- #1
Kelly Lin
- 29
- 0
Homework Statement
Homework Equations
I want to know whether my process is correct?
THANKS!
The Attempt at a Solution
1. By using Gauss's law:
[tex]E\cdot 2\pi rz = \frac{\lambda z}{\epsilon_{0}}\Rightarrow E=\frac{\lambda}{2\pi\epsilon_{0} r}[/tex]
In my coordinate system, [tex]E=\frac{\lambda}{2\pi\epsilon_{0}[y^{2}+(z-d)^{2}]^{-1/2}}[/tex]
Then,
[tex]V=-\int\mathbf{E}\cdot d\mathbf{l}=-\int_{0}^{z}\frac{\lambda}{2\pi\epsilon_{0}[y^{2}+(z-d)^{2}]^{-1/2}}dz=\frac{\lambda}{2\pi\epsilon_{0}y}\ln{\frac{\left | \sqrt{d^{2}+y^{2}}-d \right |}{\left | \sqrt{(z-d)^{2}+y^{2}}+(z-d) \right |}}[/tex]
Since [tex]C=\frac{Q}{V}[/tex], then
[tex]
C(\text{per length})=2\pi\epsilon_{0}y\ln{\frac{\left | \sqrt{(z-d)^{2}+y^{2}}+(z-d) \right |}{\left | \sqrt{d^{2}+y^{2}}-d \right |}}
[/tex]
2.
Use [tex]\sigma=-\epsilon\frac{\partial V}{\partial z}|_{z=0}[/tex] to get the answer.