Introduction:
Electrodynamometer ammeter is a type of instrument used to measure electric current in a circuit. It is based on the principle of electromagnetic forces. The deflection of the pointer in an electrodynamometer ammeter depends on various factors.

Explanation:
The correct answer to this question is option 'D', which states that the deflection of the pointer is proportional to the mean-square of currents in the fixed coil and the moving coil. Let's understand why this is the case:

1. Principle of Electrodynamometer:
The electrodynamometer works on the principle of electromagnetic forces. When a current-carrying coil is placed in a magnetic field, a force is experienced by the coil due to the interaction between the magnetic field and the current flowing through the coil.

2. Deflection of the Pointer:
The deflection of the pointer in an electrodynamometer ammeter is proportional to the net force experienced by the moving coil. This force is a result of the interaction between the magnetic field produced by the fixed coil and the current flowing through both the fixed and moving coils.

3. Mean-Square of Currents:
The mean-square of currents in the fixed coil and moving coil is related to the net force experienced by the moving coil. The mean-square value of a current is a measure of the average power dissipated by the current over a period of time.

4. Relationship between Current and Force:
According to the principle of electrodynamics, the force experienced by a current-carrying coil in a magnetic field is directly proportional to the product of the current flowing through the coil and the magnetic field strength.

5. Mean-Square of Currents and Force:
Since the deflection of the pointer is directly proportional to the net force experienced by the moving coil, it can be concluded that the deflection is proportional to the mean-square of currents in the fixed coil and moving coil. This means that the higher the mean-square value of the currents, the greater the deflection of the pointer.

Conclusion:
In an electrodynamometer ammeter, the deflection of the pointer is proportional to the mean-square of currents in the fixed coil and moving coil. This is because the net force experienced by the moving coil, and hence the deflection of the pointer, depends on the average power dissipated by the currents flowing through the coils.