A long straight beam of positive ions travels in vacuum along the axis

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 beam of positive ions travels in vacuum along the

z

axis. Its cross-section is a circle of radius

R = 3.0\ \mathrm{mm}

centred on the axis, the charge density inside the beam is uniform,

\rho = 8.0\times10^{-6}\ \mathrm{C\,m^{-3}}

, and every ion moves along

+\hat{\mathbf{z}}

with the same speed

v = 2.0\times10^{5}\ \mathrm{m\,s^{-1}}

, so the current density inside is

\mathbf{J} = \rho v\,\hat{\mathbf{z}}

. The beam is steady:

\rho

and

\mathbf{J}

do not change with time, so the fields are static. Treat the beam as infinitely long and ignore its ends. Write

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

for the distance from the axis. ((a)) Using Gauss's law in integral form on a coaxial cylinder of radius

s \le R

and length

L

, show that inside the beam

\mathbf{E} = \dfrac{\rho}{2\varepsilon_0}\left(x\,\hat{\mathbf{x}} + y\,\hat{\mathbf{y}}\right)

, and give the magnitude of

\mathbf{E}

at the edge of the beam. ((b)) Compute

\nabla\cdot\mathbf{E}

from these components and check it against Gauss's law at a point. ((c)) Using Ampère's law in integral form on a circle of radius

s \le R

(normal

+\hat{\mathbf{z}}

, right-hand sense), show that inside the beam

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

, give its magnitude at the edge of the beam, and compute

\nabla\times\mathbf{B}

, checking it against the local form of Ampère's law. ((d)) Show that

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

inside the beam, and find the potential difference

V(\text{axis}) - V(\text{edge})

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