Introduction:
Mutual inductance is a measure of the extent to which the magnetic field created by one coil induces a voltage in another coil. In this case, we are given the flux linked with a large circular coil and the current flowing through a small neighboring coil. We need to calculate the mutual inductance of the pair of coils.

Given:
- Flux linked with the large coil (Φ) = 5×10^-4 Wb
- Current flowing through the small coil (I) = 0.5 A

Formula:
The mutual inductance (M) between two coils can be calculated using the following formula:

M = Φ / I

Calculation:
We can calculate the mutual inductance by dividing the flux linked with the large coil by the current flowing through the small coil.

M = Φ / I
M = 5×10^-4 Wb / 0.5 A
M = 1×10^-3 H

Explanation:
The mutual inductance of the pair of coils is 1×10^-3 H (Henry). This means that when a current of 0.5 A flows through the small coil, the large coil induces a voltage of 1×10^-3 H.

Mutual inductance depends on the number of turns, the size of the coils, and their distance from each other. In this case, we are not given the number of turns or the distance between the coils, but we can still calculate the mutual inductance based on the given information.

The mutual inductance is a measure of the coupling between the two coils. A high mutual inductance indicates a strong coupling, meaning a change in current in one coil will induce a significant voltage in the other coil. Conversely, a low mutual inductance indicates a weak coupling.

Conclusion:
The mutual inductance between the large and small coils is 1×10^-3 H. This value represents the extent to which the magnetic field created by the large coil induces a voltage in the small coil when a current of 0.5 A flows through it.