Final answer:
The magnetic field at the center of the solenoid is 3.14 x 10^-2 T
Step-by-step explanation:
The magnetic field at the center of a solenoid can be calculated using the formula B = μ0 * n * I, where B is the magnetic field, μ0 is the permeability of free space (4π x 10^-7 T*m/A), n is the number of turns per unit length, and I is the current.
In this case, the length of the solenoid is given as 5.0 cm and the radius is 0.35 cm.
The number of turns is given as 412 and the current is 12.0 A.
First, we need to calculate the number of turns per unit length (n).
Since the solenoid is 5.0 cm long and has 412 turns, n = 412 / 0.05
= 8240 turns/m.
Now we can calculate the magnetic field:
B = (4π x 10^-7 T*m/A) * (8240 turns/m) * (12.0 A)
= 3.14 x 10^-2 T.