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**Hint:**We'll start by learning what mutual inductance is and how it works.Mutual inductance is the main operating principle of generators, motors, and transformers. In order to answer this question and determine the mutual conductance coefficient.

**Complete step by step answer:**

Let's look at what mutual conductance means: When two coils are brought close together, the magnetic field in one of them tends to interact with the magnetic field in the other. The second coil then generates voltage as a result of this. Mutual inductance is the property of a coil that influences or alters the current and voltage in a secondary coil.

Coil 2 experiences a change in magnetic flux as a result of the change in ${{I}_{1}}$. When a current ${{I}_{1}}$ passes through the first coil of ${{N}_{1}}$ turns, magnetic field B is formed. Few magnetic field lines will pass through coil 2 due to the proximity of the two coils. ${{\phi }_{21}}\to $ Due to current ${{I}_{1}}$, magnetic flux in one turn of coil 2. There will be an induced emf in coil 2 if we vary the current with respect to time.

${{\varepsilon }_{ind}}=-\dfrac{d\phi }{dt}$(According to Faraday’s law) ${{\varepsilon }_{21}}=-{{N}_{2}}\dfrac{d{{\phi }_{21}}}{dt}{{\varepsilon }_{21}}=-{{N}_{2}}\dfrac{d}{dt}\left( \bar{B}.\bar{A} \right)$

The induced emf in coil 2 directly proportional to the current passes through the coil 1.

${{N}_{2}}{{\phi }_{21}}\propto {{I}_{1}}{{N}_{2}}{{\phi }_{21}}={{M}_{21}}{{I}_{1}}...\left( 1 \right)$

The constant of proportionality is called mutual inductance. It can be written as

${{M}_{21}}=\dfrac{{{N}_{2}}{{\phi }_{21}}}{{{I}_{1}}}...\left( 2 \right)$

The SI unit of inductance is known as henry (H)

$1H=\dfrac{1\left( Tesla \right).1\left( {{m}^{2}} \right)}{1.A}$

Similarly, when the current in coil 2 varies with respect to time, it may cause an induced emf in coil 1. After that,

${{\varepsilon }_{12}}=-{{N}_{1}}\dfrac{d{{\phi }_{12}}}{dt}{{N}_{1}}{{\phi }_{12}}\propto {{I}_{2}}{{N}_{1}}{{\phi }_{12}}={{M}_{12}}{{I}_{2}}...\left( 3 \right)$

${{M}_{12}}=\dfrac{{{N}_{1}}{{\phi }_{12}}}{{{I}_{2}}}...\left( 4 \right)$

Another reciprocal inductance is this proportionality constant.

**Hence,coefficient of mutual induction is the ratio of induced e.m.f in secondary coil to the rate of change of current in primary coil.**

**Note:**It's important to remember that mutual inductance is solely determined by geometrical factors of the two coils, such as the number of turns and radii of the two coils, as well as material medium properties, such as magnetic permeability of the medium surrounding the coils.

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