Two coils \(\mathrm{A}\) and \(\mathrm{B}\) have self-inductances \(L_{1}\) and \(L_{2}\). When a steady current flows in \(A\), a quarter of its magnetic flux links \(B\). Find the proportion of the flux from a current in \(\mathrm{B}\) which links \(\mathrm{A}\). Evaluate the coefficient of coupling.

Mutual inductance is a measure of how much one coil's magnetic field affects another coil when an electrical current flows through it. When current flows through coil A, it generates a magnetic field, part of which will link with coil B. This magnetic linkage is quantitatively described by mutual inductance, denoted by the symbol M.

The mutual inductance between two coils depends on their geometry, the number of turns of wire, and the nature of the material between the coils. In our example, we consider the mutual inductance M_AB between coil A and B, and similarly M_BA when the current flows in coil B. It's crucial to understand that mutual inductance is symmetrical, meaning M_AB = M_BA, which simplifies our calculations and conceptual understanding.

Self-inductance, denoted L, is a property of a coil that describes the induced electromotive force (emf) inside the coil itself when the current flowing through the coil changes. Essentially, it's a measure of how a coil resists changes in the current passing through it due to its own magnetic field. The self-inductance L is inherent to the coil's physical properties such as its size, the number of turns, and the material it's made of.

In our exercise, self-inductance is of primary importance as it determines the magnetic flux linkage within the same coil, thereby affecting the coupling between coils. The values L₁ and L₂ for coils A and B, respectively, provide the ground upon which we evaluate the proportion of magnetic linkage to calculate the mutual inductance.

Magnetic flux linkage refers to the quantity of magnetic flux that is 'threading' through a coil. It's the product of the magnetic flux through one loop of the coil and the total number of loops (or turns) in the coil. The concept of magnetic flux linkage is integral to understanding both self-inductance and mutual inductance.

In the context of our problem, we use magnetic flux linkage to determine how much of the magnetic field produced by one coil (A or B) is interacting with the other coil. For instance, when we're told that a quarter of the flux from coil A links with coil B, we can express this as a mutual inductance M_AB that is a fraction of coil A's self-inductance L₁. Understanding magnetic flux linkage helps students visualize the concept of inductance and the interaction between coils.