LENZ'S LAW: FORMULA, EQUATIONS, APPLICATIONS, EXAMPLES - PHYSICAL - 2025

The Lenz 's law states that the polarity of the induced electromotive force in a closed circuit due to variation in the magnetic field flux is such that opposes the change in said flow.

The negative sign that precedes Faraday's law takes Lenz's law into consideration, being the reason why it is called Faraday-Lenz law and which is expressed as follows:

Figure 1. A toroidal coil is capable of inducing currents in other conductors. Source: Pixabay.

Formulas and equations

In this equation, B is the magnitude of the magnetic field (without bold or arrow, to distinguish the vector from its magnitude), A is the area of ​​the surface crossed by the field and θ is the angle between the vectors B and n.

The magnetic field flux can be varied in different ways over time, to create an induced emf in a loop - a closed circuit - of area A. For example:

-Making the magnetic field variable with time: B = B (t), keeping the area and the angle constant, then:

Applications

The immediate application of Lenz's law is to determine the direction of the induced emf or current without the need for any calculation. Consider the following: you have a loop in the middle of a magnetic field, such as that produced by a bar magnet.

Figure 2. Application of Lenz's Law. Source: Wikimedia Commons.

If the magnet and the loop are at rest relative to each other, nothing happens, that is, there will be no induced current, because the magnetic field flux remains constant in that case (see figure 2a). For current to be induced, the flux must vary.

Now, if there is a relative movement between the magnet and the loop, either by moving the magnet towards the loop, or towards the magnet, there will be induced current to measure (Figure 2b onwards).

This induced current in turn generates a magnetic field, therefore we will have two fields: the magnet B ₁ in blue and the one associated with the current created by induction B ₂, in orange.

The rule of the right thumb allows to know the direction of B ₂, for this the thumb of the right hand is placed in the direction and direction of the current. The other four fingers indicate the direction in which the magnetic field bends, according to figure 2 (below).

Magnet movement through the loop

Let's say the magnet is dropped towards the loop with its north pole directed towards it (figure 3). The field lines of the magnet leave the north pole N and enter the south pole S. Then there will be changes in Φ, the flux created by B ₁ through the loop: Φ increases! Therefore in the loop a magnetic field B _{2 is created} with the opposite intention.

Figure 3. The magnet moves towards the loop with its north pole towards it. Source: Wikimedia Commons.

The induced current runs counterclockwise, -red arrows in Figures 2 and 3-, according to the right thumb rule.

Let's move the magnet away from the loop and then its Φ decreases (figures 2c and 4), therefore the loop rushes to create a magnetic field B ₂ in the same direction, to compensate. Therefore, the induced current is hourly, as shown in figure 4.

Figure 4. The magnet moves away from the loop, always with its north pole pointing towards it. Source: Wikimedia Commons.

Reversing the position of the magnet

What happens if the position of the magnet is reversed? If the south pole points toward the loop, the field points upward, since the lines of B in a magnet leave the north pole and enter the south pole (see figure 2d).

Immediately Lenz's law informs that this vertical field upwards, rushing towards the loop, will induce in it an opposite field, that is, B ₂ downwards and the induced current will also be hourly.

Finally the magnet moves away from the loop, always with its south pole pointing towards the inside of it. Then a field B ₂ is produced inside the loop to help ensure that moving away from the magnet does not change the field flux in it. Both B ₁ and B ₂ will have the same meaning (see figure 2d).

The reader will realize that, as promised, no calculations have been made to know the direction of the induced current.

Experiments

Heinrich Lenz (1804-1865) carried out numerous experimental works throughout his scientific career. The best known are those we have just described, dedicated to measuring the magnetic forces and effects created by abruptly dropping a magnet in the middle of a loop. With his results he refined the work done by Michael Faraday.

That negative sign in Faraday's law turns out to be the experiment for which he is most widely recognized today. Nonetheless, Lenz did a lot of work in geophysics during his youth, meanwhile he was engaged in dropping magnets into coils and tubes. He also did studies on the electrical resistance and conductivity of metals.

In particular, on the effects that the increase in temperature has on the resistance value. He did not fail to observe that when a wire is heated, the resistance decreases and heat is dissipated, something that James Joule also observed independently.

To always remember his contributions to electromagnetism, in addition to the law that bears his name, inductances (coils) are denoted by the letter L.

Lenz tube

It is an experiment in which it is demonstrated how a magnet slows down when it is released into a copper tube. When the magnet falls, it generates variations in the magnetic field flux inside the tube, as happens with the current loop.

Then an induced current is created that opposes the change in flow. The tube creates its own magnetic field for this, which, as we already know, is associated with the induced current. Suppose the magnet is released with the south pole down, (Figures 2d and 5).

Figure 5. Lenz's tube. Source: F. Zapata.

As a result, the tube creates its own magnetic field with a north pole down and a south pole up, which is equivalent to creating a pair of dummy magnets, one above and one below the one that is falling.

The concept is reflected in the following figure, but it is necessary to remember that the magnetic poles are inseparable. If the lower dummy magnet has a north pole down, it will necessarily be accompanied by a south pole up.

As opposites attract and opposites repel, the falling magnet will be repelled, and at the same time attracted by the upper fictitious magnet.

The net effect will always be braking even if the magnet is released with the north pole down.

Joule-Lenz law

The Joule-Lenz law describes how part of the energy associated with the electric current that circulates through a conductor is lost in the form of heat, an effect that is used in electric heaters, irons, hair dryers and electric burners, among other appliances.

All of them have a resistance, filament or heating element that heats up as the current passes.

In mathematical form, let R be the resistance of the heating element, I the intensity of current flowing through it, and t the time, the amount of heat produced by the Joule effect is:

Where Q is measured in joules (SI units). James Joule and Heinrich Lenz discovered this effect simultaneously around 1842.

Examples

Here are three important examples where the Faraday-Lenz law applies:

Alternating current generator

An alternating current generator transforms mechanical energy into electrical energy. The rationale was described at the beginning: a loop is rotated in the middle of a uniform magnetic field, like that created between the two poles of a large electromagnet. When N turns are used, the emf increases proportionally to N.

Figure 6. The alternating current generator.

As the loop spins, the vector normal to its surface changes its orientation with respect to the field, producing an emf that varies sinusoidally with time. Suppose that the angular frequency of rotation is ω, then by substituting in the equation given at the beginning, we will have:

Transformer

It is a device that allows obtaining a direct voltage from an alternating voltage. The transformer is part of countless devices, like a cell phone charger, for example, it works as follows:

There are two coils wound around an iron core, one is called primary and the other secondary. The respective number of turns is N ₁ and N ₂.

The primary coil or winding is connected to an alternating voltage (such as a household electricity socket, for example) in the form V _P = V ₁.cos ωt, causing an alternating current of frequency ω to circulate inside it.

This current causes a magnetic field that in turn causes an oscillating magnetic flux in the second coil or winding, with a secondary voltage of the form V _S = V ₂.cos ωt.

Now, it turns out that the magnetic field inside the iron core is proportional to the inverse of the number of turns of the primary winding:

And so will V _P, the voltage in the primary winding, while the induced emf V _S in the second winding is proportional, as we already know, to the number of turns N ₂ and also to V _P.

So combining these proportionalities we have a relationship between V _S and V _P that depends on the quotient between the number of turns of each one, as follows:

Figure 7. The transformer. Source: Wikimedia Commons. KundaliniZero

The metal detector

They are devices used in banks and airports for security. They detect the presence of any metal, not just iron or nickel. They work thanks to the induced currents, through the use of two coils: a transmitter and a receiver.

A high frequency alternating current is passed in the transmitter coil, so that it generates an alternating magnetic field along the axis (see figure), which induces a current in the receiver coil, something more or less similar to what happens with the transformer.

Figure 8. Principle of operation of the metal detector.

If a piece of metal is placed between both coils, small induced currents appear in it, called eddy currents (which cannot flow in an insulator). The receiving coil responds to the magnetic fields of the transmitting coil and those created by eddy currents.

Eddy currents try to minimize the magnetic field flux in the piece of metal. Therefore, the field perceived by the receiving coil decreases when a metallic piece is interposed between both coils. When this happens an alarm is triggered that warns of the presence of a metal.

Exercises

Exercise 1

There is a circular coil with 250 turns of 5 cm radius, located perpendicular to a magnetic field of 0.2 T. Determine the induced emf if in a time interval of 0.1 s, the magnitude of the magnetic field doubles and indicate the direction of the current, according to the following figure:

Figure 9. Circular loop in the middle of a uniform magnetic field perpendicular to the plane of the loop. Source: F. Zapata.

Solution

First we will calculate the magnitude of the induced emf, then the direction of the associated current will be indicated according to the drawing.

Since the field has doubled, so has the magnetic field flux, therefore an induced current is created in the loop that opposes said increase.

The field in the figure points to the inside of the screen. The field created by the induced current must leave the screen, applying the rule of the right thumb, it follows that the induced current is counterclockwise.

Exercise 2

A square winding is made up of 40 turns of 5 cm on each side, which rotate with a frequency of 50 Hz in the middle of a uniform field of magnitude 0.1 T. Initially the coil is perpendicular to the field. What will be the expression for the induced emf?

Solution

From previous sections this expression was deduced:

References

1. Figueroa, D. (2005). Series: Physics for Science and Engineering. Volume 6. Electromagnetism. Edited by Douglas Figueroa (USB).
2. Hewitt, Paul. 2012. Conceptual Physical Science. 5th. Ed. Pearson.
3. Knight, R. 2017. Physics for Scientists and Engineering: a Strategy Approach. Pearson.
4. OpenStax College. Faraday's Law of Induction: Lenz's Law. Recovered from: opentextbc.ca.
5. Physics Libretexts. Lenz's Law. Recovered from: phys.libretexts.org.
6. Sears, F. (2009). University Physics Vol. 2.