Electro Magnetic Induction

ElectroMagnetic Induction:

A charged body is capable of producing electric charge in a neighbouring conductor. The phenomenon of induction of electricity due to electricity is called electric induction. A magnet is capable of producing magnetism in a neighbouring magnetic substance. This phenomenon is production of magnetism due to magnetism is called magnetic induction. A current flowing through a wire produces a magnetic field around itself. This phenomenon of production of magnetism due to electricity is called magnetic effect of currents. The production of electricity due to magnetism is called electro-magnetic induction. 

 Magnetic Flux:

'Flux' is a word used in a study of the quantity of certain fluids following across any area. Magnetic flux deals with the study of the number of lines of force of magnetic field crossing a certain area. 

Consider an area 'a' placed in a magnetic field having magnetic induction 'B'. Let the area be inclined to the direction of 'B' at an angle 'θ' (Fig. 1). 

Fig. 1. Lines of force of magnetic field cutting through an area. 

The area, in vector notation, can be represented by a vector directed along the normal to the area and having a length proportional to the magnitude of the area. Magnetic flux 'ΦB' through area 'a' is given by

    ΦB  = B. A = BA cos θ = A (B cos θ) 

But, B cos θ = component of B perpendicular to the area 'a'. 

Magnetic flux linked with a surface, is defined as the product of area and the component of 'B' perpendicular to the area. 

Case 1. If θ = 90°, θ = 0

When angle between 'B' and the normal to the surface is 90° 'B' will be parallel to the surface (Fig. 2).

Fig. 2. Surface held parallel to lines of force. 

∴     ΦB   = BA × 0 = 0.

No magnetic flux is linked with the surface when the field is parallel to the surface. 

Case 2. If θ = 0° , cos θ = 1.

In this case 'B' is perpendicular to the surface (Fig. 3).

        (ΦB) = B × A × 1 = BA. 

Magnetic flux linked with a surface is maximum when area is held perpendicular to the direction of field. 

     Since      B = μH

Where 'μ' is the permeability of the medium and H is magnetic intensity, i.e., the number of lines of force per unit area in free space. 

     (dΦB)max = μAH

In case the magnetic field is not uniform or the surface is not a plane one, the magnetic flux ΦB linked with a surface can be obtained as follows. 

Let '(dΦB)' be the magnetic flux linked with a small area dA of the surface. The area is so small that the field can be considered to be the same over that area. 

            dΦB = B. dA = B. n̂dA

where n̂ is a unit vector alonɡ the outward drawn normal to the surface at that point.

∴      ΦB = B0∫ dΦB = S∫ B. dA

or ΦB = S∫ B. n̂dA

Thus, magnetic flux linked with a surface in a magnetic field is defined as the surface integral of the magnetic flux density over that surface. 

Any line of force cutting an electric circuit more than once has its effect multiplied by the number of times it cuts the circuit. 

One line cutting an electric circuit five times [Fig. 4(i)] is equivalent to 5 lines cutting the electric circuit once [Fig. 4(ii)]. In both the circuits magnetic flux is same. 

Fig. 4. Magnetic flux through a coil. 

If a line of force cuts the circuit n-times, i.e., it passes through a coil having n turns, total magnetic flux linked with it is, 

              ΦB = BA × n

or ΦB = μ nAH

Units of Magnetic flux

(i) S.I. unit. 

When an area 'a' is held perpendicular to the direction of lines of force, magnetic flux linked with it is

              ΦB = BA

If      B = 1 tesla, A = 1 m², ΦB = 1 × 1 weber. 

Weber. Magnetic flux linked with an area of 1 m² held normal to the direction of lines of force of a magnetic field of strength 1tesla is called 1 weber. 

(ii) C.G.S. unit

If  B = 1 gauss,        A = 1 cm²

                                 ΦB = 1 × 1 = 1 maxwell. 

Maxwell. Magnetic flux linked with an area of 1 cm² held normal to the direction of lines of force of a magnetic field of strength 1 guess is called 1 maxwell. 

           1 maxwell = 1 gauss cm²

Relation between weber and maxwell

         1 weber = 1 tesla × 1 m²

                        = 10⁴ gauss × (100 cm)²

                        = 10⁸ gauss cm²

∴      1 weber = 10⁸ maxwell

Dimensional formula of  ΦB :

                 ΦB = B . A = BA cos θ

or             ΦB = F/qv  A cos θ

where F is the force experienced by a change q moving with velocity v in the magnetic field. 

∴           [ΦB] = [M1 L1 T-2] × [L2]/[A1T1] [L T-1]

or [ΦB] = [M1 L2 T-2 A-1]

which is the required formula. 

The dimensions of magnetic flux are 1, 2, -2 and -1 in mass, length, time and electric current respectively. 

Important notes 1. Magnetic flux may or may not be equal to the number of magnetic lines of force passing through the circuit.  2. Magnetic flux ΦB linked with a surface depends upon the angle 'θ' between B and the normal to the surface.  (a) If θ = 0°, ΦB is maximum.  (b) If θ = 90°, ΦB is zero.  3. Magnetic flux linked with a coil, placed in a magnetic field, depends upon the permeability of the material placed inside the coil.

Faraday's Laws of Electromagnetic Induction:

 Faraday's law deal with the induction of an e.m.f. in an electric circuit when magnetic flux linked with the circuit changes. They are started as follows :

Faraday's first law (qualitative) :

Whenever magnetic flux linked with a circuit changes, an e.m.f. is induced in it. 

The induced e.m.f. exists in the circuit so long as the change in magnetic flux linked with it continues. 

Faraday's second law (quantitative) :

The induced e.mf. is directly to the negative rate of change in magnetic flux linked with the circuit. 

If 'dΦB' is the change in magnetic flux linked with a circuit, that takes place in a time dt.  

       Rate of change of magnetic flux = dΦB/dt.

If 'E' is e.m.f. induced in the circuit as a result of this change, 

E ∝ -dΦB /dt       or      E = - K  dΦB/dt

By selecting unit of 'E', 'ΦB' and 't' in a proper way, we can have

   K = 1           ∴         E = - dΦB/dt

     Negative sign is due to direction of induced e.m.f., which is explained by Lenz's law. 

Experiment verification:

     Consider a coil 'L' wound over a hollow wooden cylinder and having 'n' number of turns. It has a galvanometer 'G' connected between its free terminals. A magnet  N-S is placed in its neighbourhood with its center at 'A' [in Fig. 1]. The coil has some magnetic flux linked with it when magnet is at 'A'. Now move the magnet towards the coil so that its center moves from 'A' to 'B'. As a result of this, higher intensity region of magnetic field has moved closer to the coil, thereby, increasing 'H' and hence increasing magnetic flux 'ΦB' (= μnaH) linked with the circuit.

Fig. 1. Change in magnetic flux, linked with a coil, due to motion of a magnet. 

     First law. It will be observed that as magnet is moved from 'A' to 'B', galvanometer gives a deflection. This indicates that electric current and e.m.f. has been induced in the circuit due to a change in magnet flux linked with it. This verifies the first law. It will be observed that the deflection in the galvanometer persists so long as the magnet is in motion from 'A' to 'B' or from 'B' to 'A'. As soon as the magnet comes to rest, deflection disappears. This verifies the second law. 

     Second law. Bring the magnet from A to B at such a speed that it takes 2 seconds to reach B. Note the deflection in galvanometer. Now take the magnet from B to A in 1 second. It will be observed that the deflection obtained in second case is double of that obtained in first case. Since rate of change of magnetic flux in second case is double than that in first case. Hence, e.m.f. is directly proportional to the rate of change of magnetic flux. This varifies the third law. 

     Magnetic flux linked with a circuit can be changed in a number of ways :

• By moving a magnet towards a fixed coil or moving it away from coil. 
• By moving a coil towards a fixed magnet or moving it away from magnet. 
• By increasing or decreasing current in a neighborhood circuit. 

Faraday's Experiments:

It observed by Faraday's that whenever magnetic flux linked with a circuit changes, an e.m.f. is induced in the circuit. If the circuit is closed, it will cause an electric current to flow through the circuit. There may be different reasons for the changing magnetic flux, in different experiments, but the result is same. 

     (i) Induction due to motion of magnet. Consider a coil comprising of a large number of turns of a wire wound over an insulating cylinder. A galvanometer is connected in between its free terminals (Fig. 2). 

Fig. 2. Motion of magnet near a coil. 

     Let a magnet having its North Pole pointing towards the cross-section of the coil be moved towards it [Fig. 2(i)]. As the magnet approaches the coil the strength of magnetic field around the coil increases. This results in an increase in magnetic flux. It will be observed that the galvanometer connected in the circuit gives a deflection indicating the induction of electric current. It can further be observed that the direction of deflection, in galvanometer, gets reversed if the North pole of the magnet is moved away from the face of the coil [Fig. 2(ii)]. Thus, electric current is induced due to motion of magnet. 

     (ii) Induction due to motion of coil. The phenomenon, explained above, can also be observed if the coil having a galvanometer connected in its circuit is moved in the magnetic field of a stationary magnet (Fig. 3). It can also be noted that the direction of deflection in the galvanometer gets reversed if we reverse the direction of motion of coil. In this case also, the induction of electric current takes place due to a change in magnetic flux which takes place due to motion of coil. 

Fig. 3. Motion of coil near a magnet. 

     (iii) Induction due to a current changing in the neighborhood. Consider two coils P and S would over each other with an insulating cylindrical core. The two coils are electrically insulated from each other. A source of d.c. is connected in the circuit of coli P while a galvanometer is connected in the circuit of coil S (Fig. 4).

Fig. 4. A current changing in a neighbouring circuit. 

      It will be observed that the galvanometer gives a deflection when the circuit of coil P is made or broken. The deflection at these two instances are in opposite directions and these are inspite of the fact that there is no electrical contact between P and S. This is gain due to the change in magnetic flux linked with coil S. This time the change in magnetic flux takes place due to a change in strength of magnetic field, around S, due to a changing current in P which changes from zero to maximum at make and from maximum to zero at break. If the key K is kept pressed, a constant current will flow through P. There will be no change in magnetic flux linked with S. Therefore, there is no deflection in the galvanometer. 

 Moving Coil Galvanometer - "Dead-Beat Galvanometer" (Pivoted Type) and its conversion to ammeter and voltmeter

Principle

 It is based on the principle that whenever a loop carrying current is placed in a magnetic field, it experiences a torque which tends to rotate it. 

Construction

It consists of a coil 'C' having a large number of turns of thin copper wire. The coil is suspended from a torsion head T by means of a fine suspension fibre of phosphor bronze in such a way that it hangs in between the cylindrical pole pieces N and S of a horse shoe magnet. A soft iron cylindrical core 'K' is adjusted inside the coil in such a way that the coil touches neither the pole pieces nor the core. One end of the coil is soldered to a suspension fibre F through a small plane mirror M while the other end is connected to a delicate spring S of phosphor bronze. The torsion head and the spring are internally connected to the terminals A and B to pass current through the galvanometer [Fig. 1(i)].

Theory

When a closed loop is suspended in a magnetic field, it experiences a torque τ which tends to rotate it about a vertical axis, 

           τ = nBiA cos θ

     Here, n is the number of turns of the coil of area A, i is the current passing through it. B is the strength of magnetic field due to magnet NS and θ is the angle which the plane of the coil makes with the magnetic field. 

Fig. 1. Moving coil galvanometer. 

     Since the pole pieces are of cylindrical form, the lines of force will be along the radius. Therefore as the coil rotates, its plane will always be parallel to the  direction of lines of force, in all its positions, as shown in Fig. 1(ii). Such a field is known as radial field. 

      For a radial field,       

                 θ = 0

∴               τ = nBiA                (∵  cos θ = 1) 

     As a result of this couple, the coil gets deflected. This produces a twist in the suspension fibre, due to which the coil is acted upon by the another couple called restoring couple which tends to take it back to the original position. 

     If C is the torsional couple per unit angular twist of the fibre, then

     Moment of the restoring couple = Cθ

     In equilibrium position, the torque due to deflecting couple must balance that due to the restoring couple. 

i.e.,             nBiA = Cθ

or                      i = C/nBA . θ

or                      i = Kθ

where  K = C/nBA is known as the reduction factor of the moving coil galvanometer and is constant for a galvanometer. 

     Hence, the current passing through the galvanometer is directly proportional to the direction. 

     If  θ = 1,   then   i = K. 

     Thus, the reduction factor of moving coil galvanometer is defined as the current required to produce a unit deflection in the galvanometer. 

Working

The levelling screws are adjusted for levelling the instrument. The torsion head is set so that the coil hangs freely without touching the poles or the soft iron core. The current to be measured is now passed through the terminals A and B. It will be observed that the coil gets deflected. The deflection can be measured by using a lamp and scale arrangement which consists of  lamp L mounted on a vertical stand. A fine collimated beam from the lamp is focussed on to the mirror M. The reflected beam is received on a semi-transparent scale 'S' mounted on the same stand [Fig. 2]. As the mirror gets deflected, a bright circular spot moves over the scale graduated in centimetres. Thus, the deflection of the coil can be measured. 

Fig. 2. Lamp and scale arrangement. 

Merits

A moving coil galvanometer has following merits :

1. It is a highly sensitive instrument and hence can be used to measure very small currents.                                              
2. Since the electric current is proportional to the deflection produced, a linear scale can be used for measurement purposes.                             
3. Its magnet produces a strong magnetic field within the neighborhood of coil. Therefore, the performance of the galvanometer cannot be affected by the presence of any stray magnet or magnetic substance nearby.                       
4. Like a tangent galvanometer, it does not require any specific setting. This is again due to its own strong magnetic field. 

  Demerits

1. It is not direct reading. We have to make a lot of adjustments to use this galvanometer.                                               
2. It is not portable.                                          
3. Its suspension fibre is very delicate. If due precaution is not taken, it gets broken too often. That is why the coil has to be locked while displacing the galvanometer from one place to the other.                                                               
4. A lot of space (for setting up a lamp and a scale arrangement) is required for its operation. 

SENSITIVITY OF A GALVANOMETER

The sensitivity of an instrument is measured by the minimum input required for a certain standard output. In case of a galvanometer employing the use of a lamp and a scale arrangement, the standard output is a deflection of a spot of light by 1 mm on a scale clamped at a distance of 1 metre from the reflecting mirror of galvanometer. It is generally named as a u it deflection in the case. 

     (i) Current sensitivity. Current sensitivity of a galvanometer is defined as the minimum current required to produce a deflection of a spot of light by 1 mm on a scale fixed at a distance of 1 metre from the reflecting mirror of the galvanometer. 

     If 'θ' is the deflection produced in the coil due to a current i flowing through it, current sensitivity Si is given by

                 Si = i/θ = i/d

where 'd' is deflection in mm, of the spot on a screen placed 1 m away. 

    (ii) Voltage sensitivity. Voltage sensitivity of a galvanometer is defined as the minimum voltage which when applied across the galvanometer produced a deflection of a spot of light by 1 mm on a scale fixed at a distance of 1 metre from the reflecting mirror of the galvanometer. 

     If 'V' is the potential difference applied across the galvanometer of resistance G, current i flowing through it is

                 i = V/G

     For moving coil galvanometer, 

                  i = C/nHA . θ        

∴          V/G = C/nHA . θ 

     Voltage sensitivity of the galvanometer

                Sv = V/θ = iG/d = CG/nAH

     It should be clearly noted that smaller the value of 'Si' or 'Sv', more sensitive is the galvanometer. Thus, to make a galvanometer more sensitive we should aim at having values if 'Si' and 'Sv' as small as possible. This can be achieved in a number of ways. 

     (i) By increasing n. An increase in the number of turns of galvanometer will make it more sensitive. But we can't increase n indefinitely to a very large value. This will the resistance 'G' of a galvanometer which will result in increase of Sv . 

    (ii) By increasing A. An increase in area of coil results in decrease of Si or Sv thereby making the galvanometer more sensitive. The magnetic field of the radial field is uniform over a narrow region. If the area of the coil is extraordinary large, some portion of the coil will be out of the region of uniform field. So we can't increase the area indefinitely. 

   (iii) By increasing B. To have a large B we must have a bigger magnet. This results in an abnormal increase in size of galvanometer. 

    (iv) By decreasing C. A galvanometer can be made highly sensitive by making use of a suspension fibre which has a small value of C. For a suspension fibre

                       C =  π η r⁴/2l

where l = length of fibre, r = radius of fibre and η = modulus of rigidity of the material of fibre. 

     'C' can be decreased by making use of a long (longer l), thin (smaller r) and silken (lesser η) fibre. Since C ∝ r⁴, a decrease in radius of fibre will be much more effective in increasing sensitivity of galvanometer. 

Experimental determination

     The circuit diagram for experimental determination of current and voltage sensitives of a galvanometer is shown in Fig. 3.

Fig. 3. Determination of sensitivity of galvanometer. 

     A source of e.m.f., E is connected to resistance 'S' and 'r' in series through a key. A galvanometer having high resistance R in series, is connected across BC. This ensures that effective resistance in between B and C is r. [∵  r << R + G]

     Let V and v be the potential difference (to be measured with a voltmeter) across AC and BC. 

           v/V = r/S + r        or        v = Vr/S + r 

     If i is the current flowing in the galvanometer, 

            i = v/R + G = Vr/(S + r) (R + G) 

     Current sensitivity, 

          Sv = iG/d = Vr/d(S + r) (R + G) × G

     Here 'd' is deflection of spot of light, in mm, on a screen placed at a distance of 1 metre from the mirror of galvanometer. 

WESTON TYPE GALVANOMETER

In order to avoid the difficult adjustment of D'Arsonval type moving coil galvanometer, Dr. Weston modified it to make it a direct reading instrument without making use of lamp and scale arrangement. This modified type is called pointer type galvanometer. The coil having a few number of turns of copper wire, wound on a metal frame, is mounted on a spindle. The ends of the spindle are fixed in two jewelled bearings to reduce friction. The coil is placed in between the two pole pieces of a horse shoe magnet having cylindrical pole pieces which provide a strong radial field. The restoring couple controls the rotation of the coil. This couple is provided by two very delicate springs attached with the spindle. 

     A light aluminium pointer is attached to the spindle. The whole apparatus is enclosed in a bakelite cover. As the coil rotates, the pointer slides over a scale as shown in Fig. 4.

Fig. 4. Weston type galvanometer. 

     A source of current is connected between the terminals, A and B. When the current passes through the coil, it experiences a deflecting couple which deflects it. When the deflecting couple due to the current balances the restoring couple provided by the springs, the coil comes to rest. The deflection can be measured by noting the reading of pointer on the scale. 

CONVERSION OF AN GALVANOMETER INTO AN AMMETER

An ammeter is the instrument used for measurement of current in a circuit. 

     Galvanometers are generally very sensitive. Current of small value takes the pointer out of scale. It may even damage the coil. Thus, it is dangerous to expose a galvanometer directly to an unknown current without doing any arrangement for its safety. 

     A galvanometer is rendered safe when a suitable low resistance 'S' is connected across the terminals A and B and is in parallel with the galvanometer coil. The resistance 'S' provides a bypass for the excess current. 

     The value of 'S' to be connected in parallel with coil is selected in such a way that when the combination of S and G, in parallel, is exposed to current 'i' (to be measured), the galvanometer takes a current 'ig' only [Fig. 5(i)]. Here ig is the current required for full scale deflection in the galvanometer. 

     If ig is the current passing through the shunt, then

                i = ig + is

 ∴              is = i - ig

 

Fig. 5. Conversion of a galvanometer into an ammeter. 

  

      Since G and S are in parallel, 

∴          is / ig = G/S    or       S = Gig /ig  = Gig / i - ig

     Knowing the value of G, I and ig the values of S can be calculated. A Weston type galvanometer having a calculated low resistance S connected in parallel with it is called an ammeter [Fig. 5(ii)]. The pointer is fixed in such a way that in 'no current position', it stands at zero mark on the extreme left to the scale. 

     The terminals A and B are marked '+' and '-' respectively so as to send the current through the instrument in one direction only. 

     The presence of a resistance 'S', in parallel, results in the decrease of resistance of the galvanometer. Thus, an ammeter is an ammeter is a low resistance galvanometer and is, therefore, always connected in series with the circuit. 

CONVERSION OF A GALVANOMETER INTO A VOLTMETER

A voltmeter is an instrument used for measuring the potential difference between two points. 

     To convert a galvanometer into a voltmeter, a suitable high resistance R is connected in series with the galvanometer coil. The value of 'R' is so selected that when the combination is exposed to full potential difference V, the current passing through the voltmeter should be 'ig' [Fig. 6(i)]. Hence, ig is the current required for full scale deflection. 

or              R + G = V/ig

∴                       R = V/ig - G 

Fig. 6. Conversion of a galvanometer into a voltmeter. 

     Knowing the values of V, ig and G, th3 value of 'R' can be calculated. A Weston type galvanometer having a high resistance R in series with galvanometer coil is shown in Fig. 6(ii).

     The presence of a high resistance 'R' in series with the galvanometer coil results in the increase in resistance of the galvanometer. Thus, a voltmeter is a high resistance galvanometer and is, therefore, always connected in parallel with the circuit. 

IMPORTANT NOTES

1. Galvanometer is a current detecting device. That is why its pointer is in the centre of scale since we must be prepared for any direction of current. In case of ammeter and voltmeter the zero position of the pointer is on one side of scale since we know the direction of current. This helps in increasing the sensitivity of the instrument. 2. A low resistance connected in parallel with a circuit decreases its resistance drastically. Therefore, an ammeter, being a low resistance instrument, is always connected in series with the circuit.                                                   3. A high resistance connected in series with the circuit increases its resistance drastically. Therefore, a voltmeter, being a high resistance instrument, is always connected in parallel with the circuit.

Examples

Example 1. 

     The resistance of a galvanometer is 49 ohm and the maximum current which can be passed through it is 0.001 A. What resistance must be connected to it in order to convert it into : (i) an ammeter if range 0.5 A, (ii) a voltmeter of range 5 V ? 

 Solution. 

     Given, G = 49 ohm,  ig = 0.001 A

     (i) For ammeter,    i = 0.5 A. 

     Value of shunt 'S' is,

          S = Gig/i - ig = 49 × 0.001/(0.5 - 0.001) Ω

             = 49 × 0.001/0.499 Ω

             = 0.098 ohm. 

     So the galvanometer converted into an ammeter of range 0.5 A by connecting a resistance of 0.098 ohm in parallel with the coil of galvanometer. 

    (ii) For voltmeter, 

             R = V/ig - G,           V = 5 V, 

             ig = 0.001 A,          G = 490 ohm

            R = [5/0.001 - 49] Ω

               = (5000 - 49) Ω 

               = 4951 ohm. 

     Thus, the galvanometer can be converted into a voltmeter of range 5 V by connecting a resistance of 4951 ohm in series with the galvanometer coil. 

Example 2.

     A battery of e.m.f. 1.4 V and internal resistance 2 Ω is connected to a resistor of 100 Ω through an ammeter. The resistance of the ammeter is 4/3 Ω. A voltmeter has also been connected to find the potential difference across the resistor. Draw the circuit diagram. If the ammeter reads 0.02 A, what is the resistance of the voltmeter ? The voltmeter reads 1.10 V, what is the error in the reading ? 

Solution. 

     Let R = resistance of voltmeter

     If R' = resistance of parallel combination of 100 Ω and R, 

                  R' = 100 R/100 + R                  ....(i) 

Fig. 7.

     Total resistance of the circuit = 2 + 4/3 + 100 R/100 + R

               i = E.M.F./Total resistance

         0.02 = 1.4/2 + 4/3 + 100 R/100 + R

         0.02 = 1.4 (100 + R) ×3/6 (100 + R) + 4 (100 + R) + 300 R

     0.02 × 6 × (100 + R) + 0.02 × 4 × (100 + R) + 0.02 × 300 R = 1.4 × 100 × 3 + 1.4 × 3R

     12 + 0.12 R + 8 + 0.08 R + 6 R = 420 + 4.2 R

                                                  2 R = 400

                                                     R = 200

     ∴  Resistance of voltmeter = 200 ohm. 

     Substituting for R in equation (i)

               R' = 100 × 200/100 + 200 Ω

     ∴  Reading of voltmeter = i × R' 

                                                 = 0.02 × 200/3 V

                                                 = 4/3 V

                                                 = 1.33 V

     Error in the reading of voltmeter

                   = 1.33 V - 1.10 V

                   = 0.23 V. 

Example 3.

     A galvanometer, together with an unknown resistance in series is connected across two identical batteries each of 1.5 V. When the batteries are connected in series, the galvanometer records a current of 1 A and when the batteries are connected in parallel the current is 0.6 A. What is the internal resistance of the battery ? 

Solution. 

     Let R = unknown resistance in the circuit (including that of galvanometer) 

            r = internal resistance of each battery. 

     Case (i) When the batteries are connected in series [Fig. 8(i)]

     Total internal resistance = 2r

     Total resistance of the circuit = R + 2r

     Total e.m.f. of the circuit = 1.5 V + 1.5 V

                                                  = 3 V. 

     ∴                i₁ = 3/R +2r

    Since         i₁ = 1            ∴ = 3/R + 2r

    ∴        R + 2r = 3                                  ... (i) 

Fig. 8. 

     Case (ii) When the two batteries are connected in parallel [Fig. 8(ii)]

     Total internal resistance = r × r/r + r

                                                  = r²/2r = r/2

     Total resistance of the circuit = R + r/2

     Total e.m.f. of the circuit = 1.5 V

     ∴                i₂ = 1.5/R + r/2 = 1.5 × 2/2R + 2

     Since        i₂ = 0.6

     ∴             0.6 = 1.5 × 2/2R + r

     ∴       2R + r = 3                                   ... (ii) 

     Multiplying (i) by 2 and subtracting (ii) from it. 

              2R + 4r - 2R - r = 6 - 5 = 1

                                   3R = 1

     ∴                              r = 1/3 Ω. 

Magnetic effect of electric current and Biot-savart's Law or 'Ampere's Theorem'

 Concept of Field

It is a matter of common experience that for transmission of mechanical force, from one body to the other, a physical contact between them is necessary. There are cases where a body exerts a force on the other without any contact between them e.g., gravitational force or the electro-static force. This is termed as 'interaction at a distance'. To explain this interaction we take shelter of the concept of the field. Presence of a body, at any place, modifies the space around it. This modified space, around the body, is called a field. When two bodies are placed near each other, both the bodies have a field of their own. These fields interact with each other to give rise to a force. This is the reason why two charged bodies exert a force on each other or why two bodies, in this universe attract each other due to gravitational force. 

     Electro-static force arises due to interaction of the electric field while a magnetic force arises due to the interaction of two magnetic fields. If we place a magnetic pole in an electric field or an electric charge near a magnet no force is experienced by either of them. 

Magnetic effect of electric current - 'Oersted's Experiment'

In 1820, Oersted was able to show that an electric current flowing through a wire produces a magnetic field around it. 

     Consider a wire AB connected to a battery in such a way that electric current flows from A to B on closing the key (Fig. 1). Place a compass needle just below the wire AB. 

Fig. 1. Deflection of magnetic compass due to the magnetic field of a current. 

     Close the key 'K' so that a current starts flowing through the wire. It is observed that the compass needle gets deflected. Compass needle, which is a magnet, can only be effected by a magnetic field. The experiment indicates that a magnetic field gets developed around the wire when an electric current flows through it. The phenomenon is called magnetic effect of currents. 

Rules for Direction of Deflection of N-pole

The direction in which the North Pole of the compass needle is deflected can be determined by applying any one of the following rules :

(i) 'SNOW' rule. "If the direction of current flowing through the wire is from south (S) to north (N) and the wire is placed over (O) the needle, the North pole of the needle is deflected towards west (We). 

     If the wire is placed below the needle and the current still flows from south to north, north pole of the needle is deflected towards east. 

(ii) Ampere's rule. "Imagine a men to swim along the wire in the direction of the current with his face towards the needle so that the current enters at his feet and leaves at his head. The north pole of the needle will be deflected towards his left hand (Fig. 2).

Fig. 2. Ampere's rule. 

BIOT-SAVART'S LAW OR "AMPERE'S THEOREM" 

A wire carrying current has a magnetic field all around it. The intensity at any point, in this magnetic field can be obtained with the help of "Biot-Savart's law".

     Consider a small section dl, of a wire carrying q current 'i'. Let 'p' be the observation point at a distance 'r' from the centre 'O' of the element. The position vector of P subtends an angle θ with the direction of flow of current in the element dl as shown in Fig. 3.

Fig. 3. Strength of magnetic field, at any point due to a wire carrying current. 

     If  'dB' is a magnetic intensity at P due to this small element, it has been observed that

                 dB ∝ dl,   dB ∝ i

                 dB ∝ sin θ,  dB ∝ 1/r²

Combining all these factors, we get

                  dB ∝ i dl sin θ/r²

or              dB ∝ k × i dl sin θ/r²                  ... (1) 

where k is the constant if proportionality and its value depends upon the system of units selected. 

     Equation (1) gives the scalar form of Biot-Savart's law or Ampere's theorem. 

     In S.I. unit's, 'i' is taken in ampere, dl and r in metre. 

and      k =  μ0/4π   (where μ0 = 4π × 10-7 Wb A-1 m-1) 

dB = μ0/4π × i dl sin θ/r² × tesla.  

Vector form

     Biot-Savart's law, in vector form, can be written as 

               dB = μ0/4π × i × dl × r/r³ ... (2)

dB = μ0/4π × i × dl × r̂/r²                   ... (3) 

Note - (dB, dl r is in rightward arrow) 

     dl is a vector quantity whose length is proportion to the length of wire element and whose direction is given by the direction of current, r (rightward arrow) is a vector whose length is proportional to the straight line distance between 'O' and 'Q' (point of observation) and whose direction is along OQ. 

     According to the rule of cross product, direction of dB (rightward arrow) is perpendicular to the plane containing dl (rightward arrow) and r (rightward arrow). Fig. 4, OPQ is the plane containing dl (rightward arrow) and r (rightward arrow). Applying right hand thumb rule it can be concluded that 'dB' (rightward arrow) is perpendicular to OPQ and is directed upwards. 

Fig. 4. Direction of magnetic field. 

        |dB| (rightward arrow) = μ0/4π × i × dl × sin θ/r²

Case (i) if θ = 0°, |dB| = 0

i.e., there is no magnetic intensity at any point, on the axial line of the element. 

Case (ii) if θ = 90°,

|dB| (rightward arrow) = μ0/4π × i dl/r (maximum) 

Magnetic intensity at any point lying on a line perpendicular to the length of the element is maximum. 

Direction of lines of force :

Consider a long straight wire XY passing through the centre of a cardboard 'ABCD' and perpendicular to its plane. Spread some iron fillings on the cardboard. When an electric current is passed through the wire, the iron fillings arrange themselves around the wire in concentric circles, indicating that the lines of forces of magnetic field are in the form of concentric circles (Fig. 5). Lines of force due to two straight conductors, carrying currents in opposite directions are shown in Fig. 5(i) and Fig. 5(ii). The direction of lines of force is given by applying any of the following rules. 

Fig. 5. Magnetic field due to a long straight wire carrying current. 

     (i) Maxwell's cork screw rule. Imagine a right handed cork screw placed along the conductor carrying current with its axis coinciding with the direction of current so that if the screw is twisted it travels foreword in the direction of the current. The direction in which the ends of the handle turns gives the direction of lines of force (Fig. 6).

Fig. 6. Maxwell's cork screw rule. 

     (ii) Right hand thumb rule. Imagine the wire carrying current to be held in your right hand with its thumb pointing in the direction of electric current. The direction in which the fingers curl, gives the direction of lines of force if the magnetic field (Fig. 7).

Fig. 7. Right hand thumb rule. 

Magnetic field due to a long straight conductor carrying current

Consider an element XY of a long straight conductor AB carrying current i in the direction from A to B. Let People be the observation point at a distance x from the centre of the element. 

     Draw PM perpendicular to the length of conductor such that PM = r. 

     Let θ be the angle which PO makes with the direction of current  (Fig. 8).

     If dB be the magnetic intensity at P due to a current i through the element XY, then by Biot-savart's law, 

                dB = μ₀i/4π × dl sin θ/x²             ... (4) 

Fig. 8. Magnetic field at any point due to a long straight conductor carrying current. 

     In right angle ∆ OMP, θ + ∝ = π/2

or                   θ = π/2 - ∝

or            sin θ = sin (π/2 - ∝) 

or            sin θ = cos ∝

     Again, in right angle ∆ OMP

                cos ∝ = r/x

∴                    x = r/cos ∝

     Also  tan ∝ = l/r

∴                    l = r tan ∝

     Differentiating, we get

                    dl = r sec² ∝ d∝

     Substituting for dl, sin θ and x in equation (4), we get

         dB = μ₀/4π × i × (r sec² ∝ d∝) cos ∝/r² × cos² ∝

or     dB = μ₀i/4πr cos ∝ . d∝                        ... (5) 

     Applying right hand thumb rule, it can be concluded that direction of B will be perpendicular to the plane of paper directed outward. All other elements of the conductor will also produce intensity in the same direction. Net magnetic intensity at P due to whole of the conductor AB can be obtained by integrating equation (5) within the limits θ₁ and θ₂.

     ∴       B = B0∫ dB = μ₀i/4πr θ₂θ₁∫ cos cos ∝ d∝ 

       

                     = μ₀i/4πr [sin ∝]θ₂θ₁

or  B = μ₀i/4πr [sin θ₂ - sin θ₁] ... (6)

Special cases

     Case (i) When the point is situated symmetrically with respect to the two ends of the conductor :

Fig. 9. Point Of equidistant from two ends of conductor. 

     Let 'l' be the length of the conductor. 

     In this case, ∝₁ = − ∝ and ∝₂ = + ∝

     Using equation (6), we get

           B = μ₀i/4πr [sin ∝ - sin (- ∝)]

or       B = μ₀i/4πr × 2 sin ∝

     From Fig. 9, sin ∝ = l/√l² + 4r²

     ∴          B = 2 μ₀i/4πr × l/√l² + 4r²

or             B = μ₀i/2πr × l/√l² + 4r²

     Case (ii) If the conductor is of infinite length

     For an infinitely long conductor, angle θ₁ and θ₂ tends to - π/2 and + π/2 radian respectively. 

     Substituting θ₁ = - π/2 and θ₂

                                = + π/2 in equation (6) we get

          B = μ₀i/4πr [sin (π/2) - sin (- π/2)]

             = μ₀i/4πr [sin π/2 + sin π/2]

             = μ₀i/4πr [1 + 1] = μ₀2i/4πr

or     B = μ₀/2π × i/r ⇒B ∝ 1/r

 

Fig. 10.

 Key formulae

1. Biot-savart's law

         dB = μ0/4π × i dl sin θ/r² (Scalar form)

dB = μ0/4π × i × dl × r/r³ (Vector form)

(Note - dB, dl, r is in rightward arrow)

2. Field due to a long straight conductor carrying current

(i) Conductor of finite length

B = μ₀i/4πr [sin ∝₂ - sin ∝₁]

(ii) Conductor of infinite length

B = μ₀/2π × i/r