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Electromagnetic Induction And Alternating Currents

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Electromagnetic Induction and Alternating Currents PDF Notes, Important Questions and Formulas

Alternating Current

Chapter Cover: Section A - Average and RMS value of Alternating Current

 1. ALTERNATING CURRENT

Until now, we have studied only circuits with direct current (dc) which flows only in one direction (as shown in figure I and II). The primary source of emf in such circuit is a battery. When a resistance is connected across the terminals of the battery, a current is established in the circuits, which flows in a unique direction from the positive terminal to the Negative terminal via the external resistance.

 

                                                                                 Fig-I

 

Fig-II

But most of the electric power generated and used in the world is in the form of alternating current (ac), the magnitude of which changes continuously with time and direction is reversed periodically (as shown in figure III & IV)and it is given by

i=i0sin (ωt + ϕ)

Here i is instantaneous value of current i.e., the magnitude of current at any instant of time and i0 is the maximum value of current which is called peak current amplitude and the current repeats its value after each time interval T = begin mathsize 12px style fraction numerator 2 straight pi over denominator straight omega end fraction end style as shown in figure, This time interval is called the time of period and ω is angular frequency which is equal to 2π times of frequency f.

 

 

The current is positive for half the time period and negative for remaining half period. It means that the direction of current is reversed after each half time period. The frequency of ac in India is 50 Hz.

An alternating voltage is given by

V=V0 sin (ωt +ϕ)

It also varies alternatively as shown in the figure (b), where V is instantaneous voltage and V0 is peak voltage. It is produced by ac generator also called as ac dynamo

 

Ac Circuit: An ac circuit consists of circuit element i.e., resistor, capacitor, inductor or any combination of these and a generator that provides the alternating current as shown in figure. The ac source is represented by symbol      in the circuit.

 

2. AVERAGE AND RMS VALUE OF ALTERNATING CURRENT

2.1 Average current (mean current) As we know an alternating current is given by

                                         i= i0 sin (ωt + ϕ)           …… (1)

The mean or the average value of ac over any time T is given by

        iavg = begin mathsize 12px style fraction numerator begin display style integral from 0 to straight T of idt end style over denominator begin display style integral from 0 to straight T of dt end style end fraction end style

Using equation (1)

iavg begin mathsize 12px style fraction numerator begin display style integral from 0 to straight T of straight i subscript 0 sin left parenthesis ωt plus straight ϕ right parenthesis dt end style over denominator begin display style integral from 0 to straight T of dt end style end fraction end style

In one complete cycle, the average current

iavgbegin mathsize 12px style negative straight i subscript 0 over straight T left square bracket fraction numerator cos left parenthesis ωt plus straight ϕ right parenthesis over denominator straight omega end fraction right square bracket subscript 0 superscript straight T end style

     = begin mathsize 12px style negative straight i subscript 0 over straight T left square bracket fraction numerator cos left parenthesis ωt plus straight ϕ right parenthesis minus cosϕ over denominator straight omega end fraction right square bracket end style

     = begin mathsize 12px style negative straight i subscript 0 over straight T left square bracket fraction numerator Cos left parenthesis 2 straight pi plus straight ϕ right parenthesis cosϕ over denominator straight omega end fraction right square bracket equals 0 text  (as  end text straight omega text T=2 end text straight pi text ) end text end style

Since ac is positive during the first half cycle and negative during the other half cycle so iavg will be zero for long time also. Hence the dc instrument will indicate zero deflection when connected to a branch carrying ac current. So it defined for either positive half cycle or negative half cycle.

iavg = 

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                                                                          Electromagnetic Induction

Chapter Cover: Section A - Flux, Faraday’s law, Lenz’s law

1. MAGNETIC FLUX

  • Consider a closed curve enclosing an area A (as shown in the figure). Let there be a uniform magnetic field B in that region. The magnetic flux through the area A is given by



    ϕ=B.A                  …. (1)
    =BA cos Ѳ

    Where Ѳ is the angle which the vector B makes

    With the normal to the surface. If B is perpendicular to A, then the flux through the closed area A is zero. SI unit of magnetic flux is weber (Wb).

Note

  • Area vector is ⊥ to the surface
  • For open surface choose one direction as the area vector direction and stick to it for the whole problem.
  • For close surfaces outward normal is taken as area vector direction
  • Flux is basically count of number of lines crossing a surface.
  • 0 because magnetic field lines exists in closed loop.

 
2. FARADAY’S LAE OF ELECTROMAGENTIC INDUCTION

  • Whenever the flux of magnetic field the area bounded by a closed conducting loop changes, area bounded by a closed conducting loop changes, an emf is given by

    ɛ=- begin mathsize 12px style dϕ over dt end style            …… (ii)

    Where ϕ=∫B.dA the flux of magnetic field through the area.

    The emf so produced drives an electric current through the loop. If the resistance of the loop is R, then the current

            begin mathsize 12px style straight i equals straight epsilon over straight R equals negative 1 over straight R dϕ over dt end style   …. (iii)

3. LENZ’S LAW

  • The effect of the induced emf is such as to oppose the change in flux that produces it.



    In figure (a & b) as the magnet approaches the loop, the flux through the loop increases. The induced current sets up an induced magnetic field Bind whose flux opposes this change. The direction of Bind  is opposite to that of external field Bext  due to the magnet.

    In figure (c & d) the efflux through the loop decreases as the magnet moves away from the loop, the flux due to the induced magnetic field tries to maintain the flux through the loop. The direction of Bind  is same as that of B due to magnet.

    Lenz’s law is closely related to the law of conservation of energy and is actually a consequence of this general law of nature. As the north pole of the magnet moves towards the loop an induced current is produced. This opposes the motion of N-pole of the bar magnet. Thus, in order to move the magnet toward the loop with a constant velocity an external force is to be applied. The work done by this external force gets transformed into electric energy, which induces current in the loop. There is another alternative way to find the direction of current inside the loop which is described below. Figure shows a conducting loop placed near a long, straight wire carrying a current i as shown. If the current increases continuously, then there will be an emf induced inside the loop. Due to this induced emf, an electric current is induced. To determine the direction of current inside the loop we put an arrow as shown. The right hand thumb rule shows that the normal to the loop is going into the plane. Again the same rule shows that the magnetic field at the site of the loop is also going into the plane of the diagram.




    Thus B and dA are in the same direction. Therefore ∫B.dA is positive if I increases, the magnitude of flux ϕ increases. Since magnetic flux ϕ is positive and its magnitude increases, begin mathsize 12px style dϕ over dt end style is positive. Thus ɛ is negative and hence the current is negative. Thus the current induced is opposite, to that of arrow.


    Brain Teaser

    Two identical coaxial circular loops carry equal currents circulating in the same direction. What will happen to the current in each loop if the loops approach each other?


4. CALCULATION OF INDUCED EMF

  • As we know that magnetic flux (ϕ) linked with a closed conducting loop = BA cos (Ѳ) where B is the strength of the magnetic field, A is the magnitude of the area vector and Ѳ is the angle between magnetic field vector and area vector. Hence flux will be affected by change in any of them, which is discussed in the next page.

 

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