Magnetic Vector Potential

The magnetic vector potential, denoted by A, is a vector field that arises from a current distribution. It is used to describe the magnetic field B that is produced by the current. In this case, we are given the magnetic vector potential A as A = x²yî + y²xĵ - xyzḵ.

Calculating the Magnetic Field

To find the magnetic field B at a given point, we need to take the curl of the magnetic vector potential A. The curl of a vector field is a vector that describes the rotation of the field.

Curl of the Magnetic Vector Potential
∇ x A = (∂A₃/∂y - ∂A₂/∂z)î + (∂A₁/∂z - ∂A₃/∂x)ĵ + (∂A₂/∂x - ∂A₁/∂y)ḵ

Calculating the Partial Derivatives
∂A₁/∂x = 2xy
∂A₂/∂x = 0
∂A₃/∂x = -yz

∂A₁/∂y = x²
∂A₂/∂y = 2yx
∂A₃/∂y = -xz

∂A₁/∂z = 0
∂A₂/∂z = y²
∂A₃/∂z = -xy

Substituting the Partial Derivatives
∇ x A = (-yz - 2yx)î + (x² + xz)ĵ + (2xy - x²)ḵ

Calculating the Magnetic Field at (-1,2,5)
To find the magnetic field at (-1,2,5), we substitute the values of x, y, and z into the equation for the curl of A.

∇ x A = (-yz - 2yx)î + (x² + xz)ĵ + (2xy - x²)ḵ

∇ x A = (-5 - 2(2))î + ((-1)² + (-1)(5))ĵ + (2(-1)(2) - (-1)²)ḵ

∇ x A = -9î + (-6)ĵ + (-3)ḵ

Therefore, the magnetic field B at (-1,2,5) is B = -9î - 6ĵ - 3ḵ.

Conclusion
The magnetic field B at (-1,2,5) is calculated by taking the curl of the magnetic vector potential A. By substituting the given values into the equation, we find that B = -9î - 6ĵ - 3ḵ.