Question

Obtain an expression for the potential at a point near an infinitely long charged wire.

Answer 1

Lets use Gauss's law for electric field.

Fro wire spartial symmetry, choose surface as cylider with wire as axis of symmetry, with radius r and height h.

Charge of wire per meter $\tau$, so charge, enclosed in cylinder is $q = \tau \cdot h$

Electric field directed against wire and perpendicular to it. Therefore, flux of electric field via base and top of cylinder equals to 0. Therefore:

$$E(r)\cdot 2\pi r h = q/\varepsilon_0 = \tau h /\varepsilon_0$$

$$E(r) = \frac{\tau}{ \varepsilon_0 2\pi r}$$

To obtain an expression for the potential, integrate $E(r)$ with opposite sign:

$$\phi(r)-\phi(a) = - \int_{a}^{r}\frac{\tau}{ \varepsilon_0 2\pi r}dr = -\frac{\tau}{ 2\pi \varepsilon_0}\ln{\frac{r}{a}}$$

If we choose $\phi(a) = 0$ , finally

$$\phi(r) = -\frac{\tau}{ 2\pi \varepsilon_0}\ln{\frac{r}{a}}$$

where r is the distance to wire.