Lenz’s Law

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    In the essay on Maxwell’s Equations I invoked Lenz’s law, but said very little about it. As far as physicists are concerned, Lenz’s law is a minor sidebar to the larger electromagnetic theory. It’s actually meant for electrical engineers. But as such, it gives us a good example of how we transform fundamental physical theory into practical engineering.

    We begin with the third of Maxwell’s Equations, which describes Michael Faraday’s law of electromagnetic induction. It is a vector differential equation drawn as

(Eq’n 1)

in which E represents an electric field and B represents a magnetic induction field. Those fields exist on a surface S that has directed differential elements da (which elements are vectorized by the statement that the arrow points perpendicular to the area in accordance with the right hand rule: if the fingers of your right hand curve in the direction we proceed around the boundary of the patch da, your right thumb points in the direction of the vector). We now produce the dot product of Equation 1 with each of those elemental patches and calculate the Riemannian sum of all of the contributions from the surface S to get

(Eq’n 2)

A strange theorem of George Gabriel Stokes tells us that integrating the curl of a vector field over a smooth surface gives us the same result as does integrating the vector field itself as a line integral over the boundary C of the surface; that is,

(Eq’n 3)

in which dλ represents a differential element of the length of the line tracing the boundary. Using that result to substitute into Equation 2 and noting that the operations of partial differentiation and integration commute with each other gives us

(Eq’n 4)

That equation expresses the integral form of Faraday’s law of electromagnetic induction, which tells us that a changing magnetic flux on some surface S generates a net electromotive force on the boundary C defining that area.

    That equation still tells us about fields and abstract geometry. It’s still not completely suitable for engineering. The only part of that equation that we could draw in a picture is the boundary C; the rest is just too abstract. For the purpose of engineering we want to convert the equation into something that describes how to build something that will perform a task. We want to build some kind of electrical machinery.

    We start by imagining that we have laid a wire N times around C. If we recognize the integral on the left side of Equation 4 as representing the voltage V induced in the wire and recognize the integral on the right side of the equation as representing the magnetic flux (represented by the Greek letter phi) enclosed within the area bounded by the wire, then we can rewrite Equation 4 as

(Eq’n 5)

That equation presents Faraday’s law in the form known to engineers as Lenz’s Law, which was named for Russian physicist Heinrich Friedrich Emil Lenz (1804 Feb 12 - 1865 Feb 10). It tells us that if we do anything that changes the magnetic flux through the area enclosed by the wire, that change will induce in the wire a voltage that will, in accordance with Ohm’s law (V=IR), produce an electric current in the wire. Note that we have tacitly assumed that we can build something that will produce and change a magnetic flux.

    We can extract two useful pieces of information from Equation 5. The first fact tells us that we can make our wire loop generate any voltage that we want just by making the right number of turns around the boundary C, but that we cannot by that method control the amount of current produced. We need only substitute Ohm’s law into Equation 5 to see the truth of that statement. The resistance of a wire stands in direct proportion to its length, so the resistance in our coil stands in direct proportion to N (R=NRC, in which RC equals the resistance of one complete turn of the wire around the boundary C). Thus we see that the amount of current in the wire is not related to the number of turns in the coil, although the amount of current passing any given point on the coil does equal N times the current in the wire.

    The other fact involves the minus sign on the right side of Equation 5. For a full understanding of what that algebraic sign means, we must look back at Equation 4. Imagine looking down on the surface S and seeing the area vectors of the minuscule elements of that area pointing toward us. In that view the differential length vectors on the elements of the boundary point in the counterclockwise direction. If the magnetic field on the surface points even slightly toward us, then we must use a positive number to represent the magnetic flux. And if we use a positive number to represent the voltage, it means that the electric field on the boundary (and the current in the coil) points (flows) in the counterclockwise direction.

    If we have a positive magnetic flux and its magnitude increases (positive time derivative), then a negative voltage will appear in the coil. The resulting current, flowing around the boundary C in the clockwise direction, generates a magnetic field whose vectors on the surface S point more or less away from us, thereby constituting a negative flux on the surface. That negative flux opposes the change in the applied flux. If we apply that analysis to all four possibilities with the applied flux (positive and negative, increasing and decreasing), we will come to the same conclusion: the induced flux opposes the change in the applied flux. That’s what the minus sign in Equation 5 means.

    That opposition also produces a force that repels the source of the applied flux. About fifty years ago someone suggested the idea of a stove that would exploit that repulsion to float pans above the "burners". The electric current induced in the copper-bottomed pans would both levitate the pans in the flickering magnetic fields and heat them through resistance heating. It certainly looks like a nicely futuristic idea. However, stirring hot food in a floating pan would not be recommended for amateurs and the inability to control the heating finely would have made cooking a crude affair at best. The idea never caught on.

    Thus we see one way in which we can progress from abstract physical law to practical engineering formula to potential application.


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