Electric field caused by a collection of point charges

The electrostatic potential $\varphi$ caused by a collection of point charges is,

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

Here $q_i$ are the charges and $\vec{r}_i=x_i\hat{x}+y_i\hat{y}+z_i\hat{z}$ are the positions of the point charges and $|\vec{r}-\vec{r}_i|=\sqrt{(x-x_i)^2+(y-y_i)^2+(z-z_i)^2}$. The relationship between the electric field and the electrostatic potential is, $\vec{E} = -\nabla \varphi =-\frac{\partial \varphi}{\partial x}\hat{x} -\frac{\partial \varphi}{\partial y}\hat{y} -\frac{\partial \varphi}{\partial z}\hat{z}$.

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

The form below lets you specify the charges and positions of up to 10 point charges and then calculates the electrostatic potential and the electric field at position $\vec{r}$. The zero of the electrostatic potential is far away from all of the charges.

$\vec{r} = $ $\hat{x} + $ $\hat{y} + $ $\hat{z}$ [m]

$\varphi(\vec{r}) = $ [V]

$\vec{E}(\vec{r}) = $ $\hat{x} + $ $\hat{y} + $ $\hat{z}$ [V/m]

$q_{1}=$

[C]  

$\vec{r}_{1}=$

 $\hat{x} + $ $\hat{y} + $ $\hat{z}$ [m]

$q_{2}=$

[C]  

$\vec{r}_{2}=$

 $\hat{x} + $ $\hat{y} + $ $\hat{z}$ [m]

$q_{3}=$

[C]  

$\vec{r}_{3}=$

 $\hat{x} + $ $\hat{y} + $ $\hat{z}$ [m]

$q_{4}=$

[C]  

$\vec{r}_{4}=$

 $\hat{x} + $ $\hat{y} + $ $\hat{z}$ [m]

$q_{5}=$

[C]  

$\vec{r}_{5}=$

 $\hat{x} + $ $\hat{y} + $ $\hat{z}$ [m]

$q_{6}=$

[C]  

$\vec{r}_{6}=$

 $\hat{x} + $ $\hat{y} + $ $\hat{z}$ [m]

$q_{7}=$

[C]  

$\vec{r}_{7}=$

 $\hat{x} + $ $\hat{y} + $ $\hat{z}$ [m]

$q_{8}=$

[C]  

$\vec{r}_{8}=$

 $\hat{x} + $ $\hat{y} + $ $\hat{z}$ [m]

$q_{9}=$

[C]  

$\vec{r}_{9}=$

 $\hat{x} + $ $\hat{y} + $ $\hat{z}$ [m]

$q_{10}=$

[C]  

$\vec{r}_{10}=$

 $\hat{x} + $ $\hat{y} + $ $\hat{z}$ [m]
 
Plot the plane at