A circular wire coil consists of 100 turns and is wound tightly around a very long iron cylinder with a radius of 2.5 cm and a relative permeability of 2200. The loop has a current of 7.5A in it. Determine the magnetic field strength produced by the coil (a) at the center of the coil and (b) at a location on the central axis of the iron cylinder 5.0 cm above the center of the circular coil.
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To determine the magnetic field strength produced by the coil at different locations, we can use Ampere's law. Ampere's law states that the magnetic field around a closed loop is proportional to the current passing through the loop.
(a) To find the magnetic field at the center of the coil, we can consider a circular loop of radius r1 that is inside the coil and concentric with it. The magnetic field at the center of the coil is the same as the magnetic field along this smaller loop.
Using Ampere's law, the equation becomes:
∮B · dl = μ₀ * I_enclosed
The left side of the equation represents the line integral of the magnetic field B around the loop, and the right side represents the product of the permeability of free space (μ₀) and the current enclosed by the loop.
Since we have a single circular loop with 100 turns, the current enclosed by this loop is 100 times the current in the coil, i.e., 100 * 7.5 A = 750 A.
At the center of the coil, the radius of the circular loop, r1, is 0. Therefore, the equation becomes:
B * 2πr1 = μ₀ * 750 A
Since r1 is zero, the left side of the equation becomes zero, simplifying the equation to:
0 = μ₀ * 750 A
Solving for B, we find that the magnetic field at the center of the coil is zero.
(b) To find the magnetic field at a location on the central axis of the iron cylinder 5.0 cm above the center of the circular coil, we can consider a circular loop of radius r2 that is outside the coil and concentric with it.
Using Ampere's law with this new loop, the equation becomes:
∮B · dl = μ * I_enclosed
Where μ represents the relative permeability of the iron cylinder.
At the given location, the radius of the circular loop, r2, is 5.0 cm + 2.5 cm (radius of the circular coil). Therefore, the equation becomes:
B * 2πr2 = μ * μ₀ * 750 A
Solving for B, we get:
B = (μ * μ₀ * 750 A) / (2πr2)
Substituting the given values, μ = 2200, μ₀ = 4π × 10⁻⁷ Tm/A, and r2 = 5.0 cm + 2.5 cm = 7.5 cm = 0.075 m, we can calculate the magnetic field strength.
B = (2200 * 4π × 10⁻⁷ Tm/A * 750 A) / (2π * 0.075 m)
Simplifying, the result is:
B = 0.037 T
Therefore, the magnetic field strength produced by the coil at a location on the central axis of the iron cylinder 5.0 cm above the center of the circular coil is 0.037 Tesla (T).