A solid resin rod is a cylinder of radius and length

PHYSICS · Multi-part · Fermi University Physics · Mathematical Methods for Physics · Vector calculus and field geometry · Vector calculus and field geometry

Mathematical Methods for Physics, Ch Line, surface and volume integrals

Problem

A solid resin rod is a cylinder of radius

a = 0.050\,\mathrm{m}

and length

L = 0.20\,\mathrm{m}

. Manufacturing leaves it with a volume charge density that depends only on the distance

\rho

from the rod's axis,

\rho_q(\rho) = \rho_0\left(1 - \frac{\rho}{a}\right), \qquad \rho_0 = 4.0\times10^{-6}\,\mathrm{C\,m^{-3}},

where

\rho

without a subscript is the cylindrical coordinate, and

\rho_q

and

\rho_0

are charge densities. ((a)) Find the total charge

Q

on the rod. ((b)) What fraction of

Q

lies within a distance

a/2

of the axis?

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