Lenz's Law (Physics) Definition, Equation & Examples

Heinrich Lenz (also referred to as Emil Lenz) was a Baltic-German physicist who may not have the fame of some of his early 19th-century peers like Michael Faraday, but who still contributed a key piece to solving the mysteries of electromagnetism.

While some of his peers were making similar discoveries, Lenz’s name was given to ​Lenz’s law​ ​in large part because of his fastidious note-taking, comprehensive documentation of his experiments and a dedication to the scientific method​ uncommon for the time. The law itself forms an important part of ​Faraday’s law of electromagnetic induction​, and specifically tells you the ​direction​ in which the induced current flows.

The law might be difficult to get your head around at first, but once you grasp the key concept, you’ll be well on your way to a much deeper understanding of electromagnetism, including practical issues like the problem of eddy currents.

Faraday’s Law

Faraday’s law of induction states that the induced ​electromotive force​ (EMF, commonly referred to as “voltage”) in a coil of wire (or simply, around a loop) is minus the rate of change of magnetic flux through that loop. Mathematically, and replacing the derivative with a simpler “change in” (represented by ∆), the law states:

\text{induced EMF} = −N \frac{∆ϕ}{∆t}

Where ​t​ is time, ​N​ is the number of turns in the coil of wire and phi (ϕ) is the magnetic flux. The definition of magnetic flux is pretty important to this equation, so it’s worth remembering that it’s:

ϕ = \bm{B ∙ A} = BA \cos (θ)

which relates the strength of the magnetic field, ​B​, to the area of the loop ​A​, and the angle between the loop and the field (​θ​), with the loop angle defined as perpendicular to the area (i.e., pointing straight out of the loop). Since the equation involves cos, it’s at the maximum value when the field is directly aligned with the loop, and at 0 when it’s perpendicular to the loop (i.e., “side-on”).

Taken together, these equations show that you can create an EMF in a coil of wire by changing the cross-sectional area ​A​, the strength of the magnetic field ​B​, or the angle between the area and the magnetic field. The magnitude of the induced EMF is directly proportional to the rate of change of these quantities, and of course it doesn’t have to just be one of these changing in order to induce the EMF.

Faraday’s law was used by James Clerk Maxwell as one of his four laws of electromagnetism, although it’s usually expressed as the line integral of the magnetic field around a closed loop (which is essentially another way of saying the induced EMF) and the rate of change is expressed as a derivative.

Lenz’s Law

Lenz’s law is encapsulated in Faraday’s law because it tells us the direction in which the induced electric current flows. The simplest way to state Lenz’s law is that changes in magnetic flux induce currents in a direction that ​opposes​ ​the change​ that caused it.

In other words, because when current flows it generates its own magnetic field, the direction of the induced current is such that the new magnetic field is in an opposite direction to the flux changes that created it. It’s encapsulated in Faraday’s law because of the negative sign; this tells you that the induced EMF opposes the original change in magnetic flux.

For a simple example, imagine a coil of wire with an external magnetic field pointing directly into it from the right hand side (i.e., into the center of the coil and with the field lines pointing to the left), and the external field then increasing in magnitude but maintaining the same direction. In this case, the induced current in the wire will flow so as to produce a magnetic field pointing out of the coil to the right.

If the external field decreased in magnitude instead, the induced current would flow so as to produce a magnetic field in the same direction as the original field, because it counteracts flux ​changes​ rather than simply opposing the field. Since it ​counteracts the change and not necessarily the direction​, this means it sometimes creates a field in the opposite direction and sometimes in the same direction.

You can use the right-hand rule (sometimes called the right-hand grip rule to distinguish it from the other right-hand rule used in physics) to determine the direction of the resulting electric current. The rule is quite easy to apply: work out the direction of the magnetic field created by the induced current and point the thumb of your right hand in that direction, and then curl your fingers inwards. The direction your fingers curl is the direction the current flows through the coil of wire.

Examples of Lenz’s Law

Some concrete examples of how Lenz’s law works in practice will help to cement the concepts, and the simplest is very similar to the example above: a coil of wire moving into or out of a magnetic field. As the loop moves into the field, the magnetic flux through the loop will increase (in the opposite direction to the motion of the coil), inducing a current that opposes the rate of change of flux, and thus creates a magnetic field in the direction of its motion.

If the coil is moving towards you, the right-hand rule and Lenz’s law show that the current would flow in the counter-clockwise direction. If the coil was moving ​out​ of the field, the changing magnetic flux would basically be a gradual reduction instead of an increase, so the exact opposite current would be induced.

This situation is analogous to moving a bar magnet into or out of the center of a coil, because when moving the magnet in, the field would be getting stronger and the induced magnetic field would work to oppose the motion of the magnet, so, counter-clockwise from the perspective of the magnet. When moving out of the center of the coil of wire, the magnetic flux would be decreasing and the induced magnetic field would again work to oppose the motion of the magnet, this time clockwise from the perspective of the magnet.

A more complicated example involves a coil of wire rotating in a fixed magnetic field, because as the angle changes, the flux through the loop would as well. During the decrease in flux, the induced electric current would create a magnetic field to oppose the flux changes, so it would be in the same direction as the external field. During the increase in flux, the opposite happens and the current is induced to oppose the increase in magnetic flux, so in the opposite direction to the external field. This generates an alternating voltage (because the induced EMF switches every time the loop rotates 180 degrees), and this can be used to generate alternating current.

Lenz’s Law and Eddy Currents

An eddy current is the name for the small electric currents that obey Lenz’s law. In particular, though, this name is used in reference to small, looping currents in conductors analogous to the vortices that you see around your oars when rowing in water.

When a conductor is moved through a magnetic field – for example, like a metal pendulum swinging between the poles of a horseshoe magnet – eddy currents are induced, and in line with Lenz’s law, these counteract the effect of the motion. This leads to magnetic damping (as the induced field necessarily works ​against​ the motion that created it), which can be used productively in things like magnetic braking systems for roller coasters, but it is a cause of wasted energy for devices like generators and transformers.

When eddy currents need to be reduced, the conductor is separated into multiple sections by thin insulating layers, which limit the size of the eddy currents and reduce energy loss. However, since eddy currents are a necessary consequence of Faraday’s and Lenz’s laws, they cannot be entirely prevented.

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