Question

In a current free region of relative permeability 1, the magnetic scalar potential is given as Vm = x^2y + y^2x + z. Calculate the magnitude of the magnetic flux density at (1, 1, 1)?

A) 2√2 A/m
B) 4 A/m
C) 1 A/m
D) 3 A/m

Answer 1

The magnitude of the magnetic flux density at the point (1, 1, 1) is calculated to be √19 A/m using the negative gradient of the given magnetic scalar potential. This result does not match any of the provided options, implying an error in the question or options.

Step-by-step explanation:

To find the magnetic flux density at the point (1, 1, 1) for the given magnetic scalar potential Vm = x2y + y2x + z, we need to consider that magnetic flux density B is related to the magnetic scalar potential Vm by the negative gradient. Since the region has a relative permeability of 1, we can use B = -∇Vm where ∇ is the gradient operator.

Computing the gradient of Vm, we have:

• 
  
• ∂Vm/∂x = 2xy + y2
• 
  
• ∂Vm/∂y = x2 + 2yx
• 
  
• ∂Vm/∂z = 1

At the point (1, 1, 1), the partial derivatives become:

• 
  
• ∂Vm/∂x = 3
• 
  
• ∂Vm/∂y = 3
• 
  
• ∂Vm/∂z = 1

Thus, B = -∇Vm = -(∂Vm/∂x î + ∂Vm/∂y ı + ∂Vm/∂z k) at (1, 1, 1) becomes B = -(3 î + 3 ı + 1 k) A/m.

To find the magnitude of B, we calculate |B| = √(32 + 32 + 12) A/m = √19 A/m. Therefore, the magnitude of the magnetic flux density at (1, 1, 1) is √19 A/m, which is not one of the provided options, suggesting a possible typo or mistake in the question or answer choices.