When the flux through a coil with a magnetic circuit of constant reluctance increases, the current flowing through the coil also increases. This is due to Faraday's law of electromagnetic induction, which states that a change in magnetic flux induces an electromotive force (emf) in a conductor. According to Ohm's law, the induced emf causes a current to flow through the coil.

Explanation:

1. Faraday's Law of Electromagnetic Induction:
- When the flux passing through a coil changes, it induces an emf in the coil.
- This induced emf is given by the equation: emf = -N(dΦ/dt), where N is the number of turns in the coil and dΦ/dt is the rate of change of flux.
- The negative sign in the equation indicates that the induced emf opposes the change in flux.

2. Relationship between Current and Voltage:
- According to Ohm's law, the current flowing through a coil is directly proportional to the voltage across it.
- The relationship is given by the equation: I = V/R, where I is the current, V is the voltage, and R is the resistance of the coil.

3. Flux and Current Relationship:
- When the flux passing through the coil increases, the rate of change of flux (dΦ/dt) also increases.
- As a result, the induced emf in the coil increases.
- The increased emf causes a higher voltage across the coil.
- Since the resistance of the coil remains constant, according to Ohm's law, an increase in voltage leads to an increase in current flowing through the coil.

Conclusion:
When the flux passing through a coil with a magnetic circuit of constant reluctance increases, the current flowing through the coil also increases. This is because the change in flux induces an emf in the coil, which in turn leads to a higher voltage across the coil. As a result, the current flowing through the coil increases to maintain the relationship between current and voltage according to Ohm's law.