JEE Main & Advanced Physics Electro Magnetic Induction Mutual Induction

Whenever the current passing through a coil or circuit changes, the magnetic flux linked with a neighbouring coil or circuit will also change. Hence an emf will be induced in the neighbouring coil or circuit. This phenomenon is called 'mutual induction'.

(1) Coefficient of mutual induction : Total flux linked with the secondary due to current in the primary is \[{{N}_{2}}{{\phi }_{2}}\] and \[{{N}_{2}}{{\phi }_{2}}\,\propto {{i}_{1}}\Rightarrow {{N}_{2}}{{\phi }_{2}}=M{{i}_{1}}\] where \[{{N}_{1}}-\]Number of turns in primary; \[{{N}_{2}}-\]Number of turns in secondary; \[{{\phi }_{2}}-\]Flux linked with each turn of secondary; \[{{i}_{1}}-\]Current flowing through primary; M-Coefficient of mutual induction or mutual inductance.

(2) According to Faraday's second law emf induces in secondary \[{{e}_{2}}=-{{N}_{2}}\frac{d{{\varphi }_{2}}}{dt}\]; \[{{e}_{\mathbf{2}}}=-M\frac{d{{i}_{\mathbf{1}}}}{dt}\]

(3) If \[\frac{d{{i}_{1}}}{dt}=\frac{1Amp}{sec}\] then \[|{{e}_{2}}|=M\].

Hence coefficient of mutual induction is equal to the emf induced in the secondary coil when rate of change of current in primary coil is unity.

(4) Units and dimensional formula of M : Similar to self-inductance (L)

(5) Dependence of mutual inductance 

(i) Number of turns \[({{N}_{1}},\,{{N}_{2}})\] of both coils

(ii) Coefficient of self inductances \[({{L}_{1}},\,{{L}_{2}})\] of both the coils

(iii) Area of cross-section of coils

(iv) Magnetic permeability of medium between the coils \[({{\mu }_{r}})\] or nature of material on which two coils are wound (v) Distance between two coils (As d increases so M decreases) 

(vi) Orientation between primary and secondary coil (for \[{{90}^{o}}\] orientation no flux relation \[M=0\])

(vii) Coupling factor 'K' between primary and secondary coil

(6) Relation between M, \[{{L}_{1}}\] and \[{{L}_{2}}\]: For two magnetically coupled coils \[M=k\sqrt{{{L}_{1}}{{L}_{2}}}\]; where k ' coefficient of coupling or coupling factor which is defined as

\[k=\frac{\text{Magnetic flux linked in secondary}}{\text{Magnetic flux linked in primary}}\];             \[0\le k\le 1\]

                                                                                                                            (A) k = 1          (B) 0 < k < 1                            (C) k = 0  

(7) The various formulae for M :

Condition	Figure
Two concentric coplaner circular coils \[M=\frac{\pi {{\mu }_{0}}{{N}_{1}}{{N}_{2}}{{r}^{2}}}{2R}\]
Two Solenoids   \[M=\frac{{{\mu }_{0}}{{N}_{1}}{{N}_{2}}A}{l}\]
Two concentric coplaner square coils x

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