The value of the integral along a square with side length 8 cm concentric with the wire and with sides parallel to those of the wire is 2.56 × 10⁻⁶ T·m.
The magnetic field due to a long straight wire with a square cross-section can be calculated using Ampere's law. Ampere's law states that the line integral of the magnetic field around a closed loop is equal to the permeance through the loop, which is the product of the enclosed current and the permeability of free space.
In this case, the enclosed current is 10 A, and the permeability of free space is µ₀ = 4π × 10⁻⁷ T·m/A. The loop is a square with side length 8 cm, so the perimeter of the loop is 32 cm.
The magnetic field is perpendicular to the sides of the square, so the line integral of the magnetic field along each side of the square is equal to the magnitude of the magnetic field times the length of the side.
The magnetic field at a distance r from the center of the wire is given by:
B = µ₀I / (2πr)
where I is the current in the wire. In this case, r = 2.5 cm, so the magnetic field at the edge of the square is:
B = (4π × 10⁻⁷ T·m/A) × (10 A) / (2π × 0.025 m) = 0.008 T
Therefore, the line integral of the magnetic field along each side of the square is:
0.008 T × 0.08 m = 6.4 × 10⁻⁷ T·m
The total line integral of the magnetic field around the square is:
6.4 × 10⁻⁷ T·m × 4 = 2.56 × 10⁻⁶ T·m
Therefore, the value of the integral along a square with side length 8 cm concentric with the wire and with sides parallel to those of the wire is 2.56 × 10⁻⁶ T·m.
Question
a long wire with a square cross-section 13 cm on a side carries a current of 10 a that is uniformly distributed over the cross-section of the wire. what is the value of the integral along a square with side length 8 cm concentric with the wire and with sides parallel to those of the wire?