A Line Charge: Electrostatic Potential and Field

One of the fundamental charge distributions for which an analytical expression of the electric field can be found is that of a line charge of finite length. Nevertheless, the result we will encounter is hard to follow. Two limiting cases will help us understand the basic features of the result.

Problem Statement

Calculate the electrostatic potential $$\phi\left(\mathbf{r}\right)$$ and the electric field $$\mathbf{E}\left(\mathbf{r}\right)$$ of a line charge with length $$l$$. Formally, the charge distribution is given by $$\rho\left(\mathbf{r}\right)=\eta\delta\left(x\right)\delta\left(y\right)\Theta\left(\left|z\right|-l/2\right)$$. Discuss your result in the limits of infinite line charge, $$l\rightarrow\infty$$ and for large distances $$\left|\mathbf{r}\right|\gg l$$. For simplicity, restrict yourself in both limiting cases to the x-y-plane.

Background: Forces on Molecules by an Everyday Line Charge

If you rub a plastic ruler with one of your shirts, there will be some net charge on both the ruler and your t-shirt. Now you can approach the next tap, make some nice, not too strong, water stream and hold your ruler close to it. You will notice that the water stream changes his way slightly in the direction of the ruler. How can we understand the movement of the water stream?

The reason is that water, $$H_{2}O$$, has a permanent dipole moment $$\mathbf{p}$$ which is interacting with the local electric field. The interaction potential is given by $$V_{\mathrm{dipole}}\left(\mathbf{r}\right)=-\mathbf{p}\cdot\mathbf{E}\left(\mathbf{r}\right)$$ and the force acting on the dipole is the negative gradient of the potential, $$\mathbf{F}\left(\mathbf{r}\right)=-\nabla V\left(\mathbf{r}\right)$$. So the force depends on the local derivative of the electric field. Now we can see why the water stream gets diffracted. Furthermore it matters what kind of electric field is present to influence it. We might regard the ruler as a finite line charge. In the solution we will find that the field of a long or short one are in fact different and so is their force on the water stream.

Can you explain what happens to the stream inside a parallel-plate capacitor with assumed constant electric field?