Mutual inductance is a fundamental concept in electromagnetism that describes the ability of one coil or circuit to induce an electromotive force (emf) in another coil or circuit due to changes in the magnetic field. The mutual inductance (M) between two circuits is usually represented in Henrys (H). Here are the expressions for mutual inductance in various scenarios:
Ideal Solenoid and Circular Coil:
If you have an ideal solenoid (long, tightly wound coil) and a circular coil outside the solenoid, the mutual inductance between them can be calculated using the following expression:
M = (μ₀ * N₁ * N₂ * A) / l
where:
M is the mutual inductance
μ₀ is the permeability of free space (4π × 10^-7 H/m)
N₁ and N₂ are the number of turns in the solenoid and the circular coil, respectively
A is the cross-sectional area of the solenoid
l is the length of the solenoid
Two Parallel Wires:
If you have two parallel wires carrying currents, the mutual inductance between them can be approximated as:
M = (μ₀ * I₁ * I₂ * ℓ) / (2π * d)
where:
M is the mutual inductance
μ₀ is the permeability of free space
I₁ and I₂ are the currents in the two wires
ℓ is the length of the wires over which the currents flow
d is the distance between the wires
General Case:
For more complex geometries, you can calculate the mutual inductance using the following integral expression:
M = ∫∫ (B₁ * dA₂) = ∫∫ (B₂ * dA₁)
where:
B₁ and B₂ are the magnetic flux densities produced by one circuit in the vicinity of the other
dA₁ and dA₂ are differential areas over which the integration is performed
These expressions provide a basis for calculating mutual inductance in different situations. Keep in mind that the actual calculation might involve more complex integrals and considerations depending on the geometry and arrangement of the circuits involved.