## Faraday’s law of electromagnetic induction

### Current induced by a changing magnetic field

**electromagnetic induction**.

Faraday discovered that when he moved a magnet near a wire a voltage was generated across it. If the magnet was held stationary no voltage was generated, the voltage only existed while the magnet was moving. We call this voltage the induced emf (\(\mathcal{E}\)).

A circuit loop connected to a sensitive ammeter will register a current if it is set up as in this figure and the magnet is moved up and down:

### Magnetic flux

Before we move onto the definition of Faraday’s law of electromagnetic induction and examples, we first need to spend some time looking at the magnetic flux. For a loop of area \(A\) in the presence of a uniform magnetic field, \(\vec{B}\), the magnetic flux (\(φ\)) is defined as: \[\phi = BA\cos\theta\] Where: \begin{align*} \theta & = \text{the angle between the magnetic field. B. and the normal to the loop of area A}\\ A & = \text{the area of the loop}\\ B & = \text{the magnetic field} \end{align*}

The S.I. unit of magnetic flux is the weber (Wb).

You might ask yourself why the angle \(\theta\) is included. The flux depends on the magnetic field that passes through surface. We know that a field parallel to the surface can’t induce a current because it doesn’t pass through the surface. If the magnetic field is not perpendicular to the surface then there is a component which is perpendicular and a component which is parallel to the surface. The parallel component can’t contribute to the flux, only the vertical component can.

In this diagram we show that a magnetic field at an angle other than perpendicular can be broken into components. The component perpendicular to the surface has the magnitude \(B\cos(\theta)\) where \(\theta\) is the angle between the normal and the magnetic field.

### Definition: Faraday’s Law of electromagnetic induction

The emf, \(\mathcal{E}\), produced around a loop of conductor is proportional to the rate of change of the magnetic flux, φ, through the area, A, of the loop. This can be stated mathematically as:

\[\mathcal{E} = -N\frac{\Delta\phi}{\Delta t}\]

where \(\phi =B·A\) and B is the strength of the magnetic field. \(N\) is the number of circuit loops. A magnetic field is measured in units of teslas (T). The minus sign indicates direction and that the induced emf tends to oppose the change in the magnetic flux. The minus sign can be ignored when calculating magnitudes.

Faraday’s Law relates induced emf to the rate of change of flux, which is the product of the magnetic field and the cross-sectional area through which the field lines pass.

### Important:

It is not the area of the wire itself but the area that the wire encloses. This means that if you bend the wire into a circle, the area we would use in a flux calculation is the surface area of the circle, not the wire.

In this illustration, where the magnet is in the same plane as the circuit loop, there would be no current even if the magnet were moved closer and further away. This is because the magnetic field lines do not pass through the enclosed area but are parallel to it. The magnetic field lines must pass through the area enclosed by the circuit loop for an emf to be induced.