A long straight carbon rod of radius lies along the axis and carries a steady current…

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

A long straight carbon rod of radius

R = 1.5\ \mathrm{mm}

lies along the

z

axis and carries a steady current

I = 2.0\ \mathrm{A}

along

+\hat{\mathbf{z}}

, spread uniformly over its cross-section, so that

\mathbf{J} = J\hat{\mathbf{z}}

with

J

constant inside. Probes show that the electric field inside the rod is uniform and axial,

\mathbf{E} = E_0\hat{\mathbf{z}}

with

E_0 = 0.80\ \mathrm{V\,m^{-1}}

. Everything is steady, so the fields are static. Treat the rod as infinitely long. Write

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

. ((a)) Find

J

, and use Ampère's law in integral form on a circle of radius

s \le R

(normal

+\hat{\mathbf{z}}

, right-hand sense) to show that inside the rod

\mathbf{B} = \tfrac12\mu_0 J\left(-y\,\hat{\mathbf{x}} + x\,\hat{\mathbf{y}}\right)

. Give

|\mathbf{B}|

at the surface of the rod. ((b)) Compute

\nabla\times\mathbf{B}

from these components and check it against the local form of Ampère's law. ((c)) Compute

\nabla\cdot\mathbf{E}

inside the rod. What volume charge density does Gauss's law at a point give there, and what does the integral form of Gauss's law then say about the flux of

\mathbf{E}

out of any small closed box drawn wholly inside the rod? ((d)) Show that

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

inside the rod, write down a potential

V

with

\mathbf{E} = -\nabla V

, and find the potential difference between two cross-sections

0.25\ \mathrm{m}

apart.

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