Understanding Coefficient of Coupling

The coefficient of coupling, denoted by the symbol $k$, is a fundamental concept in the study of magnetically coupled circuits. It quantifies the degree to which the magnetic flux produced by one coil links with another coil. When two coils are placed near each other, a change in current in one coil can induce a voltage in the other due to the interaction of their magnetic fields. This phenomenon is known as mutual inductance.

Coefficient of Coupling Definition

The coefficient of coupling ($k$) is mathematically defined as the ratio of the mutual inductance ($M$) between the two coils to the geometric mean of their individual self-inductances ($L_1$ and $L_2$). The formula is:

$\qquad k = \frac{M}{\sqrt{L_1 L_2}}$

Here:

• $M$ is the mutual inductance between coil 1 and coil 2.
• $L_1$ is the self-inductance of coil 1.
• $L_2$ is the self-inductance of coil 2.

Coefficient of Coupling Values

The value of the coefficient of coupling $k$ always lies between 0 and 1, inclusive. This range represents the possibilities of magnetic interaction between the coils:

• If $k = 0$, it means there is no magnetic coupling between the coils. None of the flux from one coil links with the other.
• If $0 < k < 1$, it indicates partial magnetic coupling. A portion of the flux from one coil links with the other.
• If $k = 1$, it signifies perfect magnetic coupling. This happens when the entire magnetic flux produced by one coil links with the other coil.

Coefficient of Coupling when Entire Flux Links

The question states a specific condition: "the entire flux produced by one coil is linked with another coil". As discussed above, this condition corresponds to the case of perfect magnetic coupling.

When perfect magnetic coupling occurs, the mutual inductance ($M$) reaches its maximum possible value, which is equal to $\sqrt{L_1 L_2}$. Substituting this into the formula for $k$:

$\qquad k = \frac{M}{\sqrt{L_1 L_2}} = \frac{\sqrt{L_1 L_2}}{\sqrt{L_1 L_2}}$

Therefore, when the entire flux produced by one coil is linked with another coil, the coefficient of coupling $k$ is equal to 1.

This situation is ideal and often approximated in practical devices like transformers, where cores are used to ensure most of the flux is confined and links both coils.

Based on the definition and the given condition, the coefficient of coupling $k$ is 1.