AcademyInductance
Academy
Mutual Inductance
Level 1 - Physics topic page in Inductance.
Principle
Changing current in one circuit induces emf in another by changing shared flux linkage.
Notation
\(M\)
mutual inductance between two circuits
\(\mathrm{H}\)
\(I_1\)
current in source circuit
\(\mathrm{A}\)
\(N_2\Phi_{21}\)
flux linkage in circuit 2 due to current in circuit 1
\(\mathrm{Wb}\)
\(\mathcal E_2\)
emf induced in circuit 2
\(\mathrm{V}\)
\(k\)
coupling coefficient
1
\(L_1,L_2\)
self-inductances of the two circuits
\(\mathrm{H}\)
Method
Derivation 1: Define mutual inductance
If current in circuit 1 creates flux through circuit 2, the flux linkage in circuit 2 is proportional to \(I_1\) in a linear magnetic system.
Flux linkage
\[N_2\Phi_{21}=MI_1\]
Mutual inductance
\[M=\frac{N_2\Phi_{21}}{I_1}\]
Derivation 2: Induced emf
Use Faraday's law on the flux linkage in the second circuit.
Faraday's law
\[\mathcal E_2=-\frac{d}{dt}(N_2\Phi_{21})\]
Linear coupling
\[\mathcal E_2=-M\frac{dI_1}{dt}\]
Reciprocity
\[M_{12}=M_{21}=M\]
Derivation 3: Coupling limit
Not all flux from one coil must link the other. The coupling coefficient measures this.
Coupling form
\[M=k\sqrt{L_1L_2}\]
Allowed range
\[0\le k\le1\]
Rules
Mutual inductance
\[\displaystyle M=\frac{N_2\Phi_{21}}{I_1}\]
Induced emf
\[\displaystyle \mathcal E_2=-M\frac{dI_1}{dt}\]
Reciprocity
\[M_{12}=M_{21}\]
Coupling coefficient
\[\displaystyle M=k\sqrt{L_1L_2}\]
Examples
Question
Two coils have
\[M=0.20\,\mathrm{H}\]
The current in coil 1 changes at \[5.0\,\mathrm{A\,s^{-1}}\]
Find \[|\mathcal E_2|\]
Answer
\[|\mathcal E_2|=M\left|\frac{dI_1}{dt}\right|=(0.20)(5.0)=1.0\,\mathrm{V}\]
Checks
- Mutual inductance is geometric and material-dependent in a linear system.
- Constant current produces no induced emf in the other circuit.
- The minus sign gives the Lenz-law direction, not the magnitude.