- #1

Karol

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- 22

## Homework Statement

A capacitor made of a round solid bar and a thin round tube are coaxial, they are very long. the radii of the bar is r

_{a}and of the tube r

_{b}. they are both charged with λ, the charge per unit length, with opposite charges. what is the potential difference.

## Homework Equations

The field round a long wire: ##E=\frac{\lambda}{2\pi\varepsilon_0 r}##

The work to bring a charge to a point in a field:

$$W=\int_{r_a}^{r^b} E dr$$

## The Attempt at a Solution

The field round a solid round bar and a tube is equal to the field generated by a thin wire.

The potential due to the inner bar:

$$V_{ab}=\int_{r_a}^{r^b} E dr=\frac{\lambda}{2\pi\varepsilon_0}\int_{r_a}^{r^b}\frac{1}{r}dr=\frac{\lambda}{2\pi\varepsilon_0}\ln\left(\frac{r_b}{r_a}\right)$$

This is the answer in the book, but the field due to the pipe isn't taken in consideration, why not?

The potential inside the pipe, due to the pipe, is:

$$V=\frac{\lambda}{2\pi\varepsilon_0}\frac{1}{r_b}$$

The potential difference should be:

$$\Delta V=\frac{\lambda}{2\pi\varepsilon_0}\left(\ln\left(\frac{r_b}{r_a}\right)-\frac{1}{r_b}\right)$$