The Faraday law in electromagnetism establishes a changing magnetic field flux is able to induce an electrical current in a closed circuit.

In 1831, the English physicist Michael Faraday experimented with moving conductors within a magnetic field and also varying magnetic fields that passed through fixed conductors.

Figure 1. Faraday induction experiment

Faraday realized that if he varied the magnetic field flux over time, he was able to establish a voltage proportional to that variation. If ε is the voltage or induced electromotive force (induced emf) and Φ is the magnetic field flux, it can be expressed mathematically:

-ε- = ΔΦ / Δt

Where the symbol Δ indicates variation of the quantity and the bars in the emf indicate the absolute value of this. Since it is a closed circuit, the current can flow in one direction or the other.

Magnetic flux, produced by a magnetic field across a surface, can vary in a number of ways, for example:

-Moving a bar magnet through a circular loop.

-Increasing or decreasing the intensity of the magnetic field that passes through the loop.

-Leaving the field fixed, but through some mechanism change the area of ​​the loop.

-Combining the previous methods.

Figure 2. English physicist Michael Faraday (1791-1867).

Formulas and Units

Suppose we have a closed circuit area A as a circular coil or winding equal to that of Figure 1, and which has a magnet that produces a magnetic field B.

The magnetic field flux Φ is a scalar quantity that refers to the number of field lines that cross area A. In figure 1 they are the white lines that leave the north pole of the magnet and return through the south.

The intensity of the field will be proportional to the number of lines per unit area, so we can see that at the poles it is very intense. But we can have a very intense field that does not produce flux in the loop, which we can achieve by changing the orientation of the loop (or the magnet).

To take into account the orientation factor, the magnetic field flux is defined as the scalar product between B and n, where n is the unit normal vector to the surface of the loop and that indicates its orientation:

Φ = B • n A = BA.cosθ

Where θ is the angle between B and n. If, for example, B and n are perpendicular, the magnetic field flux is zero, because in that case the field is tangent to the plane of the loop and cannot pass through its surface.

On the other hand, if B and n are parallel, it means that the field is perpendicular to the plane of the loop and the lines pass through it as much as possible.

The International System unit for F is the weber (W), where 1 W = 1 Tm ² (read “tesla per square meter”).

Lenz's Law

In figure 1 we can see that the polarity of the voltage changes as the magnet moves. Polarity is established by Lenz's law, which states that the induced voltage must oppose the variation that produces it.

If, for example, the magnetic flux produced by the magnet increases, a current is established in the conductor that circulates creating its own flux, which opposes this increase.

If, on the contrary, the flux created by the magnet decreases, the induced current circulates in such a way that the flux itself counteracts said decrease.

To take this phenomenon into account, a negative sign is prepended to Faraday's law and it is no longer necessary to place the absolute value bars:

ε = -ΔΦ / Δt

This is the Faraday-Lenz law. If the flow variation is infinitesimal, the deltas are replaced by differentials:

ε = -dΦ / dt

The above equation is valid for a loop. But if we have a coil of N turns, the result is much better, because the emf is multiplied N times:

ε = - N (dΦ / dt)

Faraday experiments

In order for the current to light the bulb to be produced, there must be relative movement between the magnet and the loop. This is one of the ways in which the flux can vary, because in this way the intensity of the field passing through the loop changes.

As soon as the movement of the magnet ceases, the bulb turns off, even if the magnet is left still in the middle of the loop. What is needed to circulate the current that turns on the bulb is that the field flux varies.

When the magnetic field varies with time, we can express it as:

B = B (t).

By keeping the area A of the loop constant and leaving it fixed at a constant angle, which in the case of the figure is 0º, then:

Figure 4. If the loop is rotated between the poles of a magnet, a sinusoidal generator is obtained. Source: F. Zapata.

Thus, a sinusoidal generator is obtained, and if instead of a single coil a number N of coils are used, the induced emf is greater:

Figure 5. In this generator, the magnet is rotated to induce current in the coil. Source: Wikimedia Commons.

Referencias

1. Figueroa, D. 2005. Serie: Física para Ciencias e Ingeniería. Volumen 6. Electromagnetismo. Editado por Douglas Figueroa (USB).
2. Giambattista, A. 2010. Physics. Second Edition. McGraw Hill.
3. Giancoli, D. 2006. Physics: Principles with Applications. 6th. Ed. Prentice Hall.
4. Resnick, R. 1999. Física. Vol. 2. 3ra Ed. en español. Compañía Editorial Continental S.A. de C.V.
5. Sears, Zemansky. 2016. University Physics with Modern Physics. 14th. Ed. Volume 2.