Correct Answer - Option 1 :
\(\sqrt{L_1L_2}\)
Concept:
Self-induction:
1. Self-inductance is the property of the current-carrying coil that resists or opposes the change of current flowing through it.
2. This occurs mainly due to the self-induced emf produced in the coil itself.
\(⇒ L=N\frac{ϕ}{I}\)
Where N = number of turns in the coil, L = self-inductance of the coil, ϕ = flux associated with the coil and I = current in the coil
Mutual-inductance
1. When two coils are brought in proximity with each other the magnetic field in one of the coils tends to link with the other. If this magnetic field of the first coil is changed then the magnetic flux associated with the second coil changes and this leads to the generation of voltage in the second coil.
2. This property of a coil that affects or changes the current and voltage in a secondary coil is called mutual inductance.
\(⇒ M_{12}=N_{1}\frac{ϕ_{1}}{I_{2}}\)
Calculation:
Let us first consider a case when the total flux associated with one coil links with the other, i.e. a case of maximum flux linkage. Consider two coils placed adjacent to each other. Thus,
\(⇒ M_{12}=N_{1}\frac{ϕ_{1}}{I_{2}}\) -----(1)
\(⇒ M_{21}=N_{2}\frac{ϕ_{2}}{I_{1}}\) -----(2)
The self-inductance of the coil can be written as,
\(⇒ L_{1}=N_{1}\frac{ϕ_{1}}{I_{1}}\) -----(3)
\(⇒ L_{2}=N_{2}\frac{ϕ_{2}}{I_{2}}\) -----(3)
By multiplying equation 1 and equation 2, we get
\(⇒ M_{12}M_{21}=N_{1}\frac{ϕ_{1}}{I_{2}}N_{2}\frac{ϕ_{2}}{I_{1}}\)
\(⇒ M_{12}M_{21}=L_{1}L_{2}\)
if M12 = M21 = M,
\(⇒ M_{12}M_{21}=M^2=L_{1}L_{2}\)
\(\Rightarrow M=\sqrt{L_{1}L_{2}}\)
Hence, option 1 is correct.
