Question 3.19: Find the charge distribution on the cylindrical conductor wh...
Find the charge distribution on the cylindrical conductor whose radius is a=0.01 m and whose length is L=1 m. The potential on the surface is V=1 V. You may assume that the charge is uniformly distributed in each section. Assume that the number of the sections is N=5 and the step-size is ∆l = 0.20 m.
The matrix equation relating the potentials to the charges is (3.82), where the off-diagonal and the diagonal elements are given by (3.85) and (3.86) respectively. The solution for the unknown charge distribution is
[P][Q] = [V] (3.82)
P _{ i , j }=\frac{2 \pi a }{4 \pi \varepsilon_{0}} \frac{\Delta l _{ j }}{\left| x _{ i }- x _{ j }\right|}=\frac{ a \Delta l _{ j }}{2 \varepsilon_{0}\left| x _{ i }- x _{ j }\right|} (3.85)
P _{ j , j } \cong \frac{a}{\varepsilon_{0}} \ln \left[\frac{\Delta l _{ j }}{ a }\right] (3.86)
\rho_{ L , 1}=\rho_{ L , 5}=.2556 \varepsilon_{0}, \rho_{ L , 2}=\rho_{ L , 4}=.2222 \varepsilon_{0}, \rho_{ L , 3}=.2170 \varepsilon_{0}
Note that the charge density in the center of the line is smaller than at either end. We should expect this nonuniform distribution since there is a loss of symmetry at either end.