Final answer:
The mutual inductance of a solenoid when replaced by a coil of 55 turns with a larger radius can be found using the formula that incorporates the number of turns and cross-sectional area. It is determined by how much of the solenoid's magnetic field lines pass through the coil after accounting for the new number of turns and dimensions.
Step-by-step explanation:
The question asks about calculating the mutual inductance when the outer coil of a solenoid system is replaced by another coil with a greater number of turns and radius. Mutual inductance is a measure of how a change in current in one coil induces an electromotive force (emf) in another coil.
To find the mutual inductance, we use the formula M = (μ0 × N1 × N2 × A) / l, where μ0 is the permeability of free space, N1 is the number of turns of the solenoid, N2 is the number of turns of the coil, A is the cross-sectional area through which the magnetic field lines pass unaltered (and for a coil surrounding the solenoid, it is the cross-sectional area of the solenoid itself), and l is the length of the solenoid.
In the given problem, the mutual inductance of the configuration with a coil of 55 turns and a radius three times that of the solenoid can be calculated based on the proportion of the solenoid's magnetic field lines that will pass through the larger coil. Assuming the larger coil encompasses the solenoid completely and the field lines are uniform across the solenoid's cross-section, the cross-sectional area (A) remains the same as in the original solenoid, which is the area through which the field lines pass unaltered.
Using the formula and the provided values, the mutual inductance for the new configuration is as follows: M = (μ0 × N1 (450 turns) × N2 (55 turns) × A (8.0 × 10−3 m2)) / (l (0.40 m)). Note that μ0, the permeability of free space, is a constant valued at approximately 4π× 10−7 T·m/A. Plugging in these values would provide the mutual inductance for this specific solenoid and coil configuration in Henrys (H).