An electromagnet has a steel core (κM≈ 2500) with an approximately circular cross sectional area of 8.0 cm2. The radius of the magnet is 7.0 cm; there is a small air gap of only 3.2 mm (see sketch). The current through the magnet's N= 100-turn coil is 38 A.
What will the magnetic field strength be inside the air gap? Express your answer in Tesla. Assume that the magnetic field is azimuthal (i.e. points around the circle formed by the steel) everywhere and d≪ the radius of the electromagnet.
To find the magnetic field strength inside the air gap, we need to use Ampere's law. Ampere's law states that the magnetic field circulating around a closed loop is proportional to the current passing through the loop.
The formula for the magnetic field strength inside the air gap (B) can be written as:
B = (μ0 * N * I) / (2π * r)
Where:
μ0 is the permeability of free space (4π * 10^-7 T m/A)
N is the number of turns in the coil (100 turns)
I is the current passing through the coil (38 A)
r is the radius of the magnet (7.0 cm + 3.2 mm = 7.32 cm)
Now, let's substitute the values into the formula to calculate B:
B = (4π * 10^-7 T m/A * 100 * 38 A) / (2π * 0.0732 m)
B = (4 * 10^-7 T m/A * 100 * 38 A) / (2 * 0.0732)
B ≈ 1.03 T
Therefore, the magnetic field strength inside the air gap is approximately 1.03 Tesla.
To find the magnetic field strength inside the air gap, we can use Ampere's law.
Ampere's law states that the line integral of the magnetic field along a closed path is equal to the product of the permeability of free space (μ0) and the total current passing through the area bounded by the path. In this case, since the coil has N turns, the total current passing through the area bounded by the path is N times the current through the coil.
Mathematically, Ampere's law can be written as:
∮ B · dl = μ0 * N * I
Where ∮ B · dl represents the line integral of the magnetic field B along a closed path, μ0 is the permeability of free space (equal to 4π * 10^-7 T·m/A), N is the number of turns in the coil, and I is the current through the coil.
In this problem, we want to find the magnetic field strength inside the air gap, so we need to choose a closed path that encloses the air gap. Since the magnetic field is azimuthal (points around the circle formed by the steel) everywhere, we can choose a circular path with radius equal to the radius of the electromagnet.
Let's calculate the line integral of B · dl along this path. The line integral will be equal to the product of the magnetic field strength inside the air gap (B) and the circumference of the circular path (2π * radius of the electromagnet).
∮ B · dl = B * 2π * radius
Now we can rewrite Ampere's law for this specific configuration as:
B * 2π * radius = μ0 * N * I
Solving for B, we find:
B = (μ0 * N * I) / (2π * radius)
Now we can substitute the given values into this equation to find the magnetic field strength inside the air gap:
μ0 = 4π * 10^-7 T·m/A
N = 100 turns
I = 38 A
radius = 7.0 cm = 0.07 m
B = (4π * 10^-7 T·m/A * 100 * 38 A) / (2π * 0.07 m)
= 4 * 10^-5 T
Therefore, the magnetic field strength inside the air gap is approximately 4 * 10^-5 Tesla.