# What is Mutual Induction?

The mechanism in which the changing of current in one coil induces an **Electromotive Force (emf)** in another coil is called **Mutual Induction**.

**What is Electromotive Force (emf)?**

Electromotive force (EMF) is equal to the difference in voltage between the two ends of the circuit when there is no current flowing. Both electromotive force (EMF) and terminal potential difference (V) are measured in **volts**, but they are not the same thing. **Electromotive Force** is the amount of energy (E) that the battery gives for every **Coulomb **of charge (Q) that moves through it.

## History of Mutual Induction

**Joseph Henry **(1797–1878) and** Michael Faraday** found out about mutual inductance at about the same time, but independently. Faraday got credit for the discovery because he shared his results a little bit earlier. But because of how much and how well he published his research, Henry became well-known very quickly. He was six years younger than Faraday and 11 years older than Faraday. Both were great at trying new things, kept good records, and wrote for a wide audience.

Henry built **electromagnets **over and over again. He made bigger models with better windings and thought of new uses, like the doorbell and the magnetic relay. **Samuel Morse** used these ideas to make his revolutionary telegraph.

Henry explained what **self-inductance** is and how it works. (Oliver Heaviside used the word “inductance” for the first time in 1886.) Any conductor that lets current flow through it has self-inductance, and if another conductor is close by, it has mutual inductance. The effect is stronger if the conductor is in the shape of a coil and if the coil is wrapped around a material that lets magnetic flux through, like soft iron. The unit of inductance is called a “**henry**” after Joseph Henry.

## Constant of Mutual Induction

The constant of Mutual induction is called **Mutual Inductance** which is denoted by “**M**“.

### What is Mutual Inductance?

**Mutual inductance** between two coils is the property of one coil that makes it oppose changes in the current in the other coil, or in the coil next to it. When the current in the neighboring coil changes, the flux builds up in the coil. This causes a change in the flux emf in the coil, which is called Mutually Induced emf. This is called **Mutual Inductance**.

### Working Principle of Mutual Inductance

When two coils are close to each other, the magnetic field in one coil tends to connect with the other. This leads to more electricity being made in the second coil. Mutual inductance is a property of a coil that causes a secondary coil’s current and voltage to change.

## Derivation of Mutual Inductance

Let us consider two coils which are shown in the given figure:

When I1 changes, the magnetic flux in coil 2 also changes.

When a current **I _{1}** flows through the first coil of

**N**turns, magnetic field

_{1 }**B**is made. As the two coils get closer together, fewer magnetic field lines will also go through

**coil 2**.

If we change the current over time, an induced emf will be created in **coil 2**.

As Faraday’s law says;

$\begin{array}{l}{\epsilon}_{21}=-{N}_{2}\frac{d{\varphi}_{21}}{dt}\end{array}$ $\begin{array}{l}{\epsilon}_{21}=-{N}_{2}\frac{d}{dt}(\stackrel{\u2015}{B}.{\textstyle \phantom{\rule{0.167em}{0ex}}}\stackrel{\u2015}{A})\end{array}$The induced emf in coil 2 is directly related to the amount of current that flows through coil 1.

$\begin{array}{l}{N}_{2}{\textstyle \phantom{\rule{0.167em}{0ex}}}{\varphi}_{21}\propto {I}_{1}\end{array}$ $\begin{array}{l}{N}_{2}{\textstyle \phantom{\rule{0.167em}{0ex}}}{\varphi}_{21}={M}_{21}{I}_{1}\dots .(1)\end{array}$Mutual inductance is the name given to the constant of proportionality. We can write it as;

$\begin{array}{l}{M}_{21}=\frac{{N}_{2}{\varphi}_{21}}{{I}_{1}}\dots .(2)\end{array}$In a similar way, when the current in coil 2, I2, changes over time, it can cause an induced emf in coil 1. Then,

$\begin{array}{l}{\epsilon}_{12}=-{N}_{1}\frac{d{\varphi}_{12}}{dt}\end{array}$ $\begin{array}{l}{N}_{1}{\varphi}_{12}\propto {I}_{2}\end{array}$ $\begin{array}{l}{N}_{1}{\varphi}_{12}={M}_{12}{I}_{2}\dots .(3)\end{array}$ $\begin{array}{l}{M}_{12}=\frac{{N}_{1}{\varphi}_{12}}{{I}_{2}}\dots .(4)\end{array}$Another example of mutual inductance is this constant of proportionality.

According to Faraday’s Law;

So,

### SI unit of Mutual Inductance

The SI unit which is used for Inductance is **Henry (H)**

or

**M = Vs . A ^{-1}**

or

**M= kg. m ^{2}.s^{-2}.A^{-2}**

#### Definition of Henry (H)

One Henry is the mutual inductance of the pair of coils in which the rate of change of current of one Ampere per second in the primary causes an induced emf of one volt in the secondary coil.

#### Dimension of Mutual Inductance

When two or more coils are magnetically linked together with the same magnetic flux, the voltage induced in one coil is proportional to how fast the current in another coil is changing. This kind of thing is called “mutual inductance.”

Suppose how much inductance there is between the two coils is **L since M = √(L _{1}L_{2}) = L**

The size of this is equal to the ratio of the difference in potential to the rate of change in current. It’s written as;

**Since M = √L _{1}L_{2 }= L**

**L = € / (dI / dt)**

where;

Induced emf=** Work done/electric charge with respect to time**

Induced emf= **M. L ^{2}. T-^{2}/ IT = M.L^{2}.T-3. I^{-1}**

**or**

**€ = M. L ^{-2} . T-3. A^{-1} ** (Since I = A)

The dimension of mutual inductance when L_{1 }and L_{2} are the same is given as;

**M = L /(T- ^{2}L^{2}.A^{-2})**

**M = LT ^{2}L^{2}.A^{-2}**

### Factors affecting Mutual Inductance

The mutual inductance of two coils next to each other depends on the **size of the two coils**, **how many turns are in each coil and how far apart they are, where the axes of the two coils are in relation to each other, how well the cores let air through.**

## Mutual Inductance of two coils

Consider two coils placed close to each other One coil connected with a battery through a switch and a **rheostat **is called the “primary” and the other one connected to the galvanometer is called the “secondary”. If the current in the primary is changed by varying the resistance of the rheostat, the magnetic flux in the surrounding region changes. Since the secondary coil is in the magnetic field of the primary, the changing flux also links with the secondary. This causes an induced emf in the secondary.

The Mutual Inductance of two coils will be:

**M _{12} = (N_{2} ϕ_{ 12}) / I_{1}**

**M _{21}= (N_{1} ϕ _{21}) / I_{2}**

Where;

** M _{12}**=mutual inductance of the first coil to the second coil

**M _{21}**= mutual inductance of the second coil to the first coil

**N _{2}**= number of turns of the second coil

**N _{1}**= number of turns of the first coil

**I _{1}**=current flowing around the first coil

**I _{2}**=current flowing around the second coil.

## Mutual Inductance in Transformer

A **transformer **is a device that uses mutual induction to raise or lower the voltage in alternating current (AC) circuits. It is made up of two coils wrapped around the same core. The coil that is connected to the source (i.e., where the power comes in) is called the primary coil, and the coil that is connected to the load (i.e., where the power goes out) is called the secondary coil.

The flux through the core is always changing because of the way the alternating current flows through the primary. This changing flux causes the secondary emf to go back and forth. Since magnetic lines of force are closed curves, the flux per turn of the primary must be the same as the flux per turn of the secondary.

There exist two types of Winding in transformers:

- Primary Winding
- Secondary Winding

By the principle of mutual inductance, when the current in the primary coil changes, it also changes the current in the secondary coil. Since the current in the primary coil changes, the magnetic flux in the core also changes. This magnetic flux in the core causes the secondary winding to have a changing voltage. This is called mutual inductance in a transformer.

### Formula for Transformers

**Power input = Power output**

**V _{P}I_{P} = V_{S}I_{S}**

**V _{S} / V_{P} = I_{P} /I_{S}**

### Loses in Actual Transformer

The losses in the transformer occur in the core and winding.

#### Types of Losses in Transformer

**Copper Losses in Winding**

Due to the resistance of the windings of primary and secondary coils, some electrical energy is always lost in the form of**heat energy**.**Flux Losses**

The coupling of the primary and secondary is never perfect and the whole of the magnetic flux produced in the primary coil does not link to the secondary coil. This results in some energy loss.**Iron Losses in Core**

Iron losses are of two types: Eddy current loss and hysteresis loss.

**Eddy Current Loss**

Due to the periodically varying nature of A.C. supplied in primary, the flux associated with core changes and, therefore, eddy currents is induced it. Eddy currents induced in the core are undesirable as they heat the core and result in energy loss. The core is laminated to cut down on eddy current losses.**Hysteresis Loss**

The alternating current flowing through the coils magnetizes and demagnetizes the iron core again and again. Hysteresis makes it so that some energy is lost during each cycle of magnetization. To minimize this loss we choose a material of the core of smaller hysteresis loss generally soft iron.

### Efficiency of Transformer

Ideal transformer, efficiency is 100 % or 1. but in actual transformer output power is Always less than 100%. In general always less than the input power, so the efficiency of a transformer is very high (and is of the order of 90%).

**E = ( Output Power / Input Power ) * 100**

## Applications Of Mutual Induction

- The latest way to charge phones without wires is called “mutual induction,” and it works on this idea. One of the charging bases is hooked up to the power line. When we put our phone on the charging base, it starts charging.
- Pacemakers, which are placed in the heart to keep the heartbeat steady and save a person’s life, also work on the same principle. The pacemaker works because it makes small electrical and magnetic currents.
- Mutual inductance is also used to explain how metal detectors work, which are used to keep up security.
- The phenomenon of
**Mutual Induction**is also used in transformers, Electric Motors, Generators, and Other electrical devices which work with a magnetic field and is also used in the calculation of eddy currents, Digital signal processing.

## Limitation of Mutual Inductance

The biggest problem with mutual inductance is that if the inductance of one coil leaks, it can cause another coil that uses electromagnetic induction to work. Electrical screening must be put in place to stop leaks.

## FAQ’s

### Does the mutual inductance between transmission lines decrease with length?

Yes, When the magnetic permeability of the core around which the coil is wound goes up, the inductance goes up. When the magnetic permeability of the core goes down, the inductance goes down. The inductance goes down as the length of the transmission lines goes up.

### Does mutual inductance depend on current?

Yes, Mutual induction is what happens when a change in the current in one coil causes an EMF in the other. How strong the **EMF** depends on how well the two coils interact with each other.

### Do 3-phase transformers have mutual inductances between phases?

Yes, The idea behind it is mutual induction. The 3-phase transformers do, too. Basically, power generation is usually done in three phases, with high currents and voltages like kV or MV and a high current rating.

### Can mutual inductance be negative?

Yes, The mutual attraction LM can be positive or negative depending on the direction of the inducing current and the polarity of the mutual voltage.

### What is the mutual inductance of the pair of solenoids?

Since both solenoids are wound on top of each other, all of the magnetic field lines from the primary coil go through the secondary coil. Let N1 be the number of turns in the primary coil and N2 be the number of turns in the secondary coil. The expression M=**μ0N1N2πr12l** gives the mutual inductance of the coils.

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