# Mutual Inductance – Definition, Formula, Significance, Examples

The topic of induction was first given by Oersted by discovering that a magnetic field is produced through an electric field. After this, different scientists started to think can the reverse is possible .i.e. can we produce electric current from a magnetic field.

Michael Faraday and Joseph Henry simultaneously established that an electric current is generated in a coil when the coil and a magnet are in relative motion. The current produced here is called induced current and e.m.f produced are called induced e.m.f. In this topic, we will see what is Electromagnetic Induction and one of its types which is Mutual induction.

### Electromagnetic Induction

Electromagnetic induction is the phenomenon of changing magnetic flux or field causing an induced e.m.f. in a conductor or conducting coil. The idea of magnetic flux or field, which may be represented by magnetic field lines, is important to understand while researching electromagnetic induction. The direction of magnetic induction at any given place is determined by the tangent to the magnetic field lines.

There are two types of Electromagnetic Induction they are,

- Self Induction
- Mutual Induction

### Mutual Induction

When two coils are kept near each other and current change in one coil is causing e.m.f change in other then this phenomenon is called mutual induction.

Consider the circuit given below,

As shown in the above figure, let there be two coils placed close to each other namely P and S. The coil P has a power source (E) and a key (k) and coil S has a galvanometer to detect the deflection. Here, let coil P be Primary coil and coil S be Secondary coil.

**Working:**

The galvanometer indicates momentary deflection in one direction and deflection in the other direction when key ‘K’ is closed. When the galvanometer shows that the current flowing through primary (P) is constant or zero when the key is turned open, no deflection is created in the galvanometer. However, when the current flowing through the main coil changes, a phenomenon is known as “Mutual Induction” occurs, and the secondary coil’s e.m.f. is induced. We can see the induced e.m.f by checking the deflection in the galvanometer.

Magnetic Flux (φ): The number of magnetic field lines passing through a closed surface is known as magnetic flux. It calculates the total magnetic field that travels across a specific surface area.

**Explanation:**

The magnetic flux w.r.t coil S at any instance is related to the current flowing through the primary coil at that instance.

φ_{s} α i_{p}

∴ φ_{s =} M×i_{p}

Here, M is the coefficient of mutual inductance of the coil

So, we can find e.m.f. induced in the secondary coil (S) at any instant,

e_{s} = – dφ_{s} / dt

e_{s} = -d(M×i_{p})/dt

e_{s} = -M di_{p} /dt

|e_{s}| = |-M di_{p}/dt|

|e_{s}| = M di_{p} / dt|

**M = e _{s} / |di_{p} / dt|**

**Coefficient of mutual induction: **The coefficient of mutual induction can be defined as the ratio of e.m.f. induced in one coil to the rate of change of current in the next coil.

Mutual inductance depends on the number of turns on the coil, size of the coil, separation between each turn and the angle of the turns, and the medium where the coils are placed.

- S.I Unit of mutual inductance is Henry (H)
- Dimensions of mutual induction are [L
^{2}M^{1}T^{-2}I^{-2}]

**Examples of Mutual inductance**

- An electric clothes dryer’s heating coils can be counter-wound such that their magnetic fields cancel out, considerably lowering mutual inductance with the dryer’s housing.
- The essential functioning principle of transformers, motors, generators, etc.
- All the electric components deal with a magnetic field.

**Significance of Mutual Induction**

- As discussed in previous examples the mutual induction is essential to all the electric equipment that uses a magnetic field so it is very significant in the modern world.
- Electric motors also use their principal and are a very important piece of equipment.
- A transformer’s mutual inductance, also known as the coefficient of coupling, is a measurement of the efficiency with which power is transmitted from the primary to secondary coils.
- When two coils are placed close together, the magnetic field in one of them tends to connect with the magnetic field in the other. The second coil then generates a voltage as a result of this. Mutual inductance is the property of a coil that influences or modifies the current and voltage in a secondary coil.
- When two coils are placed close together, the magnetic field in one of them tends to connect with the magnetic field in the other. The second coil then generates a voltage as a result of this.

### Eddy current and its relation to mutual induction

According to Faraday’s law of induction, eddy currents are loops of electrical current induced within conductors by a changing magnetic field in the conductor. Within conductors, eddy currents flow in closed loops in planes perpendicular to the magnetic field.

Example of eddy current, speedometer in our vehicle.

Mutual induction is the principal cause of eddy current formation and generation in the test material. An alternating current is transmitted through a wire coil as the test material. An eddy cope device is then attached to the probe. Any conductive substance can be used for the next circuit. A magnetic field is generated in and around a coil when electricity is conducted through it. When a probe is introduced close to a conducting substance, the alternating magnetic field of the probe causes current to flow through it. These currents run in closed loops at right angles to the magnetic flux in a plane. Eddy currents are what we call them.

These eddy currents generate their magnetic field, which interacts with the coil’s primary magnetic field. Analyzing the variances in the resistance and inductive reactance of the provided coil might provide information about the test material.

**Coupling Coefficient: **It’s the open circuit to actual voltage ratio, as well as the ratio obtained if the flux is connected from one circuit to the other. It is related to mutual inductance and is a simple approach to comprehend the relationship between certain inductor orientations and arbitrary inductance.

### Sample Questions

**Question 1: Why is this inductance called “mutual Induction”?**

**Answer:**

This induction is produced by two the induction produced on each other by two adjacent circuits thus it is the current produced Mutually between two circuits thus it is named mutual induction.

**Question 2: What is the induced e.m.f. in a circular coil kept in a magnetic field?**

**Answer:**

There is no variation in magnetic flux created since the coil is in a homogeneous magnetic field. As a result, the coil has no induced e.m.f.

**Question 3: What are the S.I Unit and Dimensions of Mutual Induction?**

**Answer: **

S.I Unit of mutual inductance is Henry (H) and Dimensions of mutual induction are [L

^{2}M^{1}T^{-2}I^{-2}]

**Question 4: What factors have an impact on mutual inductance?**

**Answer: **

The factors that effects the mutual inductance are:

- Cross-sectional area
- Number of turns
- the amount of space between the two coils.
- Medium permeability between the two coils
- Dimensions

**Question 5: What is magnetic flux?**

**Answer: **

Magnetic Flux (φ): Magnetic flux is the number of magnetic field lines traveling through a closed surface. It determines the overall magnetic field that goes across a certain region.

**Question 6: Where do we use the principle of mutual induction?**

**Answer: **

We mostly use mutual induction in, transformers, generators, and all the electronic devices which use a magnetic field.

**Question 7: Two coaxial coils are very closer to each other and their mutual inductance is 10mH. If a current (60 A) sin 600t is passed in one of the coils, then find the peak value of induced e.m.f. in the secondary coil. **

**Answer:**

i = 60 sin 600t

e.m.f. here can be given by,

e = -M di/dt

e = -(10 × 10

^{3}) d (60 sin 600t)/dte = -10 × 10

^{3}× 60 cos 600t ×600e = -360 cos 600t

Thus according to the Mutual induction formula

e = M×i

_{p}

Induced e.m.f. in secondary coil is 360V

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