Two long conductors share the axis: an inner wire of radius and an outer conducting…

PHYSICS · Multi-part · Fermi University Physics · Mathematical Methods for Physics · Electricity, Magnetism & Electromagnetism · Vector calculus and field geometry

Electricity, Magnetism and Electromagnetism, Ch Divergence, curl and the local form of the field laws

Problem

Two long conductors share the

z

axis: an inner wire of radius

a = 1.0\ \mathrm{mm}

and an outer conducting cylinder whose inner surface is at radius

b = 4.0\ \mathrm{mm}

; the gap

a < s < b

is vacuum, where

s = \sqrt{x^2+y^2}

. The inner wire carries a steady current

I = 4.0\ \mathrm{A}

along

+\hat{\mathbf{z}}

and a uniform line charge

\lambda = +2.5\times10^{-9}\ \mathrm{C\,m^{-1}}

; the outer cylinder carries

4.0\ \mathrm{A}

back along

-\hat{\mathbf{z}}

and

-\lambda

per unit length. Everything is steady, so the fields are static. Treat the two conductors as infinitely long. In the gap the fields are

\mathbf{E} = \frac{\lambda}{2\pi\varepsilon_0}\,\frac{x\,\hat{\mathbf{x}} + y\,\hat{\mathbf{y}}}{x^2+y^2},\qquad \mathbf{B} = \frac{\mu_0 I}{2\pi}\,\frac{-y\,\hat{\mathbf{x}} + x\,\hat{\mathbf{y}}}{x^2+y^2}.

((a)) Name the integral law, and the surface or loop, from which each of these two fields was obtained in Chapters 2 and 8, and give

|\mathbf{E}|

and

|\mathbf{B}|

s = 2.5\ \mathrm{mm}

. ((b)) Compute

\nabla\cdot\mathbf{E}

in the gap. What charge density does Gauss's law at a point therefore give there, and is that what you expect? ((c)) Compute

\nabla\times\mathbf{B}

in the gap. What current density does the local form of Ampère's law give there? The circulation of

\mathbf{B}

round a circle of radius

2.5\ \mathrm{mm}

in the gap is

\mu_0 I \ne 0

: explain, using Stokes's theorem, why this does not contradict your result. ((d)) Show that

\nabla\times\mathbf{E} = \mathbf{0}

in the gap.

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