Question

1 Using the vector potential Consider a very long straight wire carrying a current I that points in the positive z direction. (a) Determine the vector potential A. (b) Use the vector potential you compute to obtain the magnetic field B Hint: if you find that B vanishes go back to part (a) (c) Check that the divergence A vanishes. d) The general vanishing of ▽ . A is a consquence of magnetostatics not merely a choice of gauge) when the current distribution is localized. Prove this statement

Answer 1

as current I points in +z direction vector potential also point in +z (K) Now the vector potential in z is A = mu_0 I/4 pi K integral_(-infinity)^infinity dz/(r^2 + z^2) where r is distance of point P from conductor therefore A = mo_0 I/4 pi K ln (z + squareroot (z^2 + r^2))_(-infinity)^infinity A = (- mo_0 I/2pi) ln r k (b) B = nabla times A As A is in K direction so vector potential has Phi component B = (partial differential A_r/ partial differential z) - (partial differential A_z/ partial differential r) = 0 - (partial differential/ partial differential r) (- mu_o I/2 pi ln r) B = + mu_0 I/2 pi r Divergence of A nabla A = 1/(Rho^2 sin theta) [(partial differential/ partial differential r) (A_r r^2 sin theta) + (partial differential/ partial differential theta)(A_theta r sin theta) + (partial differential/ partial differential Phi)(A_Phi r)] A is having only Phi component so nabla A = 1/r^2 sin theta [partial differential/ partial differential Phi theta mu_0 I/2 pi ln r] longrightarrow nabla A = 0 rightarrow (d) rightarrow forward