Faraday's Law

Faraday's Law of Induction

Michael Faraday is one of the most significant scientists in history. He formulated the law of induction after his first experimental demonstration of electromagnetic induction. In this experiment, he wrapped two wires on the opposite side of an iron ring, then plugged one wire with a galvanometer and the other with a battery.

He observed the deflection when he switched on the battery, and again when he switched off the battery. This experiment showed the induction of electric current through the wire, when a change in magnetic flux (\(\phi_\mathrm{B}\)) passes through the wire while the battery is switched on and off.

For a magnetic field that is constant across an area, it can be expressed as

\[\phi_\mathrm{B}= \vec{B}\, \mathrm{d} \vec{A},\]

where \(B\) is the magnetic field and \(A\) is the surface area.

This expression can be generalized for non-uniform area by applying the surface integral over differential sections of surface area:

\[\phi_\mathrm{B}=\int \vec{B}\, \mathrm{d} \vec{A}.\]

Keeping all of this in mind, let's define the law!

Faraday's Law Definition

So, what exactly does Faraday's law entail? Whenever there is a change in magnetic flux (the number of magnetic field lines passing through the coil) linked with the circuit, an emf/electric current is induced in the circuit.

Let us consider an example, to understand it in more detail

The above example explained how magnetic field lines passing through the coil related to the induced electric current in the coil. We will learn about Faraday's law using equations in the next part.

Faraday's Law Equation

According to Faraday's law of induction, the magnitude of emf induced in the coil in terms of magnetic flux is

\[\left|\mathcal{E}\right|=\left|k\frac{\mathrm{d}\phi_\mathrm{B}}{\mathrm{d}t}\right|\] where \(k\) is the proportionality constant.

This magnetic flux through any surface of area \(\vec{A}\) is measured by the total number of magnetic lines of force crossing the surface normally:

\[\phi_\mathrm{B}=\vec{B}\cdot\vec{A}.\]

In the case of a solenoid consisting of \(N\) number of turns, the magnitude of emf induced in terms of the magnetic flux passing through it is

\[\left|\mathcal{E}\right|=N\left|\frac{\mathrm{d}\phi_\mathrm{B}}{\mathrm{d}t}\right|.\]

Fig. 3 - Magnetic field lines passing through the surface of area \(\vec{A}\) at an angle of \(\theta\) with respect to the area vector.

From the above diagram, the magnetic flux through the surface of area \(\vec{A}\) is

\[\phi_\mathrm{B}=\vec{B}\cdot\vec{A}=BA\cos{\theta},\]

where \(\theta\) is the angle between the magnetic field lines \(\left(\vec{B}\right)\) and the area vector \(\left(\vec{A}\right)\) of the coil.

In Figure 3, we can see that the area vector of the coil is perpendicular to the coil's surface.

1. When a coil is placed perpendicular to the direction of the magnetic field, then the angle between its area vector and the magnetic field is \(\theta=0^\circ\). \[\therefore \phi_\mathrm{B}=BA\cos{\left(0^\circ\right)}=BA,\] which is the maximum value of magnetic flux passing through the coil.

2. When a coil is placed parallel to the direction of the magnetic field, then the angle between its area vector and the magnetic field is \(\theta=90^\circ\). \[\therefore \phi_\mathrm{B}=BA\cos{\left(90^\circ\right)}=0,\] which is the minimum value of magnetic flux passing through the coil.

Therefore, the magnitude of an emf induced in the coil when the surface area of the coil is at an angle of \(\theta\) with respect to the magnetic field lines is

\[\left|\mathcal{E}\right|=\left|\frac{\mathrm{d}}{\mathrm{d}t}\left(\vec{B}\cdot\vec{A}\right)\right|=\left|\frac{\mathrm{d}}{\mathrm{d}t}BA\cos{\theta}\right|.\]

This equation shows what induced emf depends upon:

1. the magnetic field strength \(B\),

2. the surface area of a coil \(A\), and

3. alignment of a coil with respect to the magnetic field lines.

So, an e.m.f. is induced in the coil if these three parameters change.

Let's assume the surface area of a coil parallel to the direction of magnetic field lines remains constant, then the magnitude of an emf induced in the coil due to the change in magnetic field strength is \[\begin{align*}\left|\mathcal{E}\right|&=\left| \frac{\mathrm{d}\left(BA\cos{\left(0^\circ\right)}\right)}{\mathrm{d}t}\right|\\\left|\mathcal{E}\right|&=\left|\frac{\mathrm{d}\left(BA\right)}{\mathrm{d}t}\right|\\\left|\mathcal{E}\right|&=\left|A\frac{\mathrm{d}B}{\mathrm{d}t}\right|\end{align*}\]

According to Faraday's law, the magnitude of an induced emf in the coil is the product of area and the rate of change in the component of a magnetic field with respect to time when the coil with constant surface area is placed perpendicular to the direction of magnetic field lines.

If the magnetic field remains constant but the surface area of the coil changes, then \(\left|\mathcal{E}\right|=\left|B\frac{\mathrm{d}A}{\mathrm{d}t}\right|\).

In the next part, we will understand electromagnetic induction using some experiments by Faraday and Joseph Henry.

Faraday's Law Experiment

Several experiments can explain Faraday's law of induction, so in this section, we'll learn about two of them.

Current Induced by Current

In this experiment, a coil \(\left(\mathrm{C}\right)\) connected to a galvanometer is placed near another coil \(\left(\mathrm{C'}\right)\) connected to a battery.

Fig. 4 - Due to the movement of coil \(\mathrm{C'}\) towards the coil \(\mathrm{C}\), an opposing emf is induced in the coil \(\mathrm{C'}\).

In the above diagram, we can see that the coil \(\mathrm{C'}\) is connected to a battery, due to which an electric current flows through the coil. This current flowing through the coil then induces a magnetic field enclosed by the coil. The coil \(\mathrm{C'}\) is then connected to a galvanometer which shows a deflection whenever an electric current flows through this coil.

When the coil \(\mathrm{C'}\) move towards the coil \(\mathrm{C}\), the magnetic flux passing through the coil \(\mathrm{C}\) increases. Due to this change of magnetic flux, an opposing emf/electric current is induced in the coil \(\mathrm{C}\), shown by the galvanometer's deflection.

Similarly, when the coil \(\mathrm{C'}\) moves away from the coil \(\mathrm{C}\), the magnetic flux passing through the coil decreases. This decrease in magnetic flux through the coil \(\mathrm{C}\) induces an electric current in the same direction as the current in the coil \(\mathrm{C'}\), which shows by the deflection of the galvanometer being in the opposite direction as in the previous case.

This induced emf due to the change in magnetic flux observed in the experiment proves Faraday's law of induction.

Also, the galvanometer shows large deflection when the coil \(\mathrm{C'}\) moves faster towards or away from the coil \(\mathrm{C}\), which proves that the emf induced in the coil \(\mathrm{C}\) depends upon the rate of change of magnetic flux.

Current Induced by Changing Current

In this experiment, instead of moving the coil \(\mathrm{C'}\) towards or away from the coil \(\mathrm{C}\), the magnetic flux is changed in the coil \(\mathrm{C}\) due to the change in electric current passing through the coil \(\mathrm{C'}\).

Fig. 5 - When the key is closed in the circuit containing coil \(\mathrm{C'}\), an emf is induced in the coil \(\mathrm{C}\) due to the change in magnetic flux through it.

In the above diagram, we can see that a key is connected in the circuit containing \(\mathrm{C'}\). When the key is closed, an electric current flow through the circuit. Due to this current, the magnetic flux starts passing through the coil \(\mathrm{C}\) placed near the coil \(\mathrm{C'}\). This change in magnetic flux through the coil \(\mathrm{C}\) from zero to some value induces an emf/electric current in the coil, which is shown by the deflection in the galvanometer.

Once the electric current passing through the coil \(\mathrm{C'}\) becomes stable, then the galvanometer stops showing any deflection, which indicates that the emf/electric current through coil \(\mathrm{C'}\) becomes zero. Similarly, when we open the key of the circuit, the magnetic flux passing through the coil \(\mathrm{C}\) decreases, which induces an emf/electric current in the coil \(\mathrm{C}\).

This induced emf/ electric current in the coil \(\mathrm{C}\) due to the change in magnetic flux once again proves Faraday's law of induction.

Examples of Faraday's Law of Induction

All electrical gadgets in which an electric current is induced are examples of applications of Faraday's law of induction. Some examples are listed below.

1. Cooking using an induction hob or induction cooker where a pan is heated by electrical induction.

2. Transformers consist of windings that work on the principle of Faraday's law of induction.

3. Headphones in which the variation of electric current is due to the variation in a magnetic field produced by electromagnets.

4. Electric motors use Faraday's law of induction to convert electric energy into mechanical energy.