Lenz’s Law and Electromagnetic Induction

Introduction: Magnetic Flux and a Conducting Loop

Consider a circular conducting coil placed in a magnetic field. The loop encloses a fixed area, and magnetic field lines pass through that area.

The magnetic flux through the coil depends on two factors: the strength of the magnetic field and how much of that field passes through the area of the loop.

Mathematically, magnetic flux is defined as the surface integral of the magnetic field over the area of the loop. In vector form, it is expressed as the dot product of the magnetic field vector and the area vector. This definition captures both magnitude and geometric orientation.

Increasing Magnetic Field: Flux Increases

First, suppose the magnetic field passing through the coil increases in strength while the coil itself remains stationary.

As the magnetic field becomes stronger, more magnetic field passes through the area of the loop. As a result, the magnetic flux through the coil increases.

A changing magnetic flux induces an electromotive force, or EMF, around the loop. This induced EMF drives a current in the conducting coil.

The induced current produces its own magnetic field. The direction of this induced magnetic field is such that it opposes the increase in magnetic flux that caused it. The induced field counters the change in flux, not the original magnetic field itself.

Graphs of magnetic flux and induced EMF versus time reflect this relationship. The flux increases over time, while the induced EMF is proportional to the negative of the rate at which the flux is increasing.

Decreasing Magnetic Field: Flux Decreases

Next, consider what happens when the magnetic field strength decreases.

As the field weakens, less magnetic field passes through the area of the loop, and the magnetic flux decreases. The change in flux now has the opposite sign from the previous case.

This reversed change in flux leads to a reversal of the induced EMF and a reversal of the induced current in the loop.

Once again, the induced current produces its own magnetic field. That induced field opposes the change in magnetic flux that created it. Although the direction of the current differs from the earlier case, the underlying physical principle is the same.

Changing Geometry: Rotating the Coil

Magnetic flux can also change even when the magnetic field strength remains constant.

If the coil rotates within a steady magnetic field, the orientation of the loop’s area changes relative to the field direction. As the angle between the magnetic field vector and the area vector changes, the dot product between them changes as well.

This changing orientation alters the magnetic flux through the coil over time, even though the magnetic field itself does not change in strength.

As before, the changing magnetic flux induces an EMF and a current in the loop. This same physical mechanism is the basis for electrical generators.

Flux, EMF, and Their Graphical Relationship

The relationship between magnetic flux and induced EMF is often represented using graphs plotted as functions of time.

A graph of magnetic flux versus time shows how the flux through the loop changes, while a corresponding graph of EMF versus time shows the electrical response.

At each moment, the value of the induced EMF is proportional to the negative slope of the flux curve. When the flux changes rapidly, the EMF is larger. When the flux is constant, the induced EMF is zero.

Lenz’s Law: The Governing Principle

Lenz’s law provides the physical meaning of the minus sign in Faraday’s law of induction.

Lenz’s law ensures that induced currents always create their own induced magnetic fields that oppose the changes in magnetic flux which created them.

This principle applies regardless of whether the flux changes because the magnetic field strength changes, the geometry of the loop changes, or both.