Faraday's Law of Induction: Definition, Formula & Examples

Around the turn of the 19th century, physicists were making a lot of progress in understanding the laws of electromagnetism, and Michael Faraday was one of the true pioneers in the area. Not long after it was discovered that an electric current creates a magnetic field, Faraday performed some now-famous experiments to work out if the reverse was true: Could magnetic fields induce a current?

Faraday’s experiment showed that while magnetic fields alone couldn’t induce current flows, a ​changing​ magnetic field (or, more precisely, a ​changing magnetic flux​) could.

The result of these experiments is quantified in ​Faraday’s law of induction​, and it’s one of Maxwell’s equations of electromagnetism. This makes it one of the most important equations to understand and learn to use when you’re studying electromagnetism.

Magnetic Flux

The concept of magnetic flux is crucial to understanding Faraday’s law, because it relates flux changes to the induced ​electromotive force​ (EMF, commonly called ​voltage​) in the coil of wire or electric circuit. In simple terms, magnetic flux describes the flow of the magnetic field through a surface (although this “surface” isn’t really a physical object; it’s really just an abstraction to help quantify the flux), and you can imagine it more easily if you think about how many magnetic field lines are passing through a surface area ​A​. Formally, it’s defined as:

ϕ = \bm{B ∙ A} = BA \cos (θ)

Where ​B​ is the magnetic field strength (the magnetic flux density per unit area) in teslas (T), ​A​ is the area of the surface, and ​θ​ is the angle between the "normal" to the surface area (i.e., the line perpendicular to the surface) and ​B​, the magnetic field. The equation basically says that a stronger magnetic field and a bigger area lead to more flux, along with a field aligned with the normal to the surface in question.

The ​B​ ​∙ ​A​ in the equation is a scalar product (i.e., a “dot product”) of vectors, which is a special mathematical operation for vectors (i.e., quantities with both a magnitude or “size” ​and​ a direction); however, the version with cos (​θ​) and the magnitudes is the same operation.

This simple version works when the magnetic field is uniform (or can be approximated as such) across ​A​, but there is a more complicated definition for cases when the field isn’t uniform. This involves integral calculus, which is a bit more complicated but something you’ll need to learn if you’re studying electromagnetism anyway:

ϕ = \int \bm{B} ∙ d\bm{A}

The SI unit of magnetic flux is the weber (Wb), where 1 Wb = T m2.

Michael Faraday’s Experiment

The famous experiment performed by Michael Faraday lays the groundwork for Faraday’s law of induction and conveys the key point that shows the effect of flux changes on the electromotive force and consequent electric current induced.

The experiment itself is also quite straightforward, and you can even replicate it for yourself: Faraday wrapped an insulated conductive wire around a cardboard tube, and connected this to a voltmeter. A bar magnet was used for the experiment, first at rest near the coil, then moving towards the coil, then passing through the middle of the coil and then moving out of the coil and further away.

The voltmeter (a device that deduces voltage using a sensitive galvanometer) recorded the EMF generated in the wire, if any, during the experiment. Faraday found that when the magnet was at rest close to the coil, no current was induced in the wire. However, when the magnet was moving, the situation was very different: On the approach to the coil, there was some EMF measured, and it increased until it reached the center of the coil. The voltage reversed in sign when the magnet passed through the center point of the coil, and then it declined as the magnet moved away from the coil.

Faraday’s experiment was really simple, but all of the key points it demonstrated are still in use in countless pieces of technology today, and the results were immortalized as one of Maxwell’s equations.

Faraday’s Law

Faraday’s law of induction states that the induced EMF (i.e., electromotive force or voltage, denoted by the symbol ​E​) in a coil of wire is given by:

E = −N \frac{∆ϕ}{∆t}

Where ​ϕ​ is the magnetic flux (as defined above), ​N​ is the number of turns in the coil of wire (so ​N​ = 1 for a simple loop of wire) and ​t​ is time. The SI unit of ​E​ is volts, since it’s an EMF induced in the wire. In words, the equation tells you that you can create an induced EMF in a coil of wire either by changing the cross-sectional area ​A​ of the loop in the field, the strength of the magnetic field ​B​, or the angle between the area and the magnetic field.

The delta symbols (∆) simply mean “change in,” and so it tells you that the induced EMF is directly proportional to the corresponding rate of change of magnetic flux. This is more accurately expressed through a derivative, and often the ​N​ is dropped, and so Faraday’s law can also be expressed as:

E = −\frac{dϕ}{dt}

In this form, you’ll need to find out the time-dependence of either the magnetic flux density per unit area (​B​), the cross-sectional area of the loop ​A,​ or the angle between the normal to the surface and the magnetic field (​θ​), but once you do, this can be a much more useful expression for calculating the induced EMF.

Lenz’s Law

Lenz’s law is essentially an extra piece of detail in Faraday’s law, encompassed by the minus sign in the equation and basically telling you the direction in which the induced current flows. It can be simply stated as: The induced current flows ​in a direction that opposes the change​ in magnetic flux that caused it. This means that if the change in magnetic flux was an increase in magnitude with no change in direction, the current will flow in a direction that will create a magnetic field in the opposite direction to the field lines of the original field.

The right-hand rule (or the right-hand grip rule, more specifically) can be used to determine the direction of the current that results from Faraday’s law. Once you’ve worked out the direction of the new magnetic field based on the rate of change of magnetic flux of the original field, you point the thumb of your right hand in that direction. Allow your fingers to curl inwards as if you’re making a fist; the direction your fingers move in is the direction of the induced current in the loop of wire.

Examples of Faraday’s Law: Moving Into a Field

Seeing Faraday’s law put into practice will help you see how the law works when applied to real-world situations. Imagine you have a field pointing directly forwards, with a constant strength of ​B​ = 5 T, and a square single-stranded (i.e., ​N​ = 1) loop of wire with sides of length 0.1 m, making a total area ​A​ = 0.1 m × 0.1 m = 0.01 m2.

The square loop moves into the region of the field, traveling in the ​x​ direction at a rate of 0.02 m/s. This means that over a period of ∆​t​ = 5 seconds, the loop will go from being completely out of the field to completely inside it, and the normal to the field will be aligned with the magnetic field at all times (so θ = 0).

This means that the area in the field changes by ∆​A​ = 0.01 m2 in ​t​ = 5 seconds. So the change in magnetic flux is:

\begin{aligned} ∆ϕ &= B∆A \cos (θ) \\ &= 5 \text{ T} × 0.01 \text{ m}^2 × \cos (0) \\ &= 0.05 \text{ Wb} \end{aligned}

Faraday’s law states:

E = −N \frac{∆ϕ}{∆t}

And so, with ​N​ = 1, ∆​ϕ​ = 0.05 Wb and ∆​t​ = 5 seconds:

\begin{aligned} E &= −N \frac{∆ϕ}{∆t}\\ &= − 1 ×\frac{0.05 \text{ Wb}}{5} \\ &= − 0.01 \text{ V} \end{aligned}

Examples of Faraday’s Law: Rotating Loop in a Field

Now consider a circular loop with area 1 m2 and three turns of wire (​N​ = 3) rotating in a magnetic field with a constant magnitude of 0.5 T and a constant direction.

In this case, while the area of the loop ​A​ inside the field will remain constant and the field itself won’t change, the angle of the loop with respect to the field is constantly changing. The rate of change of magnetic flux is the important thing, and in this case it’s useful to use the differential form of Faraday’s law. So we can write:

E = −N \frac{dϕ}{dt}

The magnetic flux is given by:

ϕ = BA \cos (θ)

But it’s constantly changing, so the flux at any given time ​t​ – where we assume it starts at an angle of ​θ​ = 0 (i.e., aligned with the field) – is given by:

ϕ = BA \cos (ωt)

Where ​ω​ is the angular velocity.

Combining these gives:

\begin{aligned} E &= −N \frac{d}{dt} BA \cos (ωt) \\ &= −NBA \frac{d}{dt} \cos (ωt) \end{aligned}

Now this can be differentiated to give:

E = NBAω \sin (ωt)

This formula is now ready to answer the question at any given time ​t​, but it’s clear from the formula that the faster the coil rotates (i.e., the higher the value of ​ω​), the greater the induced EMF. If the angular velocity ​ω​ = 2π rad/s, and you evaluate the result at 0.25 s, this gives:

\begin{aligned} E &= NBAω \sin (ωt) \\ &= 3 × 0.5 \text{ T} × 1 \text{ m}^2 × 2π \text{ rad/s} × \sin (π /2) \\ &= 9.42 \text{ V} \end{aligned}

Real World Applications of Faraday’s Law

Because of Faraday’s law, any conductive object in the presence of a changing magnetic flux will have currents induced in it. In a loop of wire, these can flow in a circuit, but in a solid conductor, little loops of current called ​eddy currents​ form.

An eddy current is a small loop of current that flows in a conductor, and in many cases engineers work to reduce these because they’re essentially wasted energy; however, they can be used productively in things like magnetic braking systems.

Traffic lights are an interesting real-world application of Faraday’s law, because they use wire loops to detect the effect of the induced magnetic field. Under the road, loops of wire containing alternating current generate a changing magnetic field, and when your car drives over one of them, this induces eddy currents in the car. By Lenz’s law, these currents generate an opposing magnetic field, which then impacts the current in the original wire loop. This impact on the original wire loop indicates the presence of a car, and then (hopefully, if you’re mid-commute!) triggers the lights to change.

Electric generators are among the most useful applications of Faraday’s law. The example of a rotating wire loop in a constant magnetic field basically tells you how they work: The motion of the coil generates a changing magnetic flux through the coil, which switches in direction every 180 degrees and thereby creates an ​alternating current​. Although it – of course – requires ​work​ to generate the current, this allows you to turn mechanical energy into electrical energy.

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