Question

1. If the magnetic vector potential A is conservative, then which of these is true about the magnetic field B? (A) B is non-zero (B) B is zero (C) B = ∇·A (D) Something else​

Answer 1

Answer:How do you know if A vector field is conservative?

This condition is based on the fact that a vector field F is conservative if and only if F=∇f for some potential function. We can calculate that the curl of a gradient is zero, curl∇f=0, for any twice continuously differentiable f:R3→R. Therefore, if F is conservative, then its curl must be zero, as curlF=curl∇f=0.

Explanation:The line integral over multiple paths of a conservative vector field. The integral of conservative vector field F(x,y)=(x,y) from a=(3,−3) (cyan diamond) to b=(2,4) (magenta diamond) doesn't depend on the path. Path C (shown in blue) is a straight line path from a to b. Paths B (in green) and E (in red) are curvy paths, but they still start at a and end at b. Each path has a colored point on it that you can drag along the path. The corresponding colored lines on the slider indicate the line integral along each curve, starting at the point a and ending at the movable point (the integrals alone the highlighted portion of each curve). Moving each point up to b gives the total integral along the path, so the corresponding colored line on the slider reaches 1 (the magenta line on the slider). This demonstrates that the integral is 1 independent of the path.