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.