# Faraday’s Law: formula, units, experiments, exercise

The **Faraday’s law** in electromagnetism establishes a change in the flow of the magnetic field can induce an electric current in a closed circuit.

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

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

| ε | = ΔΦ / Δt

Where the symbol Δ indicates quantity variation and the bars on the emf indicate its absolute value. Because it is a closed circuit, current can flow in one direction or another.

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

– Moving a bar magnet through a circular curve.

-Increase or decrease the intensity of the magnetic field that passes through the loop.

-Leaving the field fixed, but using some mechanism to change the loop area.

-Combine the previous methods.

__Formulas and Units__

__Formulas and Units__

Suppose they have a closed circuit area A like a circular coil or winding like Figure 1, and that they have a magnet that produces a magnetic field **B** .

The flux of the magnetic field Φ is a scalar quantity that refers to the number of field lines that pass through area A. In Figure 1, it is the white lines that leave the north pole of the magnet and return from the south.

The field strength will be proportional to the number of lines per unit area, so we can see that it is very intense at the poles. But we can have a very intense field that doesn’t 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** , with **n** being the unit normal vector for the loop surface and indicating 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 flux of the magnetic field is zero, because in that case the field is tangent to the plane of the loop and cannot cross its surface.

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

The unit in the International System 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 voltage polarity 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, which circulates creating its own flux, which opposes this increase.

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

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

ε = -ΔΦ / Δt

This is the Faraday-Lenz law. If the flux variation is infinitesimal, the deltas will be replaced by differentials:

ε = -dΦ / dt

The above equation is valid for one turn. 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’s experiences__

__Faraday’s experiences__

For current to light the lamp, there must be relative movement between the magnet and the coil. This is one of the ways in which the flow can vary, as the strength of the field passing through the loop changes in this way.

The moment the magnet’s movement stops, the lamp turns off, even if the magnet is left standing in the middle of the loop. What is needed for current to flow to the lamp is for the field flux to vary.

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

*B** = B (t).*

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

If it is possible to change the area of the loop, keeping its orientation fixed and placing it in the middle of a constant field, the induced emf is given by:

One way to do this is to place a bar that slides on a conductor rail at a certain speed, as shown in the following figure.

The bar and the rail, in addition to a lamp or a resistor connected to conducting wires, form a closed circuit in the form of a rectangular spiral.

As the bar slides, the length *x* increases or decreases and with it the coil area changes, which is enough to create a variable flux.

**Variation of magnetic flux per rotation**

As we said before, if the angle between **B** and the loop normal is varied, the field flux changes accordingly:

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

A circular coil of N rotates and radius R rotates with angular frequency ω in the middle of a magnetic field of magnitude B. Find an expression for the maximum emf induced in the coil.

**Solution**

The expression for the rotation-induced emf is applied when the coil has N turns, knowing that:

-The coil area is A = πR ^{2}

-The angle θ varies with time as θ = ωt

It is important to note that θ = ωt is first substituted in Faraday’s law and *then* derived with respect to time:

ε = -NBA (cos θ) ‘= -NB (πR ^{2} ). [cos (ωt)]’ = NBω (πR ^{2} ) sin (ωt)

Since the maximum emf is requested, this occurs whenever sin ωt = 1, so finally:

ε _{max} = NBω (πR ^{2} )