In 1820 Hans Christian Oersted realized that electric current creates magnetic fields. So, it was natural for physicists of the time to wonder whether the opposite is possible - for a magnetic field to create electric current.

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Fig. 1 - The winding of copper wire inside an electric motor that uses electromagnetic induction to convert electrical energy into mechanical energy.

Michael Faraday, an English scientist, worked for about ten years to prove the relationship between magnets and electricity. Therefore, the famous law of induction is named after him. He used his discovered properties of electromagnets to demonstrate the electric effect caused by a changing magnetic field. In this article, we will study Faraday's law of induction, its definition, equation, examples, and experiments proving the law.

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.

A galvanometer is a tool used to measure small electric currents.

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.

Magnetic flux is the total magnetic field passing through a certain area.

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!

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.

Faraday's law of induction states that the magnitude of the induced emf/electric current is directly proportional to the rate of change of magnetic flux linked with the circuit.

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

Imagine we have a magnet and a coil. We move the magnet while keeping the coil in a fixed position.

Fig. 2 - An electric current is induced in the coil due to the change in magnetic field lines passing through it.

The above setup proves us the following:

• The number of magnetic field lines passing through the coil increases when the magnet is moved toward the coil. This change in magnetic flux induces emf/electric current in the coil.
• Similarly, when a magnet is moved away from the coil, the magnetic field lines passing through the coil decrease. This change in magnetic flux induces emf/electric current in the coil.
• When the magnet is moved quickly towards or away from the coil, the amount of emf/electric current induced in the coil is large.
• In contrast, when the magnet is moved slowly towards or away from the coil, the amount of emf/electric current induced in the coil is small.

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.

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.

While Faraday's law gives us the magnitude of the emf induced by the current, the direction is found using Lenz's law, which transforms the equation above into $$\mathcal{E}=-k\frac{\mathrm{d}\phi_\mathrm{B}}{\mathrm{d}t}$$.

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.

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 $$\mathrm{C}$$.

The current in coil $$\mathrm{C'}$$ changes rapidly from zero to some constant current. As a result $$\frac{\mathrm{d}\phi_\mathrm{B}}{\mathrm{d}t}$$ becomes very high in coil $$\mathrm{C}$$, so the current through $$\mathrm{C}$$ will be huge for a very short time.

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.

## Faraday's Law - Key takeaways

• Magnetic flux is the total magnetic field passing through a certain area, mathematically equal to $$\phi_\mathrm{B}=\int \vec{B}\, \mathrm{d} \vec{A}$$.
• Faraday's law of induction states that the magnitude of the induced emf/electric current is directly proportional to the rate of change of magnetic flux linked with the circuit.
• The magnitude of emf induced in the coil in terms of magnetic flux is $$\left|\mathcal{E}\right|=\left|k\frac{\mathrm{d}\phi_\mathrm{B}}{\mathrm{d}t}\right|$$.
• The direction is found using Lenz's law, which mathematically is equal to $$\mathcal{E}=-k\frac{\mathrm{d}\phi_\mathrm{B}}{\mathrm{d}t}$$.
• For a solenoid with $$N$$ number of turns, Faraday law becomes:

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

• Current-induced current and current induced by changing current are two important experiments that explain the induced current in a coil using Faraday's law of induction.

## References

1. Fig. 1 - An electric motor with all its copper windings! (https://unsplash.com/photos/SkUkZ2auN4E) by Mika Baumeister (https://unsplash.com/@mbaumi) under the Unsplash license (https://unsplash.com/license).
2. Fig. 2 - Faraday's law of induction, StudySmarter Originals.
3. Fig. 3 - Magnetic flux passing through the coil, StudySmarter Originals.
4. Fig. 4 - Current induced by current experiment, StudySmarter Originals.
5. Fig. 5 - Current induced by changing current, StudySmarter Originals.

###### Learn with 15 Faraday's Law flashcards in the free StudySmarter app

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Faraday's law of induction states that the magnitude of the induced e.m.f./electric current is directly proportional to the rate of change of magnetic flux linked with the circuit.

When a coil is placed in a varying magnetic field, an e.m.f. is induced in the coil. Faraday's law of induction is used to find the magnitude of an induced em.f. by calculating the rate of change of magnetic flux.

What is Faraday's law of electromagnetic induction?

Faraday's law of electromagnetic induction explains that whenever there is a change in magnetic flux (the number of magnetic field lines passing through the coil) linked with the circuit, an e.m.f./electric current is induced in the circuit.

What are the examples of Faraday's law?

Some real-life examples of applications of Faraday's law of induction are: induction cooker, transformer, party balloons, headphones.

Faraday's law has the following formula: | E |=| dΦ/dt|.

## Test your knowledge with multiple choice flashcards

According to Faraday's law of induction, the magnitude of an e.m.f. induced in the coil is proportional to the _____.

Which of the following formula is correct according to Faraday's law of induction?

Which of the following parameters does an induced emf not depend on?

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