For the uniform field

PHYSICS · Subjective · Fermi University Physics · Electricity, Magnetism & Electromagnetism · Magnetostatics · Magnetostatics

The Boundary-Value Problem, Ch The vector potential and the magnetostatic boundary-value problem

Problem

For the uniform field B=B0z^\mathbf{B}=B_0\hat{\mathbf{z}}, A1=B0x y^\mathbf{A}_1=B_0x\,\hat{\mathbf{y}} is a valid vector potential. Someone claims that A2=12B0(xy^+yx^)\mathbf{A}_2=\tfrac12B_0\big(x\hat{\mathbf{y}}+y\hat{\mathbf{x}}\big) is obtained from A1\mathbf{A}_1 by the gauge transformation A2=A1+∇λ\mathbf{A}_2=\mathbf{A}_1+\nabla\lambda with λ=12B0xy\lambda=\tfrac12B_0xy, and therefore produces the same B\mathbf{B}. Check the claimed gauge transformation directly, find the error, and state what field A2\mathbf{A}_2 actually produces.

Browse the problem library