[tex]\nablaV[/tex]Answer:
[tex]V = \frac{1}{2 \epsilon} { \frac{a\rho l}{\sqrt{a^{2} +z^{2} }} }[/tex]
(b) [tex]E = \frac{a\rho l}{2 \epsilon} {\frac{z}{({a^{2} +z^{2}})^\frac{3}{2}} }\hat{Z}[/tex]
Explanation:
[tex]V = \frac{1}{4\pi \epsilon} \int\limits^a_b {\rho l/R'} \, dl' \\\\V = \frac{1}{4\pi \epsilon} \int\limits^{2\pi }_0 { \frac{\rho l}{\sqrt{a^{2} +r^{2}-2arcos(\phi'-\phi)+z^{2} }} } \, ad\phi'[/tex]
(0, 0, z) implies (r, Φ, z) in cylindrical coordinates
[tex]V = \frac{1}{4\pi \epsilon} \int\limits^{2\pi} _0 { \frac{\rho l}{\sqrt{a^{2} +z^{2} }} } \, ad\phi'\\\\V = \frac{1}{4\pi \epsilon} { \frac{\rho l}{\sqrt{a^{2} +z^{2} }} } \ a (2\pi -0)\\\\V = \frac{1}{2 \epsilon} { \frac{a\rho l}{\sqrt{a^{2} +z^{2} }} }[/tex]
(b) Corresponding electric field E = -[tex]\nabla V[/tex]
[tex]-\nabla V = -\frac{a\rho l}{2 \epsilon} { \frac{\delta }{\delta z} \frac{1}{\sqrt{a^{2} +z^{2} }} } \hat{Z}\\\\-\nabla V = \frac{a\rho l}{2 \epsilon} {\frac{z}{({a^{2} +z^{2}})^\frac{3}{2}} }\hat{Z}[/tex]
