he magnetic coils of a tokamak fusion reactor are in the shape of a toroid having an inner radius of 0.700 m and an outer radius of 1.30 m. The toroid has 800 turns of large-diameter wire, each of which carries a current of 12.0 kA.
(a) Find the magnitude of the magnetic field inside the toroid along the inner radius.
T
(b) Find the magnitude of the magnetic field inside the toroid along the outer radius.
T
B₁= μ₀NI/2πr = 4π•10⁻⁷•800•12000/2π•0.7= …
B₂= μ₀NI/2πR = 4π•10⁻⁷•800•12000/2π•1.3= …
http://hyperphysics.phy-astr.gsu.edu/hbase/magnetic/toroid.html
To solve these problems, we can use Ampere's Law, which states that the magnetic field along a closed loop is proportional to the current enclosed by the loop.
(a) To find the magnitude of the magnetic field inside the toroid along the inner radius, we will consider a circular loop of radius r = 0.700 m inside the toroid. The loop encloses all the wire turns with a total current of I = 800 * 12.0 kA = 9.60 MA.
Using Ampere's Law, we have:
∮B * dl = μ₀ * Ienc
where B is the magnetic field, dl is the small element of the loop, μ₀ is the vacuum permeability (4π × 10^(-7) T * m/A), and Ienc is the current enclosed by the loop.
Since B and dl are parallel, the integral simplifies to:
B * 2πr = μ₀ * Ienc
Rearranging the equation to solve for B:
B = (μ₀ * Ienc) / (2πr)
Substituting the given values:
B = (4π × 10^(-7) T * m/A) * (9.60 MA) / (2π * 0.700 m)
Simplifying the expression:
B = (4π × 10^(-7) T * m/A) * (9.60 × 10^6 A) / (2π * 0.700 m)
B ≈ 20.51 T
Therefore, the magnitude of the magnetic field inside the toroid along the inner radius is approximately 20.51 T.
(b) To find the magnitude of the magnetic field inside the toroid along the outer radius, we will consider a circular loop of radius r = 1.30 m inside the toroid. The loop encloses all the wire turns with the same total current of I = 9.60 MA.
Using the same formula as before:
B = (μ₀ * Ienc) / (2πr)
Substituting the given values:
B = (4π × 10^(-7) T * m/A) * (9.60 MA) / (2π * 1.30 m)
Simplifying the expression:
B = (4π × 10^(-7) T * m/A) * (9.60 × 10^6 A) / (2π * 1.30 m)
B ≈ 11.42 T
Therefore, the magnitude of the magnetic field inside the toroid along the outer radius is approximately 11.42 T.
To find the magnitude of the magnetic field inside the toroid along the inner radius, you 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 closed path.
The formula for the magnetic field at a distance r from a wire carrying a current I is given by B = (μ₀ * I) / (2π * r), where B is the magnetic field, μ₀ is the permeability of free space, I is the current, and r is the distance from the wire.
In this case, the current is given as 12.0 kA (kiloamperes) and the distance is the inner radius of the toroid, which is 0.700 m. The permeability of free space (μ₀) is a constant with a value of 4π × 10⁻⁷ T·m/A.
Substituting the given values into the formula, we have:
B = (4π × 10⁻⁷ T·m/A * 12.0 × 10³ A) / (2π * 0.700 m)
Simplifying the expression, we get:
B = (4 × 10⁻⁷ * 12.0 × 10³) / (2 × 0.700) T
B = (48 × 10⁻⁴) / (1.4) T
B ≈ 34.29 × 10⁻⁴ T
Therefore, the magnitude of the magnetic field inside the toroid along the inner radius is approximately 34.29 × 10⁻⁴ T.
To find the magnitude of the magnetic field inside the toroid along the outer radius, we can use the same formula, but with the distance being the outer radius of the toroid, which is 1.30 m.
Substituting the given values into the formula, we have:
B = (4π × 10⁻⁷ T·m/A * 12.0 × 10³ A) / (2π * 1.30 m)
Simplifying the expression, we get:
B = (4 × 10⁻⁷ * 12.0 × 10³) / (2 × 1.30) T
B = (48 × 10⁻⁴) / (2.6) T
B ≈ 18.46 × 10⁻⁴ T
Therefore, the magnitude of the magnetic field inside the toroid along the outer radius is approximately 18.46 × 10⁻⁴ T.