Question

Find the equation of the equipotentials for an infinite cylinder of radius r₀, carrying charge of linear density λ.

Answer 1

To find the potential at distance r from the line consider the electric field. We note that from symmetry the field lines must be radially outward. Draw a cylindrical Gaussian surface of radius r and length l. Then

${\displaystyle \oint E.dS=\frac{1}{{\epsilon }_0}\lambda 1}$

Or ${\displaystyle E_r.2\pi rl=\frac{1}{{\epsilon }_0}\lambda 1}$

⇒ ${\displaystyle E_r=\frac{\lambda }{2\pi {\epsilon }_{0}r}}$

Hence, if r₀ is the radius,

${\displaystyle V\left(r\right)-V\left(r_0\right)=-{\int }_{r_0}^{r}E.dl=\frac{\lambda }{2\pi {\epsilon }_0}\ln \frac{r_0}{r}}$

For a given V,

ln ${\displaystyle \frac{r}{r_0}=-\frac{2\pi {\epsilon }_0}{\lambda }[V\left(r\right)-V\left(r_0\right)]}$

⇒ r = r₀e –2πε₀Vr₀/λ_e + 2πε₀V(r)/λ

The equipotential surfaces are cylinders of radius r = r₀e –2πε₀[V(r) – V(r₀)]/λ