Question 4.25: 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 \mathscr{L} = 1 m. The potential on the surface is V = 1 V. You may assume that the charge is distributed uniformly in each section. Assume that the number of the sections is N = 5 and the step size is Δ \ell = 0.20 m.
The matrix equation relating the potentials to the charges is (4.101), where the off-diagonal and the diagonal elements are given by ( 4.104) and ( 4.105) respectively. The solution for the unknown charge distribution is
[P][Q] = [V] (4.101)
P_{i,j} = \frac{2\pi a}{4\pi \varepsilon _{0} } \frac{\Delta \ell_{j}}{\left|x_{i} – x_{j}\right| } = \frac{a{\Delta \ell_{j}}}{2\varepsilon _{0} \left|x_{i} – x_{j}\right|} (4.104)
P_{j,j}\cong \frac{a}{\varepsilon _{0}} ln\left[\frac{\Delta \ell_{j}}{a} \right] (4.105)
\rho _{\ell , 1}= \rho _{\ell , 5} = 0.2556 \varepsilon _{0}, \rho _{\ell , 2} = \rho _{\ell , 4} = 0.2222\varepsilon _{0}, \rho _{\ell , 3} = 0.2170 \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.