To find the minimum number of windings needed, we can use the formula for the magnetic field inside a toroid, given by B = (μ₀ * N * I) / (2π * r), where B is the magnetic field, μ₀ is the permeability of free space (4π x 10^-7 Tm/A), N is the number of windings, I is the current, and r is the radius.
In this case, we want the magnetic field in the center of the windings to be 0.0020 T. Since the center of the windings is the same as the radius of the toroid, we can rewrite the formula as B = (μ₀ * N * I) / (2π * R), where R is the radius.
We also know that the wire can carry a current of at most 0.5 A and that the magnetic field anywhere inside the windings must be at least 0.0016 T.
To find the minimum number of windings, we can rearrange the formula to solve for N:
N = (2π * R * B) / (μ₀ * I)
Substituting the given values, we have:
N = (2π * 0.03 m * 0.0020 T) / (4π x 10^-7 Tm/A * 0.5 A)
Simplifying the expression, we get:
N = (0.06 * 0.0020) / (4 x 10^-7 * 0.5)
N = 0.00012 / 2 x 10^-7
N = 0.00012 x 5 x 10^6
N = 600 windings
Therefore, the minimum number of windings needed is 600.