Two long straight wires

PHYSICS · Multi-part · Fermi University Physics · Mathematical Methods for Physics · Electricity, Magnetism & Electromagnetism · Magnetostatics

Invariance, Ch Integrating forms: Stokes's theorem, holes and Maxwell's equations

Problem

Two long straight wires, perpendicular to a plane, carry steady currents

I_1=4.0\,\mathrm{A}

and

I_2=2.5\,\mathrm{A}

. A closed loop

\gamma

lying in that plane winds

n_1=3

times anticlockwise round wire

1

and

n_2=2

times clockwise round wire

2

(so

n_2=-2

in the convention where anticlockwise is positive), enclosing no other current. Outside a wire carrying current

I

, the magnetic field one-form is

(\mu_0I/2\pi)\,\mathrm{d}\phi

about that wire, where

\phi

is the angle measured round it; take

\mu_0=4\pi\times10^{-7}\,\mathrm{T\,m\,A^{-1}}

. ((a)) Compute the circulation

\oint_\gamma\mathbf{B}\cdot\mathrm{d}\mathbf{l}

. ((b)) Away from the two wires,

\mathbf{B}

has zero curl everywhere on

\gamma

. Explain, using the language of closed and exact forms, why this does not imply that

\mathbf{B}

has a single-valued potential on the whole region traversed by

\gamma