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What is mutual inductance
Mutual Inductance is the property of a coil to induce an emf in a nearby coil by principle of magnetic induction when the current in the first coil changes. The two coils should lie into close proximity with each other so the magnetic field from one links with the other. It is denoted by “M”. The S.I. unit of mutual inductance is Henry (H)
Explanation of Mutual inductance
Consider two coils “A” and “B” lying close to each other. Connect the coil “A” to battery through a switch “S” and a vary resistor “R”. The coil “B” is connected to Sensitive voltmeter as shown in figure below.
When switch “S” is closed the current in the coil “A” is established and set up the magnetic field around itself and partly links with the coil “B”. With the change in current in coil “A” , the flux linked with coil “B” is also change. Hence the mutually induced emf is produced in coil “B” whose magnitude is given by Faraday’s Laws and direction by Lenz’s Law.
Mutual Inductance Formulae
Consider the figure below, If changing current “I1 ” is flowing in coil “A” which produces changing magnetic flux. Since the two coils are close to each other, some of the magnetic flux produced by the coil “A” will pass through the coil “B”. The emf is induced in the coil “B”.
E_{21}=-\frac{d\phi_{21}}{dt}\;\;\;\;(\text{As per Faraday's Law})\\\;\\\textrm{Where}\\\phi_{21}= \textrm{Flux per turn due to current}\; I_{1}\\E_{21}=\textrm{ emf induced in coil "B" due to }\phi_{21}
If coil “B” has “N2 ” turns then \\\;\\E_{21} =-N_2 \frac{d\phi_{21}}{dt}\;\;\;\;\;\;(1)\\\;\\The induced emf in coil “B” directly proportional to the current passes through the coil “A” i.e. I1
Since the total flux linking with coil “B” due to current I1 in coil “B” is directly proportional to current I1 \\\;\\N_{2}\phi_{21}\propto I_{1}\\\;\\N_{2}\phi_{21}=M_{21}I_{1}\;\;\;\;\;\;(2)\\\;\\M_{21}=\frac{N_{2}\phi_{21}}{I_{1}}\;\;\;\;\;\;\;(a)\\\;\\Where M21 is the constant of proportionality called Mutual inductance of coil ‘B” with respect to coil “A”
Similarly Mutual inductance with respect of coil “A” with respect to coil “B” is M12 \\\;\\M_{12}=\frac{N_{1}\phi_{12}}{I_2}\;\;\;\;\;\;(b)
Differentiating equation (2) with respect to time we get\\\;\\N_{2}\frac{d\phi_{21}}{dt}=M_{21}\frac{dI_{1}}{dt}\;\;\;\;\;(3)\\\;\\\text{From equation (1) and (3)}\\\;\\E_{21}=-M_{21}\frac{dI_1}{dt}\;\;\;\;\;(4)
According to Ampere’s law and Biot-Savart law confirm that the two constants, M21 and M12, are equal if the material medium (Material of Core) is same in the two coils. \\Thus M12 = M21 = M \\\;\\\textrm{Thus equation (4) becomes}\\\;\\E_{21}= -M\frac{dI_{1}}{dt}
Mutual Inductance Between Two Coaxial Solenoids
Consider two coaxial solenoids “S1 ” and “S2 ” of length \textit{"l"} and area of cross section “A1 ” and “A2 ” respectively. Let “N1 ” and “N2 ” are the turns on the solenoids “S1 ” and “S2 ” . Let “I2 ” is the current in the solenoid “S2 ” .
Magnetic flux density due to current I2 in the solenoid S2 is given by \\\;\\B_2 = \mu n_2 I_2=\mu \frac{N_2}{l}I_2
Total Magnetic flux in Solenoid “S1 ” is \\\;\\N_1\phi_2=N_1B_2A_1=N_1\mu\frac{N_2}{l}I_2A_2\;\;\;\;\;(5)\\\;\\\textrm{Also}\\\;\\M_{12}I_2=N_1\phi_{12}\;\;\;\;\;(6)\\\;\\\textrm{From equation (5 )and (6)}\\\;\\M_{12}=\frac{\mu N_1 N_2 A_1}{l}
In General The Mutual Inductance of two coaxial solenoids is\\\;\\M=\frac{\mu N_1 N_2 A}{l}
Coefficient of Coupling (K)
The Coefficient of coupling can be defined as the fraction or percentage of the magnetic flux produced by the current in one coil that links with the other coil. It is represented by the symbol (K). It is expressed as a fractional number between 0 and 1 instead of a percentage (%) value, where 0 indicates zero or no inductive coupling, and 1 indicating full or 100% or maximum inductive coupling. The coefficient of coupling will always less than the unity and the maximum value of the coefficient of coupling can be 1 or 100%.
From equation (a) and equation (b)\\\;\\M_{21}=\frac{N_2 \phi_{21}}{I_1}\;\;\;\;(7)\;\textrm{Mutual inductance of coil "B" with respect to coil "A"}\\\;\\And\\\;\\M_{12}=\frac{N_1 \phi_{12}}{I_2}\;\;\;\;\;(8)\;\textrm{Mutual inductance of coil "A" with respect to coil "B"}
Let coeffiecent of coupling between two coils is “K”, then \\\;\\\phi_{21}=K\phi_1\\\;\\\textrm{And}\\\;\\\phi_{12}=K\phi_2\\\;\\\textrm{putting the value of }\phi_{21} \;and \;\phi_{12} \textrm{in above equation (7) and (8) respectively}
M_{21}=\frac{N_2 K \phi_1}{I_1}=\frac{KN_2\phi_1}{I_1}\;\;\;\;\;\;(9)\\\;\\M_{12}=\frac{N_1K\phi_2}{I_2}=\frac{KN_1\phi_2}{I_2}\;\;\;\;\;\;(10)
M_{21}\times M_{12}=(\frac{KN_2\phi_1}{I_1})\times(\frac{KN_1\phi_2}{I_2})=K^2 \times {(\frac{N_1\phi_1}{I_1})\times(\frac{N_2\phi_2}{I_2})}\;\;\;\;\;\;(11)\\\;\\\textrm{we know}\\\;\\M_{12} =M_{21}=M\\\;\\\textrm{then equation (11) becomes }\\\;\\M^2 = K^2\times (\frac{N_1\phi_1}{I_1})\times(\frac{N_2\phi_2}{I_2})\;\;\;\;\;\;(12)
Let Self inductance of coil “A” and coil “B” are “L1 ” and “L2 ” respectively\\\;\\L_1=\frac{N_1\phi_1}{I_1}\\\;\\L_2=\frac{N_2\phi_2}{I_2}
Puuting The Value of “L1 ” and “L2 ” in equation (12) we get\\\;\\M^2 = K^2 L_1 L_2\\\;\\K =\frac{M}{\sqrt{L_1L_2}}
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