A SIMPLE NON-MATHEMATICAL PROOF OF LENZ’S LAW

Editor: B. Somanathan Nair

ABSTRACT: In 1831, Michael Faraday enunciated the law of electromagnetic induction. This law states that whenever a conductor cuts a magnetic field, an electromotive force (EMF) is induced in it. In 1835, Heinrich Lenz enunciated the Lenz's law, which states that when an EMF is generated by a change in magnetic flux as per Faraday's Law, the polarity of the induced EMF (or, voltage) is such that it produces a current whose magnetic field opposes the change which has produced it. This law also has been accepted (just like the Faraday’ law) by the scientific world as such without any modification for nearly two centuries now. The proof of this law is usually given on the basis of the Law of Conservation of Energy, which involves complex mathematical explanations. This paper gives a very simple, naturally logical, and non-mathematical proof of the Lenz’s law. In this connection, it may be noted that this paper is an extension of our previous blog on Faraday’s law..

1.    INTRODUCTION

       Lenz’s law has been regarded as similar to Newton’s third law of motion, which states that for every action there is an equal and opposite reaction. Considering this law, we find that, if a current produces a magnetic field, it is natural to assume that this action has a reaction by which the generated magnetic field produces a reverse current which then naturally is opposite to the first current. The proofs given so far have been based on this concept and scientists used the Law of Conservation of Energy to prove the Lenz’s law. In one of the articles on Lenz’s law, the concepts of pressurized aether and electron-positron dipole are used for proving the Lenz’s law¹. We now state that Lenz’s law is not at all that complicated and can be proved non-mathematically by using simple and natural logic.

2. MAGNETIC INDUCTION²

      Consider an experiment in which a straight copper conductor AB being applied with an ac voltage across its terminals, as shown in Fig. 1. A straight-conductor concept is used here for simplifying the explanation.

      During positive half-cycles (PHCs) of the input ac voltage, when the top terminal A of the conductor is positive with respect to its bottom terminal B, free electrons in the conductor move upwards through it towards A and an electron current I_1e flows through the primary conductor from B to A. This is indicated by green-coloured dotted-line block arrow in Fig. 1. The conventional current I_1p, corresponding to I_1e, indicated by the orange-coloured block arrow in Fig. 1, flows in a direction opposite to that of I_1e.

     Now, since electrons are also tiny magnets (dual property of electrons), as they move upwards, the magnetic field (indicated by blue-coloured circular dotted-line thin arrows) associated with them will also move upwards. It may also be noted that this magnetic field is oriented in a direction perpendicular to that of the electron flow. This is illustrated in Fig. 1. It may further be noted that since the applied ac voltage is sinusoidal, the current and hence the magnetic field generated are also sinusoidal.

      Let us now assume that a second conductor CD (with terminals C and D being shorted through a suitable load resistance R_L) be placed inside the same magnetic field, as shown in Fig. 1. It can be easily observed that in this case, the magnetic field produced by I_1e in the first conductor AB in turn interacts with the free electrons in the second conductor CD and deflects them so that they move in a downward direction through it. This reversal of current flow is quite natural because it is the upward motion of electrons in AB that produced the magnetic field; this field in turn produces the motion of electrons in CD. Since this is a reverse process, naturally the direction of current flow in CD must be opposite to that in AB. The reversed electron current I_2e and corresponding conventional current (I_2p) are indicated in Fig. 1 using the cyan-coloured and blue-coloured block arrows, respectively.

     From the discussions given above, it has now been proved that a current will be induced in any secondary coil placed inside the magnetic field generated by a primary current. This principle may be extended in the form of a general statement:

Current will be induced in all the secondary conductors placed inside the magnetic field generated by the current flowing through a primary conductor. This is true for all the conductors located near or far away from the primary provided that the effect of the magnetic field generated by it is sensed at these locations.

     The general statement given above is illustrated in Fig. 2. In this figure, P is the primary conductor, which carries conventional primary current I₁ (represented by the longest orange-coloured block arrow). There are n secondary conductors S₁ to S_n, located at different distances inside the same magnetic field produced by I₁. The induced secondary conventional currents I₂₁ to I_2n are indicated by green-coloured block arrows drawn on each secondary conductor. The lengths of these arrows are shown as decreasing with increasing distance from P. This indicates that the magnitude of the current induced in a secondary conductor decreases as the distance between that conductor and P increases. The directions of the currents shown in Fig. 2 are for positive half-cycles of the input voltage (for negative half-cycles, these directions reverse). 

     Since the statement given above is true, it also suggests that there will be a current induced in the primary conductor itself, considering it as a secondary conductor placed inside the same magnetic field generated by the primary current. Since we are now considering the primary conductor as a secondary conductor, the current I₂ induced in it by the magnetic field due to primary current I₁ will be in a direction opposite to that of I₁. These actions are illustrated in Fig. 3. Here, I₂ is produced as a self-induced current and hence this process is called self-induction.

       The actions explained above is in effect is the statement of the Lenz’s law. It can be seen that this effect is produced because the same coil can act as the primary and secondary coils. Thus the proof of the Lenz’s law is very simple and straightforward; it does not require the support of the Law of Conservation of Energy (which is currently used for its proof) to prove it.

3.    SELF-INDUCTANCE REDEFINED

Lenz’s law suggests that an ac current is generated within a conductor when an externally applied ac voltage drives an ac current through it; the direction of flow of this induced current is opposite to that which has generated it. This in effect suggests that the new induced current opposes the very current that has generated it. This gives rise to what is known as self-inductance.

Note: In this article, we have frequently used the term ac current for alternating current. Even though, in reality ac means alternating current, we have used it here just to mean it as alternating or varying. This is in conformity with usages such as ac voltage, ac magnetic field, and ac light.

4.    SUMMARY

We have presented through this article a very simple and logical proof of the Lenz’s law. This has the following features:

      1. There is no mathematics involved in this proof.

      2. It does not make use of the Law of Conservation of Energy.

      3. It also does not use any aether concept.

      4. It does not make use of the electron-positron dipole concept.

5. It makes use of the Faraday’s law of magnetic induction in its modified form, which states that the primary itself can be regarded as a secondary.

5.    ACKNOWLEDGEMENT

We state that the figures shown in this article are drawn using the DRAWING TOOLS available in the Microsoft word software. It may be noted that these tools are extremely useful to prepare drawings in two and three dimensions. In all our articles, we shall be using these tools. We acknowledge our gratitude to the Microsoft Corporation for creating this excellent software package.

  

6. REFERENCES

1. Frederick David Tombe: “Lenz’s Law”, The General Science Journal, 2009.

2. B. Somanathan Nair, P. S. Chandramohan Nair, S. R. Deepa, N. Anand: "Some New Perceptions on the Magnetic Field and the Radiating Properties of Antennae", IEEE International Workshop on Optical Networking Technologies and Data Security (ONTDS), 2014.

A RE-LOOK INTO FARADAY’S LAW OF ELECTROMAGNETIC INDUCTION BASED ON ELECTRON THEORY

EDITOR: B. SOMANATHAN NAIR

ABSTRACT: Michael Faraday enunciated the law of electromagnetic induction in 1831. This law states that whenever a conductor cuts a magnetic field, an electromotive force (EMF) is induced in it. This law has been accepted by the scientific world as such without any modification for nearly two centuries now. In this paper, we investigate the completeness of the statement of Faraday’s law and suggest some modifications in its statement so that it explains the action of what is known as (electro) magnetic induction. 

1.    INTRODUCTION

Faraday’s law of electromagnetic induction (or simply, magnetic induction¹) states that whenever a conductor cuts a magnetic field, an electromotive force (EMF) is induced in it. Everybody in the scientific world has accepted this law as such without any question being asked about its completeness. There is no doubt about the truthfulness and validity of this law. However, a couple of questions arise in this case which nobody has raised or answered so far:

      1. What is electromagnetic induction?

      2. What is the speed with which induction takes place in a conductor?

In this paper, we attempt to give answers to these important basic questions and suggest some more aspects that may be included in the Faraday’s law.

2. EXPERIMENTAL DEMONSTRATION OF THE PROCESS KNOWN AS ELECTROMAGNETIC INDUCTION

      Consider an experiment in which an ac voltage (or, EMF) is applied across the terminals of a straight metallic (say, copper) conductor AB as shown in Fig. 1. A straight-wire conductor is used here for the simplification of explanation.

    During positive half-cycles (PHCs) of the input voltage, when the top terminal A of the conductor is positive with respect to its bottom terminal B, free electrons in the conductor move upwards from B to A, which results in a primary electron current I_1e flowing from B to A. This is illustrated in Fig. 1 by the green-colored dotted-line block arrow shown as superimposed over conductor AB. In Fig. 1, we have also shown the conventional primary current I_1p corresponding to I_1e, indicated by the orange-colored block arrow. It must be remembered that the direction of electron current (due to the flow of negative charges) and that of the conventional current (due to the flow of positive charges) corresponding to this electron current are mutually opposite to each other. This is the reason why the green-colored and orange-colored block arrows are shown in opposite directions.

      Now, since electrons are also tiny magnets (their natural dual property), as they move upwards, the magnetic field associated with them (indicated by the blue-colored dotted-line circular arrows) will also move upwards. It may be noted that this magnetic field is oriented in a direction perpendicular to that of the electron flow, as illustrated in Fig. 1. It may also be noted that since the applied ac voltage is sinusoidal, the resultant primary current and the generated primary magnetic field are sinusoidal with their respective frequencies the same as that of the input voltage. This means that, if the input voltage is of 50 Hz, then the generated alternating current and ac magnetic field are also of 50 Hz.

     Let us now assume that a second copper conductor CD (with terminals C and D being connected through a suitable load resistance R_L) be placed inside the same magnetic field, as shown in Fig. 1. It can be easily observed that in this case, the magnetic field produced by the electron current flow I_1e in the primary conductor AB in turn interacts with the free electrons in the secondary conductor CD and deflects them so that they move in a downward direction through it (i.e., CD). This reversal of current flow is quite natural because it is the upward motion of electrons in AB that produced the magnetic field; this field in turn produces the motion of electrons in CD. Since this is a reverse process of the first action, naturally the direction of the current flow in CD must be opposite to that in AB. The reversed secondary electron current I_2e and the conventional current (I_2p) corresponding to it are indicated in Fig. 1 using the cyan-coloured and the blue-coloured block arrows, respectively. It may further be noted that both I_2e and I_2p possess the same frequency as that of I_1e.

It can also be seen that the generated primary magnetic field is an alternating quantity, it spreads around the primary conductor with the same frequency as that of the primary current and at a velocity equal to that of the light. In this connection, we remember that the magnetic force generated by a magnet can be felt at places far away from its original location (natural property of a magnet); the more powerful the magnet, the more the distance at which its effect is felt. Also, if this magnetic field is alternating with a frequency of f Hz, then its effect will be felt at a distance of 3×10⁸ meters at the same frequency of f Hz after 1 second. This means that current is generated (or, induced) in the secondary coil with a frequency of f Hz and at a speed equal to that of the light. It may further be noted that it is the magnetic field (in the form of waves) that is spreading through space; there is no electric-field component attached to this. Actually, electric part of these magnetic waves occurs only when they are intercepted (or cut) by a conductor and then they induce a current in it. Thus the term electromagnetic wave is a misnomer. It must be actually redesignated as magnetic wave and not as electromagnetic wave.¹

From the discussions given above, we observe that when a conductor is placed inside a varying magnetic field, a varying current of the same frequency is produced in it due to the deflection of free electrons in it by that magnetic field. This clearly suggests that it is an alternating current, and not an EMF, that is generated in a conductor when it is placed in an alternating magnetic field. The EMF mentioned in the Faraday’s law can be seen to be the potential drop that is produced in the conductor due to this current flow. However, due to long-term usage, the term magnetic induction can be used to indicate the generation of an electromotive force (voltage) also.

Figure 2 shows the situation during negative half-cycles (NHCs) of the applied ac voltage. During the negative half-cycles, terminal B becomes positive with respect to terminal A and hence electrons in the conductor move downwards reversing the direction of current I_1e through AB. In this condition, we notice that the associated magnetic field has also reversed. This in turn results in an upward motion of electrons through conductor CD. Thus we find that current I_2e also gets reversed in this case, as shown in Fig. 2. 

          

3.    STATE OF AFFAIRS WHEN THE SECONDARY IS OPEN

In Section 2, we had assumed that the secondary is shorted through a load resistance. A pertinent question arises here: What will happen if the secondary terminals are kept open? In fact, original Faraday’s law states that a voltage is induced in the secondary when a magnetic field is cut by that conductor. This statement suggests that the secondary is open.

According to Section 2, to have a current flowing through it, the secondary must form a closed path; then only electrons in that coil can move to produce the current. However, how can a current flow through a conductor if it is open?

  The problem given above can be solved by considering the fact that there always exists a very low-value parasitic capacitance with air as dielectric across the terminals of a conductor carrying currents of opposite polarity (a basic property of a capacitance). Based on this property, we find that there always exists a parasitic capacitance C_P across secondary terminals C and D as shown in Fig. 3. It can be easily seen that it is this capacitance C_P that completes the required secondary path through which the secondary current is flowing. This current in turn develops a voltage drop across CD, which forms the open-circuit secondary voltage (as per Faraday’s law).

A similar action as given above takes place during the negative half-cycles. The only difference is that, as illustrated earlier, the directions of currents reverse in this case from those shown in Fig. 3. However, no figure is given for this case, since it is similar to Fig. 2, with R_L replaced with C_P.

It is found that C_P will be usually in the range of a few picofarads so that the secondary current will be usually very small (maybe on the order of a few picoamperes). The following calculations will prove this fact.

Let the secondary voltage be equal to 10 volts and let C_P be equal to 10 pF. Then, for a 50-Hz ac, the capacitive reactance X_CP = 1/2πfC = 318.3×10⁶ ohms. The secondary current, therefore, will be 10/318.3×10⁶ = 31 nanoamperes (approximately). This current is usually considered as negligible. However, it may be noted that if the input frequency is raised to 1 GHz, the secondary current will be about 0.6 ampere, which is a large value compared to 31 nA.

4.    SUGGESTED NEW ADDITIONS TO BE INCORPORATED INTO THE EXPLANATION OF ORIGINAL FARADAY’S LAW

      We now suggest that the following points may be incorporated into the explanation of original Faraday’s law so that it may become more clarified.

      Whenever a conductor cuts a magnetic field, free electrons in it get deflected by the field producing a current flow through it. This current will be a transient direct-current spike if the magnetic field is a DC field; it will be an alternating current if the field is an ac magnetic field and continuous if the field is continuous. This current completes its path through the parasitic capacitance existing across the terminals of the conductor and develops a potential (or EMF) across its terminals. 

      As an alternative to the above, we may state: An ac current flowing through a conductor produces an ac magnetic field which spreads in the space surrounding it with a velocity equal to that of the light and deflects the free electrons in another conductor placed inside the same magnetic field; this produces an ac motion of the electrons resulting in an ac current being driven through it in a direction opposite to that in the first conductor; this ac current in turn produces an ac voltage drop across the terminals of the second conductor, which becomes the induced EMF in it.

The statements given above can be elaborated into the following actions, which may be used in explaining the Faraday’s law.

1. A DC magnetic field will produce a dc current spike or transient in conductor that cuts it.

2.  If the magnetic field is alternating and continuous, the generated current will also be alternating and continuous.

3.  In both the cases mentioned above, the current will flow through a path completed by the parasitic capacitance existing across the terminals of the conductor; producing a voltage drop (or EMF) in it.

4. Alternatively, if the magnetic field is generated by applying an ac voltage across a conductor (called the primary conductor), then free electrons in it gets deflected by the generated field producing an alternating motion of the electrons. This produces an ac current that flows through the primary conductor.

5. Since electrons are also tiny magnets (dual property), when they move through a conductor, they produce a moving ac magnetic field surrounding it in a direction perpendicular to the axis of the conductor.

6. The generated ac magnetic field spreads through the space surrounding the primary conductor with a velocity equal to that of the light.

7. Since this magnetic field is alternating in nature, it will interact with the electrons in a second conductor (called the secondary conductor) placed inside the same field and deflect them magnetically.

8. The deflection of electrons in the secondary conductor in turn produces an alternating current in it, when the secondary is closed through a connected load.

9. The direction of this secondary current can be seen to be opposite to that of the primary current. This is because the spreading magnetic field is produced by the primary current; this magnetic field in turn produces the secondary current. Since these two actions are mutually opposite to each other, direction of the current flow in the secondary has to be (and will be) naturally opposite to that of the current flow in the primary.

10. If the secondary is open (instead of a closed one, as proposed in the original Faraday’s law), then the secondary can be assumed to form a closed path through the invisible parasitic air capacitance of very low value that will always exist across the open-circuited secondary terminals. The secondary ac current (of very low value) will then flow through this completed path and produce a voltage drop across the secondary terminals, which will then act as the open-circuit secondary voltage (since the parasitic capacitance is invisible).

11. From the above arguments, it is also clear that current will be generated in all the conductors placed inside the same magnetic field.

12. In actual practice, straight conductors are replaced with multi-turn coils.

5.    SUMMARY

Through this article, we have introduced a few links that we feel are missing in the original Faraday’s law. In particular, we have:

1.  Given some more explanation required to define the process known as electromagnetic induction.

2.   Specified the speed with which induction takes place in a second conductor.

3. Explained on how the direction of the induced secondary current gets reversed with respect to the direction of its generating current.

4.   Given an explanation on how magnetic induction produces a voltage across an open-circuited secondary.

5.   Mentioned the fact that current gets induced in all the conductors placed inside the same primary magnetic field.

6.    ACKNOWLEDGEMENTS

It is specially mentioned here with thanks that the figures shown in this article are drawn using the DRAWING TOOLS available in the Microsoft word software. It may be noted that these tools are extremely useful to prepare drawings in two and three dimensions. In all our articles, we shall be using these tools. We acknowledge our gratitude to the Microsoft Corporation for creating this excellent software package.

7. REFERENCE

1. B. Somanathan Nair, P. S. Chandramohan Nair, S. R. Deepa, N. Anand: "Some New Perceptions on the Magnetic Field and the Radiating Properties of Antennae", IEEE International Workshop on Optical Networking Technologies and Data Security (ONTDS), 2014.