Electric field produced by a curved line of charges

Consider a wire of length $L$ that has a uniform charge density $\lambda$. This wire can be bent into different shapes. The electrostatic potential $V$ produced by this wire can be approximated by breaking the wire into short segments and adding all of the contributions of these segments together. The segments have a length $\Delta s$ and a charge $\Delta q=\lambda\Delta s$. The contribution to the electrostatic potential at position $\vec{r}$ is,

$\Large V(\vec{r})=\sum \limits_{i=1}^{N} \frac{\Delta q}{4\pi \epsilon_0 |\vec{r}-\vec{r}_i|}=\sum \limits_{i=1}^{N} \frac{\Delta q}{4\pi \epsilon_0 \sqrt{(x-x_i)^2+(y-y_i)^2+(z-z_i)^2}}$ [V].

Here $\vec{r}_i=x_i\hat{x}+y_i\hat{y}+z_i\hat{z}$ are the positions of the point charges along the wire. The relationship between the electric field and the electrostatic potential is, $\vec{E} = -\nabla V=-\frac{\partial V}{\partial x}\hat{x} -\frac{\partial V}{\partial y}\hat{y} -\frac{\partial V}{\partial z}\hat{z}$.

$\Large \vec{E}(\vec{r})=\sum \limits_{i=1}^{N} \frac{q_i(\vec{r}-\vec{r}_i)}{4\pi \epsilon_0 |\vec{r}-\vec{r}_i|^3}$ [V/m]

In terms of the $x$, $y$, and $z$ coordinates, the electric field is,

$\Large \vec{E}(\vec{r})=\sum \limits_{i=1}^{N} \left[ \frac{q_i(x-x_i)}{4\pi \epsilon_0 \left((x-x_i)^2+(y-y_i)^2+(z-z_i)^2\right)^{3/2}}\hat{x}+\frac{q_i(y-y_i)}{4\pi \epsilon_0 \left((x-x_i)^2+(y-y_i)^2+(z-z_i)^2\right)^{3/2}}\hat{y}+\frac{q_i(z-z_i)}{4\pi \epsilon_0 \left((x-x_i)^2+(y-y_i)^2+(z-z_i)^2\right)^{3/2}}\hat{z}\right]$ [V/m].

The position and shape of the wire can be specified by parametric equations in terms of a parameter $s$ that measures the distance along the wire. For instance, a straight wire from $\vec{r}_1$ to $\vec{r}_2$ is described by,

$\large \vec{r}_{wire}=(r_{1x}+s(r_{2x}-r_{1x}))\hat{x} + (r_{1y}+s(r_{2y}-r_{1y}))\hat{y} + (r_{1z}+s(r_{2z}-r_{1z}))\hat{z}$  where $s=[0,1]$.

For a wire loop of radius $R$ in the $x$-$y$ plane at $z=0$,

$\large \vec{r}_{wire}=R\cos(2\pi s)\hat{x} + R\sin(2\pi s)\hat{y} + 0\hat{z}$  where $s=[0,1]$.

For a 10 turn spiral coil,

$\large \vec{r}_{wire}=R\cos(2\pi s)\hat{x} + R\sin(2\pi s)\hat{y} + \frac{s}{n} \hat{z}$  where $s=[0,10]$,

where $n$ is the number of turns per meter of the coil. The form below can be used to specify the shape of the wire and to calculate the electric field at position $\vec{r}$ .

The longer the wire, the more segments are needed for an accurate answer. If the wire is a coil, about 300 segments per turn is appropriate.

The mathematical functions that can be used are list below. Multiplication must be specified with a '*' symbol, 3*cos(x) not 3cos(x). Powers are specified with the 'pow' function: x² is pow(x,2) not x^2.