
    • Replacing the loop by a coil of many turns

Info

As magnetic flux is $$ \phi=BA\cos\theta $$ So the change in flux can be caused as:

  • Change in magnetic field strength
  • Change in projected area.
  • Change of orientation of coil in field.

Faraday's Law

First Law

Induced EMF lasts till the flus remains changing.

Second Law

Induced EMF is directly proportional to the time rate of change of magnetic flux i.e $$ \epsilon=-N\frac{\Delta\phi_B}{\Delta t} $$

  • Direction of induced f is given by Lenz's law, which states

Lenz's Law

EMF is induced in such a way that it opposes its own cause of generation.

  • The mechanism of generation of induced EMF is such that energy fed to change flux, changes to magnetic force, which moves free electrons to cause the current and hence induced EMF.
  • Following ways can change magnetic flux / can produce induced EMF.
    • Motion of magnet
    • Motion of coil
    • Relative motion of coil and magnet
    • Change of current
  • If there does not exist Lenz's law then perpetual motion machine of 1st kind was possible. Thus the process will be self-perpetuating.

Motional EMF

  • The EMF induced by the motion of a conductor across a magnetic field is called motional EMF. $$ \epsilon_{(motional)}=-vBL $$ where \(v\) is speed of conductor, \(L\) is length of conductor

  • If there is some orientation of the coil to magnetic field then $$ \epsilon_{(motional)}=-vBL\sin\theta $$ where \(\theta\) is the angle between \(v\) & \(B\).

  • The unit of motional EMF is \(volt\).
  • Magnetic field strength is considered to be uniform while deriving the above expression.

Mutual Induction & Self Induction

  • Mutual Induction is phenomenon in which a changing current in one coil induces and EMF in another coil. $$ \epsilon_{sec}=-M\frac{\Delta I_p}{\Delta t} $$ Where \(M=N_s\phi_s/I_p\) is called Mutual Inductance.

Mutual Inductance

The ratio of back EMF in the secondary to the reate of change of current in primary coil. $$ M=-\frac{\epsilon}{\Delta I_p/\Delta t} $$

  • Self Induction is a phenomenon of generation of induced EMF in a coil due to change of current in itself. $$ \epsilon_{self}=-L\frac{\Delta I}{\Delta t} $$ where \(L\) is called self Induction. Also \(N\phi=LI\)

Self Inductance

The ration of back EMF to rate of change of current in the same coil $$ L=\frac{-\epsilon_{\text{self}}}{\Delta I/\Delta t} $$

  • SI unit of mutual and self inductance is \(\text{Henry} (H)\).

    • \(M=1H\) if \(\epsilon=1V\) and \(\Delta I_p/\Delta t=1A/s^{-1}\).
      Mutual Inductance is said to be \(1H\) if a current change in primary coil at rate of \(1As^{-1}\) produces EMF of \(1V\) in secondary.
    • \(L=1H\) if \(\epsilon=1V\) and \(\Delta I_p/\Delta t=1A/s^{-1}\).
      Mutual Inductance is said to be \(1H\), if current of changing rate of \(1As^{-1}\) produces an EMF of \(1V\) in the coil itself.

Energy Stored in an Inductor

Inductor is simply an insulated coil that offers very small resistance to DC current but offers high resistance to AC current.

  • In case of capacitor, energy is stored in electric field between the plates. Similarly, energy is stored in magnetic field of the inductor.
  • Work done by the battery to move charges against the induced EMF is $$ W=\frac{1}{2}LI^2 $$.

  • This work is stored as potential energy in the inductor as $$ U=\frac{1}{2}LI^2 $$ Also \(L=\mu_o n^2 Al\) $$ U=\frac{1}{2}(\mu_o n^2 Al)I^2 $$ As, \(B=\mu_o nl\) Then, $$ U=\frac{1}{2} B^2 \frac{Al}{\mu_o} $$ As, \(Al=\text{Volume}\) Then energy density is $$ =\frac{U}{Al}= \frac{IB^2}{2\mu_o} $$ OR $$ =\frac{1}{2}\frac{B^2}{\mu_o} \implies U\propto B^2 $$

  • Inductor in an AC circuit always resists the abrupt change in current.

Info

The density of energy storage in an inductor is directly proportional to the square of the magnetic field strength provided all the other factors remains constant.

Inductance in Series & Parallel

Series

If two coils of of inductance \(L_1\) and \(L_2\) are connected in series then: $$ \epsilon=\epsilon_1+\epsilon_2\;\;\implies\;\; L_s\frac{\Delta I}{\Delta t}= L_1\frac{\Delta I_1}{\Delta t} + L_2\frac{\Delta I_2}{\Delta t} $$ But in series current remains same \(I=I_1=I_2\) Therefore $$ L_s = L_1+L_2 $$

Parallel

In Parallel combination current divides so, $$ I=I_1+I_2\;\; \implies \;\; \frac{\Delta I}{\Delta t} = \frac{\Delta I_1}{\Delta t} + \frac{\Delta I_2}{\Delta t} $$ As, \(\frac{\Delta I}{\Delta t}=-\frac{\epsilon}{L}\) so, $$ \frac{\epsilon}{L_p}= \frac{\epsilon_1}{L_1} + \frac{\epsilon_2}{L_2} $$ Thus, \(\epsilon=\epsilon_1=\epsilon_2\) Thus, $$ \frac{1}{L_p}= \frac{1}{L_1} +\frac{1}{L_2} \;\; \implies \;\; L_p=\frac{L_1L_2}{L_1+L_2}$$

AC & DC Generator

Current generator is a device, which converts mechanical energy into electrical energy in the presence of magnetic field.

Principle

EMF is induced across a coil rotating in a magnetic field due to change of magnetic flux.

Main Parts

  • Pole Pieces (U-Shape magnet) with concave poles.
  • Armature (assembly of coil on iron cylinder)
  • Slip rings (as connector)
  • Carbon brush (external supply)

Working

  • EMF is induced by the side of loop intersecting the magnetic field.

  • Total EMF for N number of loops is given as: $$ \epsilon=N(2vB\sin \theta) \;\;\; \text{also} \;\;\; \epsilon=N\omega AB sin \omega t$$ $$ \epsilon_{max}=N\omega AB \;\;\; \text{when} \;\;\; \theta = 90\degree $$ So, $$ \epsilon=\epsilon_{max}\sin \omega t $$ if \(\omega=2\pi f\) then $$ \epsilon=\epsilon_{max} \sin 2\pi ft $$ In terms of potential difference, \(V=V_o\sin 2\pi ft\)
    In terms of current \(I=I_o\sin 2\pi ft\)
    where f is the frequency with which the direction of current is changing.

  • In Pakistan \(f=50Hz\), it means 50 times in a second direction is changed.

  • In DC generator split rings or commutators, inverted by William Sturgeon are used in place of slip rings.
  • DC generator produce uni-directional pulsating DC, not pure DC.
  • For pure Dc many coils are wound around cylindrical core.

Back Motor Effect in Generator

  • The device in electric circuit that consumes electric energy is known as load.
  • The greater the load, the larger the current is supplied by the generator.
  • When the circuit is closed, a current is drawn through the coil. The magnetic field exerts force on the current carrying coil. This force produces a counter torque that opposes the rotational motion of coil. This effect is called back motor effect of generator.
  • According to law of conservation of energy, the energy consumed by the load must come from the energy source used to drive turbines.
  • The larger the current drawn by the load, the greater is the counter torque produced.

DC Motor

A motor is a device which converts electrical energy into mechanical energy. * A generator running backward is called motor. * Principle of motor is that whenever a current carrying coil is placed in a magnetic field it experiences a couple of forces, which causes torque.

Back EMF Effect in Motor

  • The self induced emf (\(\epsilon\)) in a motor opposes the voltage \(v\) running the motor. Thus induced emf is called the back emf of motor.
  • The magnitude of back emf increases with the speed of motor.
  • \(V\) & \(\epsilon\) are in opposite polarity.

  • Current flowing in motor is: $$ I=\frac{(V-\epsilon)}{R} \;\; \implies \;\; V=\epsilon + IR $$

  • As the motor speeds up, the back emf increases and the current become smaller and smaller. So current should be kept sufficient to provide torque.

  • If motor is overloaded, the back emf decreases and allow the motor to draw more current.
  • If motor is overloaded beyond its limits, the current could be so high that it may burn the motor coil.

Transformer

Transformer is an electromagnetic device used to step up or step down AC voltage or AC current, not DC.

  • Principle of transformer is the mutual induction.
  • Coil of transformer where input is applied is called primary coil while other where output is obtained is called secondary coil.
  • Core consist of laminated punched sheets of sotf iron on which coil is wound.
  • Lamination of core is done to reduce eddy current losses (eddy current is and induced current produced in a core).
  • Transformer can not step up or step down the energy or power.
  • Transformer consumes very little part of input power so it can kept running hours without considerable loss of power. $$ \frac{V_s}{V_p} =\frac{N_s}{N_p}= \frac{I_p}{I_s} $$

Info

Transformer is one of the static devices in the electrical appliances. So there is no loss of energy in friction. Hence they can have higher efficiencies. We can achieve up to 95% efficient transformer.

Energy Losses

  • Copper Losses in winding: Due to resistance of the winding of primary and secondary coil, some electrical energy is always lost in form of heat energy.
  • Flux Losses: The coupling of the primary and secondary is never perfect and whole of magnetic flux produced in the primary coil does not link the secondary coil. This results in some energy loss.
  • Iron losses in core: Iron losses are of two types:
    1. Eddy Current Loss: Due to the periodically varying nature of AC supplied in primary, the flux associated with core changes and therefore, eddy currents are induced in it. Eddy currents induced in the core are undesirable as they heat the core and result in energy loss. To minimize the eddy current losses, core is laminated.
    2. Hysterisis Loss: The alternating current flowing through the coils magnetized and demagnitizes the iron core again and again. therefore, during each cycle of magnetizatio, same energy lost due to hysterisis. To minimize this loss we choose material of core of smaller hysterisis loss generally soft iron.

Efficiency of Transformer

Ideal transformer efficiency is 100% or 1 but in actual tranformer output power is always less than inout power, so efficincy also always less than 100%. In general efficiency of transformer is very high and of the order of 90%.

  • Efficiency is given by: $$ \eta =\frac{\text{Output Power}}{\text{Input Power}}\times 100 $$
  • In terms of secondary and primary voltage and current $$ \eta = \frac{V_sI_s}{V_pI_p} $$
  • As, \(\text{Imput}=\text{Output} + \text{Losses}\) so, $$ \eta =\frac{\text{Output Power}}{\text{Output Power} + \text{Losses}} $$

Applications

  • They are used in power distribution in a safe from power stations to the utility stations.
  • The principle of transmission is the that we reduce the value of current via increasing the transmission voltage so saving the cost of copper as less think wire can full fill the purpose due to small currents.
  • Transformer also provide a degree of control of electricity distribution to a town.

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