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# Mutual Inductance: Definition & Formula

When the two coils are arranged in such a way that a change of current in one coil causes an emf to be induced in the other, the coils are said to have *mutual inductance*. The mutual inductance is denoted letter *M* and measured in *Henry*.

Consider two coils, coil 1 and coil 2 placed adjacent to each other. When a current I1 flows in the coil 1, a magnetic flux (Φ_{1}) is produced in it and some part of the Φ_{1} links with the coil 2 and is known as *mutual flux *(Φ_{m}).

Now, if the current in the coil 1 changes, the mutual flux also changes, therefore an EMF is induced in the coil 2. This induced emf in the coil 2 is known as *mutually induced emf *(𝑒_{𝑚}). This mutually induced emf is responsible for the mutual inductance between the coils. The effect of mutual inductance is to either increase or decrease the total inductance of the two coils depending upon the arrangement of the coils.

## Mutual Inductance Formula

The mutual inductance *(M)* between two coils can be determined by using any one of the following three methods depending upon the known quantities −

### Method 1

If the magnitude of mutually induced emf (𝑒_{𝑚}) in one coil and the rate of change of current in the other coil are known, then mutual inductance *(M)* is given by,

$$\mathrm{e_{m}=M\frac{dl_{1}}{dt}}$$

$$\mathrm{\Rightarrow\:M=\frac{e_{m}}{(dl_{1}/dt)}\:\:\:...(1)}$$

### Method 2

Consider two magnetically coupled coils, coil 1 and coil 2, having N_{1} and N_{2} turns respectively. If a current I_{1} flowing in the coil 1, a mutual flux (Φ_{m}) is produced that links the coil 2. Therefore,

$$\mathrm{{m}=M\frac{dl_{1}}{dt}=\frac{d}{dt}(Ml_{1})}$$

Also, the mutually induced emf is given by,

$$\mathrm{e_{m}=N_{2}\frac{d\phi_{m}}{dt}=\frac{d}{dt}(N_{2}\phi_{m})}$$

Thus, by equating these two equations, we get,

$$\mathrm{Ml_{1}=N_{2}\phi_{m}}$$

$$\mathrm{\Rightarrow\:M=\frac{N_{2}\phi_{m}}{l_{1}}\:\:\:...(2)}$$

### Method 3

If the physical dimensions of the magnetic circuit are known, then the mutual inductance of it can be determined as follows −

Let, the *‘l’* and *‘a’* be the length and cross-sectional areal of the magnetic circuit. The N_{1} and N_{2} are number of turns in the coil 1 and coil 2 respectively.

**Mutual flux,**

$$\mathrm{\phi_{m}=\frac{MMF}{Reluctance(s)}=\frac{N_{1}l_{1}}{(1/\mu_{0}\mu_{r}a)}}$$

Where,

μ

_{𝑟}= relative permeability of material of magnetic circuit,μ

_{0}= absolute permeability of vacuum or air.

$$\mathrm{\Rightarrow\:\frac{\phi_{m}}{l_{1}}=\frac{N_{1}}{(1/\mu_{0}\mu_{r}a)}}$$

$$\mathrm{(\because\:M=\frac{N_{2}\phi_{m}}{l_{1}})}$$

$$\mathrm{M=N_{2}(\frac{N_{1}}{1/\mu_{0}\mu_{r}a})}$$

$$\mathrm{\Rightarrow\:M=(\frac{N_{1}N_{2}}{1/\mu_{0}\mu_{r}a})=\frac{{N_{1}N_{2}}}{Reluctance(S)}\:\:\:\:...(3)}$$

### Method 4

If two coils have self-inductances L1 and L2, then the mutual inductance can also be given as,

$$\mathrm{M=k\sqrt{L_{1}L_{2}}\:\:\:...(4)}$$

Where, k is the *coefficient of coupling.*

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