Final answer:
To wind a solenoid that produces a 0.0250 T magnetic field with 3.0 A, calculate the number of turns per meter needed using the magnetic field formula for solenoids and multiply by the length of the solenoid, then multiply the number of turns by the circumference of the solenoid. Approximately 116996 cm or 1169.96 m of wire is needed.
Step-by-step explanation:
To find the total length of wire needed to wind a solenoid that produces a magnetic field of 0.0250 T with a current of 3.0 A, we can use the formula for the magnetic field inside a solenoid: B = μ₀ * n * I, where B is the magnetic field in teslas, μ₀ is the permeability of free space (4π x 10^-7 T·m/A), n is the number of turns per meter, and I is the current in amperes.
First, we need to solve this equation for n:
n = B / (μ₀ * I)
If we plug in the values from our scenario (B = 0.0250 T, I = 3.0 A), we get:
n = 0.0250 / (4π x 10^-7 T·m/A * 3.0 A)
n ≈ 66477.2 turns/m
We are provided with the length of the solenoid, 16 cm. We convert this to meters to match our unit for n:
L = 16 cm = 0.16 m
The total number of turns, N, is n times L:
N = n * L = 66477.2 turns/m * 0.16 m
N ≈ 10636 turns
The diameter of the solenoid is 3.5 cm, so the circumference (the length of wire per turn) is:
C = π * d
C = π * 3.5 cm ≈ 11 cm
Finally, we multiply the number of turns by the circumference to find the total length of wire:
Length of wire = N * C = 10636 turns * 11 cm/turn
Length of wire ≈ 116996 cm
The total length of wire needed is approximately 116996 cm, or about 1169.96 m.