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6.1 INTRODUCTION
Electricity and magnetism were considered separate and unrelated phenomena for a long time. In the early decades of the nineteenth century, experiments on electric current by Oersted, Ampere and a few others established the fact that electricity and magnetism are inter-related. They found that moving electric charges produce magnetic fields. For example, an electric current deflects a magnetic compass needle placed in its vicinity. This naturally raises the questions like: Is the converse effect possible? Can moving magnets produce electric currents? Does the nature permit such a relation between electricity and magnetism? The answer is resounding yes! The experiments of Michael Faraday in England and Joseph Henry in USA, conducted around 1830, demonstrated conclusively that electric currents were induced in closed coils when subjected to changing magnetic fields. In this chapter, we will study the phenomena associated with changing magnetic fields and understand the underlying principles. The phenomenon in which electric current is generated by varying magnetic fields is appropriately called electromagnetic induction. When Faraday first made public his discovery that relative motion between a bar magnet and a wire loop produced a small current in the latter, he was asked, “What is the use of it?” His reply was: “What is the use of a new born baby?” The phenomenon of electromagnetic induction
Chapter Six
ELECTROMAGNETIC
INDUCTION
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is not merely of theoretical or academic interest but also of practical utility. Imagine a world where there is no electricity – no electric lights, no trains, no telephones and no personal computers. The pioneering experiments of Faraday and Henry have led directly to the development of modern day generators and transformers. Today’s civilisation owes its progress to a great extent to the discovery of electromagnetic induction.
6.2 THE EXPERIMENTS OF F ARADAY AND
H ENRY
The discovery and understanding of electromagnetic induction are based on a long series of experiments carried out by Faraday and Henry. We shall now describe some of these experiments.
Experiment6. 1
Figure 6.1 shows a coil C 1 * connected to a galvanometer G. When the North-pole of a bar magnet is pushed towards the coil, the pointer in the galvanometer deflects, indicating the presence of electric current in the coil. The deflection lasts as long as the bar magnet is in motion. The galvanometer does not show any deflection when the magnet is held stationary. When the magnet is pulled away from the coil, the galvanometer shows deflection in the opposite direction, which indicates reversal of the current’s direction. Moreover, when the South-pole of the bar magnet is moved towards or away from the coil, the deflections in the galvanometer are opposite to that observed with the North-pole for similar movements. Further, the deflection (and hence current) is found to be larger when the magnet is pushed towards or pulled away from the coil faster. Instead, when the bar magnet is held fixed and the coil C 1 is moved towards or away from the magnet, the same effects are observed. It shows that it is the relative motion between the magnet and the coil that is responsible for generation (induction) of electric current in the coil.
Experiment6. 2
In Fig. 6.2 the bar magnet is replaced by a second coil C 2 connected to a battery. The steady current in the coil C 2 produces a steady magnetic field. As coil C 2 is
- Wherever the term ‘coil or ‘loop’ is used, it is assumed that they are made up of conducting material and are prepared using wires which are coated with insulating material.
FIGURE 6.1 When the bar magnet is pushed towards the coil, the pointer in the galvanometer G deflects.
Josheph Henry [1797 – 1878] American experimental physicist professor at Princeton University and first director of the Smithsonian Institution. He made important improvements in electro- magnets by winding coils of insulated wire around iron pole pieces and invented an electromagnetic motor and a new, efficient telegraph. He discoverd self-induction and investigated how currents in one circuit induce currents in another.
JOSEPH HENRY (1797 – 1878)
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a plane of area A placed in a uniform magnetic field B (Fig. 6.4) can be written as
Φ (^) B = B.^ A = BA cos θ (6.1)
where θ is angle between B and A. The notion of the area as a vector has been discussed earlier in Chapter 1. Equation (6.1) can be extended to curved surfaces and nonuniform fields. If the magnetic field has different magnitudes and directions at various parts of a surface as shown in Fig. 6.5, then the magnetic flux through the surface is given by Φ B = B 1 i d A 1 + B 2 i dA 2 +... = all
∑ B i^ i^ dA i (6.2)
where ‘all’ stands for summation over all the area elements dA i
comprising the surface and B i is the magnetic field at the area element dAi. The SI unit of magnetic flux is weber (Wb) or tesla meter squared (T m^2 ). Magnetic flux is a scalar quantity.
6.4 FARADAY ’S LAW OF INDUCTION
From the experimental observations, Faraday arrived at a conclusion that an emf is induced in a coil when magnetic flux through the coil changes with time. Experimental observations discussed in Section 6.2 can be explained using this concept. The motion of a magnet towards or away from coil C 1 in Experiment 6.1 and moving a current-carrying coil C 2 towards or away from coil C 1 in Experiment 6.2, change the magnetic flux associated with coil C 1. The change in magnetic flux induces emf in coil C 1. It was this induced emf which caused electric current to flow in coil C 1 and through the galvanometer. A plausible explanation for the observations of Experiment 6.3 is as follows: When the tapping key K is pressed, the current in coil C 2 (and the resulting magnetic field) rises from zero to a maximum value in a short time. Consequently, the magnetic flux through the neighbouring coil C 1 also increases. It is the change in magnetic flux through coil C 1 that produces an induced emf in coil C 1. When the key is held pressed, current in coil C 2 is constant. Therefore, there is no change in the magnetic flux through coil C 1 and the current in coil C 1 drops to zero. When the key is released, the current in C 2 and the resulting magnetic field decreases from the maximum value to zero in a short time. This results in a decrease in magnetic flux through coil C (^1) and hence again induces an electric current in coil C 1 *. The common point in all these observations is that the time rate of change of magnetic flux through a circuit induces emf in it. Faraday stated experimental observations in the form of a law called Faraday’s law of electromagnetic induction. The law is stated below.
FIGURE 6.4 A plane of surface area A placed in a uniform magnetic field B.
FIGURE 6.5 Magnetic field B i at the i th^ area element. dA i represents area vector of the i th^ area element.
- Note that sensitive electrical instruments in the vicinity of an electromagnet can be damaged due to the induced emfs (and the resulting currents) when the electromagnet is turned on or off.
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The magnitude of the induced emf in a circuit is equal to the time rate of change of magnetic flux through the circuit. Mathematically, the induced emf is given by
d
B t
Φ ε = (^) (6.3)
The negative sign indicates the direction of ε and hence the direction of current in a closed loop. This will be discussed in detail in the next section. In the case of a closely wound coil of N turns, change of flux associated with each turn, is the same. Therefore, the expression for the total induced emf is given by
N B t
ε = Φ (6.4)
The induced emf can be increased by increasing the number of turns N of a closed coil. From Eqs. (6.1) and (6.2), we see that the flux can be varied by changing any one or more of the terms B, A and θ. In Experiments 6.1 and 6.2 in Section 6.2, the flux is changed by varying B. The flux can also be altered by changing the shape of a coil (that is, by shrinking it or stretching it) in a magnetic field, or rotating a coil in a magnetic field such that the angle θ between B and A changes. In these cases too, an emf is induced in the respective coils.
Example 6.1 Consider Experiment 6.2. (a) What would you do to obtain a large deflection of the galvanometer? (b) How would you demonstrate the presence of an induced current in the absence of a galvanometer? Solution (a) To obtain a large deflection, one or more of the following steps can be taken: (i) Use a rod made of soft iron inside the coil C 2 , (ii) Connect the coil to a powerful battery, and (iii) Move the arrangement rapidly towards the test coil C 1. (b) Replace the galvanometer by a small bulb, the kind one finds in a small torch light. The relative motion between the two coils will cause the bulb to glow and thus demonstrate the presence of an induced current. In experimental physics one must learn to innovate. Michael Faraday who is ranked as one of the best experimentalists ever, was legendary for his innovative skills.
Example 6.2 A square loop of side 10 cm and resistance 0.5 Ω is placed vertically in the east-west plane. A uniform magnetic field of 0.10 T is set up across the plane in the north-east direction. The magnetic field is decreased to zero in 0.70 s at a steady rate. Determine the magnitudes of induced emf and current during this time-interval.
Michael Faraday [1791– 1867] Faraday made numerous contributions to science, viz., the discovery of electromagnetic induction, the laws of electrolysis, benzene, and the fact that the plane of polarisation is rotated in an electric field. He is also credited with the invention of the electric motor, the electric generator and the transformer. He is widely regarded as the greatest experimental scientist of the nineteenth century.
MICHAEL FARADAY (1791–1867)
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6.5 LENZ’ S L AW AND C ONSERVATION OF E NERGY
In 1834, German physicist Heinrich Friedrich Lenz (1804-1865) deduced a rule, known as Lenz’s law which gives the polarity of the induced emf in a clear and concise fashion. The statement of the law is: The polarity of induced emf is such that it tends to produce a current which opposes the change in magnetic flux that produced it. The negative sign shown in Eq. (6.3) represents this effect. We can understand Lenz’s law by examining Experiment 6.1 in Section 6.2.1. In Fig. 6.1, we see that the North-pole of a bar magnet is being pushed towards the closed coil. As the North-pole of the bar magnet moves towards the coil, the magnetic flux through the coil increases. Hence current is induced in the coil in such a direction that it opposes the increase in flux. This is possible only if the current in the coil is in a counter-clockwise direction with respect to an observer situated on the side of the magnet. Note that magnetic moment associated with this current has North polarity towards the North-pole of the approaching magnet. Similarly, if the North- pole of the magnet is being withdrawn from the coil, the magnetic flux through the coil will decrease. To counter this decrease in magnetic flux, the induced current in the coil flows in clockwise direction and its South- pole faces the receding North-pole of the bar magnet. This would result in an attractive force which opposes the motion of the magnet and the corresponding decrease in flux. What will happen if an open circuit is used in place of the closed loop in the above example? In this case too, an emf is induced across the open ends of the circuit. The direction of the induced emf can be found using Lenz’s law. Consider Figs. 6.6 (a) and (b). They provide an easier way to understand the direction of induced currents. Note that the direction shown by and indicate the directions of the induced currents. A little reflection on this matter should convince us on the correctness of Lenz’s law. Suppose that the induced current was in the direction opposite to the one depicted in Fig. 6.6(a). In that case, the South-pole due to the induced current will face the approaching North-pole of the magnet. The bar magnet will then be attracted towards the coil at an ever increasing acceleration. A gentle push on the magnet will initiate the process and its velocity and kinetic energy will continuously increase without expending any energy. If this can happen, one could construct a perpetual-motion machine by a suitable arrangement. This violates the law of conservation of energy and hence can not happen. Now consider the correct case shown in Fig. 6.6(a). In this situation, the bar magnet experiences a repulsive force due to the induced current. Therefore, a person has to do work in moving the magnet. Where does the energy spent by the person go? This energy is dissipated by Joule heating produced by the induced current.
FIGURE 6. Illustration of Lenz’s law.
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E XAMPLE
(^) 6.
Example 6. Figure 6.7 shows planar loops of different shapes moving out of or into a region of a magnetic field which is directed normal to the plane of the loop away from the reader. Determine the direction of induced current in each loop using Lenz’s law.
FIGURE 6.
Solution (i) The magnetic flux through the rectangular loop abcd increases, due to the motion of the loop into the region of magnetic field, The induced current must flow along the path bcdab so that it opposes the increasing flux. (ii) Due to the outward motion, magnetic flux through the triangular loop abc decreases due to which the induced current flows along bacb, so as to oppose the change in flux. (iii) As the magnetic flux decreases due to motion of the irregular shaped loop abcd out of the region of magnetic field, the induced current flows along cdabc, so as to oppose change in flux. Note that there are no induced current as long as the loops are completely inside or outside the region of the magnetic field.
Example 6. (a) A closed loop is held stationary in the magnetic field between the north and south poles of two permanent magnets held fixed. Can we hope to generate current in the loop by using very strong magnets? (b) A closed loop moves normal to the constant electric field between the plates of a large capacitor. Is a current induced in the loop (i) when it is wholly inside the region between the capacitor plates (ii) when it is partially outside the plates of the capacitor? The electric field is normal to the plane of the loop. (c) A rectangular loop and a circular loop are moving out of a uniform magnetic field region (Fig. 6.8) to a field-free region with a constant velocity v. In which loop do you expect the induced emf to be constant during the passage out of the field region? The field is normal to the loops.
E XAMPLE
(^) 6.
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where we have used d x /d t = – v which is the speed of the conductor PQ. The induced emf Blv is called motional emf. Thus, we are able to produce induced emf by moving a conductor instead of varying the magnetic field, that is, by changing the magnetic flux enclosed by the circuit. It is also possible to explain the motional emf expression in Eq. (6.5) by invoking the Lorentz force acting on the free charge carriers of conductor PQ. Consider any arbitrary charge q in the conductor PQ. When the rod moves with speed v , the charge will also be moving with speed v in the magnetic field B. The Lorentz force on this charge is qvB in magnitude, and its direction is towards Q. All charges experience the same force, in magnitude and direction, irrespective of their position in the rod PQ. The work done in moving the charge from P to Q is, W = qv B l
Since emf is the work done per unit charge,
W q
ε =
= Blv This equation gives emf induced across the rod PQ and is identical to Eq. (6.5). We stress that our presentation is not wholly rigorous. But it does help us to understand the basis of Faraday’s law when the conductor is moving in a uniform and time-independent magnetic field. On the other hand, it is not obvious how an emf is induced when a conductor is stationary and the magnetic field is changing – a fact which Faraday verified by numerous experiments. In the case of a stationary conductor, the force on its charges is given by F = q (E + v × B) = q E (6.6) since v = 0. Thus, any force on the charge must arise from the electric field term E alone. Therefore, to explain the existence of induced emf or induced current, we must assume that a time-varying magnetic field generates an electric field. However, we hasten to add that electric fields produced by static electric charges have properties different from those produced by time-varying magnetic fields. In Chapter 4, we learnt that charges in motion (current) can exert force/torque on a stationary magnet. Conversely, a bar magnet in motion (or more generally, a changing magnetic field) can exert a force on the stationary charge. This is the fundamental significance of the Faraday’s discovery. Electricity and magnetism are related.
Example 6.6 A metallic rod of 1 m length is rotated with a frequency of 50 rev/s, with one end hinged at the centre and the other end at the circumference of a circular metallic ring of radius 1 m, about an axis passing through the centre and perpendicular to the plane of the ring (Fig. 6.11). A constant and uniform magnetic field of 1 T parallel to the axis is present everywhere. What is the emf between the centre and the metallic ring?
E XAMPLE
(^) 6.
http://www.ngsir,netfirms.com/englishhtm/Induction.htm Interactive animation on motional emf:
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FIGURE 6. Solution Method I As the rod is rotated, free electrons in the rod move towards the outer end due to Lorentz force and get distributed over the ring. Thus, the resulting separation of charges produces an emf across the ends of the rod. At a certain value of emf, there is no more flow of electrons and a steady state is reached. Using Eq. (6.5), the magnitude of the emf generated across a length d r of the rod as it moves at right angles to the magnetic field is given by d ε = Bv d r. Hence,
0
d d
R
ε = ∫ ε=∫ Bv r
2
0
d 2
R (^) B R
= ∫ B ω r r =^ ω
Note that we have used v = ω r. This gives
ε 2
(^1) 1.0 2 50 (1 ) 2
= × × π × ×
= 157 V Method II To calculate the emf, we can imagine a closed loop OPQ in which point O and P are connected with a resistor R and OQ is the rotating rod. The potential difference across the resistor is then equal to the induced emf and equals B × (rate of change of area of loop). If θ is the angle between the rod and the radius of the circle at P at time t , the area of the sector OPQ is given by 2 1 2 2 2
π R × θ^ = R θ π where R is the radius of the circle. Hence, the induced emf is
ε = 2
d 1 d 2
B R t
× ⎡⎢^ θ⎤⎥ ⎣ ⎦
=
1 2 d^2 2 d 2
BR B^ R t
θ (^) = ω
[Note:
d (^2) d t
θ (^) = ω = π ν ] This expression is identical to the expression obtained by Method I and we get the same value of ε.
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Thus, mechanical energy which was needed to move the arm PQ is converted into electrical energy (the induced emf) and then to thermal energy. There is an interesting relationship between the charge flow through the circuit and the change in the magnetic flux. From Faraday’s law, we have learnt that the magnitude of the induced emf is,
B t
Φ ε
Δ
Δ However, Q Ir r t
ε
Δ = = Δ Thus,
Q B r
Δ Φ Δ =
Example 6.8 Refer to Fig. 6.12(a). The arm PQ of the rectangular conductor is moved from x = 0, outwards. The uniform magnetic field is perpendicular to the plane and extends from x = 0 to x = b and is zero for x > b. Only the arm PQ possesses substantial resistance r. Consider the situation when the arm PQ is pulled outwards from x = 0 to x = 2 b , and is then moved back to x = 0 with constant speed v. Obtain expressions for the flux, the induced emf, the force necessary to pull the arm and the power dissipated as Joule heat. Sketch the variation of these quantities with distance.
(a) FIGURE 6. Solution Let us first consider the forward motion from x = 0 to x = 2 b The flux ΦB linked with the circuit SPQR is ΦB = B l x 0 ≤ x < b = B l b b ≤ x < 2 b The induced emf is, d B d t
ε = −^ Φ = − Blv 0 ≤ x < b = 0 b ≤ x < 2 b
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E XAMPLE
(^) 6.
When the induced emf is non-zero, the current I is (in magnitude)
I Bl v r
=
(b) FIGURE 6.
The force required to keep the arm PQ in constant motion is I l B. Its direction is to the left. In magnitude 2 2 0 0 2
F B l v x b r b x b
= ≤ < = ≤ < The Joule heating loss is 2 P J = I r 2 2 2 0 0 2
B l v x b r b x b
= ≤ < = ≤ <
One obtains similar expressions for the inward motion from x = 2 b to x = 0. One can appreciate the whole process by examining the sketch of various quantities displayed in Fig. 6.12(b).
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(iii) Induction furnace : Induction furnace can be used to produce high temperatures and can be utilised to prepare alloys, by melting the constituent metals. A high frequency alternating current is passed through a coil which surrounds the metals to be melted. The eddy currents generated in the metals produce high temperatures sufficient to melt it. (iv) Electric power meters : The shiny metal disc in the electric power meter (analogue type) rotates due to the eddy currents. Electric currents are induced in the disc by magnetic fields produced by sinusoidally varying currents in a coil. You can observe the rotating shiny disc in the power meter of your house.
ELECTROMAGNETIC DAMPING
Take two hollow thin cylindrical pipes of equal internal diameters made of aluminium and PVC, respectively. Fix them vertically with clamps on retort stands. Take a small cylinderical magnet having diameter slightly smaller than the inner diameter of the pipes and drop it through each pipe in such a way that the magnet does not touch the sides of the pipes during its fall. You will observe that the magnet dropped through the PVC pipe takes the same time to come out of the pipe as it would take when dropped through the same height without the pipe. Note the time it takes to come out of the pipe in each case. You will see that the magnet takes much longer time in the case of aluminium pipe. Why is it so? It is due to the eddy currents that are generated in the aluminium pipe which oppose the change in magnetic flux, i.e., the motion of the magnet. The retarding force due to the eddy currents inhibits the motion of the magnet. Such phenomena are referred to as electromagnetic damping. Note that eddy currents are not generated in PVC pipe as its material is an insulator whereas aluminium is a conductor.
6.9 INDUCTANCE
An electric current can be induced in a coil by flux change produced by another coil in its vicinity or flux change produced by the same coil. These two situations are described separately in the next two sub-sections. However, in both the cases, the flux through a coil is proportional to the current. That is, ΦB α I. Further, if the geometry of the coil does not vary with time then, d d d d
B I t t
Φ ∝
For a closely wound coil of N turns, the same magnetic flux is linked with all the turns. When the flux ΦB through the coil changes, each turn contributes to the induced emf. Therefore, a term called flux linkage is used which is equal to N ΦB for a closely wound coil and in such a case
N ΦB ∝ I
The constant of proportionality, in this relation, is called inductance. We shall see that inductance depends only on the geometry of the coil
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and intrinsic material properties. This aspect is akin to capacitance which for a parallel plate capacitor depends on the plate area and plate separation (geometry) and the dielectric constant K of the intervening medium (intrinsic material property). Inductance is a scalar quantity. It has the dimensions of [M L^2 T –2^ A–2^ ] given by the dimensions of flux divided by the dimensions of current. The SI unit of inductance is henry and is denoted by H. It is named in honour of Joseph Henry who discovered electromagnetic induction in USA, independently of Faraday in England.
6.9.1 Mutual inductance
Consider Fig. 6.15 which shows two long co-axial solenoids each of length l. We denote the radius of the inner solenoid S 1 by r 1 and the number of turns per unit length by n 1. The corresponding quantities for the outer solenoid S 2 are r 2 and n 2 , respectively. Let N 1 and N 2 be the total number of turns of coils S 1 and S 2 , respectively. When a current I 2 is set up through S 2 , it in turn sets up a magnetic flux through S 1. Let us denote it by Φ 1. The corresponding flux linkage with solenoid S 1 is N 1 Φ 1 = M (^) 12 I 2 (6.9) M 12 is called the mutual inductance of solenoid S 1 with respect to solenoid S 2. It is also referred to as the coefficient of mutual induction. For these simple co-axial solenoids it is possible to calculate M 12. The magnetic field due to the current I 2 in S 2 is μ 0 n 2 I 2. The resulting flux linkage with coil S 1 is,
N 1 (^) Φ 1 = ( n l 1 ) (^) ( π r 1^2 ) ( μ 0 n I 2 2 ) 2 = μ 0 n n 1 (^) 2 π r l I 1 2 (6.10) where n 1 l is the total number of turns in solenoid S 1. Thus, from Eq. (6.9) and Eq. (6.10), M 12 = μ 0 n 1 n 2 π r (^) 12 l (6.11) Note that we neglected the edge effects and considered the magnetic field μ 0 n 2 I 2 to be uniform throughout the length and width of the solenoid S 2. This is a good approximation keeping in mind that the solenoid is long, implying l >> r 2. We now consider the reverse case. A current I 1 is passed through the solenoid S 1 and the flux linkage with coil S 2 is, N 2 Φ 2 = M 21 I 1 (6.12) M 21 is called the mutual inductance of solenoid S 2 with respect to solenoid S 1. The flux due to the current I 1 in S 1 can be assumed to be confined solely inside S 1 since the solenoids are very long. Thus, flux linkage with solenoid S 2 is
N (^) 2 Φ 2 = (^) ( n l 2 (^) ) ( π r 1^2 ) ( μ 0 n I 1 1)
FIGURE 6.15 Two long co-axial solenoids of same length l.
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Now, let us recollect Experiment 6.3 in Section 6.2. In that experiment, emf is induced in coil C 1 wherever there was any change in current through coil C 2. Let Φ 1 be the flux through coil C 1 (say of N 1 turns) when current in coil C 2 is I 2. Then, from Eq. (6.9), we have N 1 Φ 1 = MI 2 For currents varrying with time,
d ( 1 1 ) d( 2 )
d d
N MI t t
Φ
Since induced emf in coil C 1 is given by
d ( 1 1 )
N t
Φ ε 1 =
We get,
I M t
ε 1 =
It shows that varying current in a coil can induce emf in a neighbouring coil. The magnitude of the induced emf depends upon the rate of change of current and mutual inductance of the two coils.
6.9.2 Self-inductance
In the previous sub-section, we considered the flux in one solenoid due to the current in the other. It is also possible that emf is induced in a single isolated coil due to change of flux through the coil by means of varying the current through the same coil. This phenomenon is called self-induction. In this case, flux linkage through a coil of N turns is proportional to the current through the coil and is expressed as
N Φ B ∝ I
N Φ B (^) = LI (6.15) where constant of proportionality L is called self-inductance of the coil. It is also called the coefficient of self-induction of the coil. When the current is varied, the flux linked with the coil also changes and an emf is induced in the coil. Using Eq. (6.15), the induced emf is given by
d ( B)
N t
Φ ε =
d
I L t
ε = (^) (6.16)
Thus, the self-induced emf always opposes any change (increase or decrease) of current in the coil. It is possible to calculate the self-inductance for circuits with simple geometries. Let us calculate the self-inductance of a long solenoid of cross- sectional area A and length l , having n turns per unit length. The magnetic field due to a current I flowing in the solenoid is B = μ 0 n I (neglecting edge effects, as before). The total flux linked with the solenoid is
N Φ B = ( nl ) ( μ 0 n I ) ( A )
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= μ 0 n^2 Al I where nl is the total number of turns. Thus, the self-inductance is,
L I
=ΝΦ^ Β
2 = μ 0 n Al (6.17) If we fill the inside of the solenoid with a material of relative permeability μ r (for example soft iron, which has a high value of relative permiability), then, 2 L = μ μ r 0 n Al (6.18) The self-inductance of the coil depends on its geometry and on the permeability of the medium. The self-induced emf is also called the back emf as it opposes any change in the current in a circuit. Physically, the self-inductance plays the role of inertia. It is the electromagnetic analogue of mass in mechanics.
So, work needs to be done against the back emf ( ε ) in establishing the
current. This work done is stored as magnetic potential energy. For the current I at an instant in a circuit, the rate of work done is d d
W I t
= ε
If we ignore the resistive losses and consider only inductive effect, then using Eq. (6.16),
d d d d
W I L I t t
=
Total amount of work done in establishing the current I is
0
d d
I
W = ∫ W =∫ L I I
Thus, the energy required to build up the current I is,
(^12) 2
W = LI (6.19)
This expression reminds us of mv^2 /2 for the (mechanical) kinetic energy of a particle of mass m , and shows that L is analogus to m (i.e., L is electrical inertia and opposes growth and decay of current in the circuit). Consider the general case of currents flowing simultaneously in two nearby coils. The flux linked with one coil will be the sum of two fluxes which exist independently. Equation (6.9) would be modified into
N 1 Φ = 1 M 11 (^) I 1 (^) + M 12 (^) I 2 where M 11 represents inductance due to the same coil. Therefore, using Faraday’s law,
1 11 d^1 12 d^2 d d
I I M M t t
ε = − −
