Equipotentials and Electric Field Lines: Collections of Point Charges and the Method of Images

Developed by J. D. McDonnell

In this set of exercises, the student will numerically calculate and produce plots of the electric field and equipotential surfaces for collections of point charges. The student will also learn how to use these plots of equipotential surfaces to help solve problems with the method of images.
Subject Area Electricity & Magnetism First Year and Beyond the First Year IPython/Jupyter Notebook Students who perform these exercises will - be able to describe in words or pseudocode a procedure to calculate the electric field and electric potential at a point in space due to an arbitrary collection of point charges (**Exercises 1 and 2**); - be able to implement a numerical algorithm that calculates the electric field and electric potential at many points in space due to given collections of point charges (**Exercises 1, 2, and 3**); - be able to describe, interpret, and validate their numerically computed electric field lines and equipotentials (**Exercises 1, 2, and 3**); - develop their understanding of the superposition principle for electric fields and electric potential (**Exercise 3**); - and develop their understanding of the method of images (**Exercise 4**). 120 min
### Exercise 1: Calculate the electric field of a single point charge at many grid points Consider a $1.0$nC point charge at the origin. Construct a 100x100 grid in the $xy$-plane. Let both $x$ and $y$ be in the range from $-5.0$ to $5.0$. 1. Describe (in words or pseudocode) and then implement a procedure to calculate the components of the electric field at each point on the grid according to $$\vec{E}(\vec{r}) = \frac{1}{4\pi\epsilon_0} \frac{q\left( \vec{r} - \vec{r}\ ' \right)}{\left\vert\vec{r} - \vec{r}\ ' \right\vert^3},$$ where $\vec{r}$ is a point on the 100x100 grid and $\vec{r}\ '$ is the location of the point charge (in this case, the origin). 2. Choose several points at which to validate your numerical calculation of the electric field with an analytical calculation done by hand. 2. Construct a vector plot of the resulting electric field. What are the major features you see? Do they align with your expectations from what you already know about electric field lines? 3. Repeat this process for a negative point charge - compare and contrast your plots. ### Exercise 2: Calculate the electric potential of a single point charge at many grid points Consider the same $1.0$ nC point charge at the origin, and the same 100x100 grid in the $xy$-plane. 1. At each point on the grid, describe (in words or pseudocode) and then implement a procedure to calculate the electric potential according to $$V(\vec{r}) = \frac{1}{4\pi\epsilon_0} \frac{q}{\left\vert\vec{r} - \vec{r} \ ' \right\vert}.$$ 2. Choose several points at which to validate your numerical calculation of the electric potential with an analytical calculation done by hand. 3. Construct a contour plot of the resulting electric potential. Since the contour lines represent lines of equal electric potential, they are known as **equipotential lines**. What are the major features you see? Do they align with your expectations from what you already know about electric potential? 4. Repeat this process for a negative point charge again - compare and contrast your plots. 5. Now, overlay your previous electric field vector plot with your equipotential lines on the same plot. What relationship do you see between the electric field lines and the equipotential lines? ### Exercise 3: Plot electric field and equipotential lines for collections of multiple charges Now, adapt the above methods to produce plots that contain both the electric field and equipotential lines for each of the following collections of point charges. **Note**: In your plots, it will turn out to be helpful to highlight the $V=0$ equipotentials, for example by plotting that line extra thickly. 1. A positive charge of $1$nC at $(x,y) = (1.0, 0.0)$, and an equal but opposite charge at $(x,y) = (-1.0, 0.0)$. 2. A positive charge at $1nC$ $(x,y) = (1.0, 0.0)$, and another positive charge of equal magnitude at $(x,y) = (-1.0, 0.0)$. 3. A positive charge of $1$nC at $(x,y) = (4.0,0.0)$, and a negative charge of half the magnitude at $(x,y) = (1.0,0.0)$. 4. A set of four point charges: $+1$nC at $(+1.0, +1.0)$, $-1$nC at $(-1.0, +1.0)$, $+1$nC at $(-1.0, -1.0)$, and $-1$nC at $(+1.0, -1.0)$. Describe the shapes of the electric field lines and the equipotential lines for each case. Compare and contrast the features of these plots with the single-particle cases above. ### Exercise 4: The method of images The motivation for the method of images is the **uniqueness theorem** for electrostatics - it does not matter *how* a solution to an electrostatics problem is found, as long as the solution satisfies the problem's boundary conditions. 1. Consider an infinite plane conductor, situated at $x=0$, that is held at a fixed potential $V=0$. A charge $1.0$nC is placed at $(1.0, 0.0, 0.0)$. What are the electric field and the electric potential everywhere in the region $x>0$? - Review the plots you made in Exercise 3. *Do any of the plots satisfy this current problem's specifications* (namely, a charge at $(1.0, 0.0, 0.0)$, and a $V=0$ equipotential plane at $x=0$)? - Now, use the method of images to plot the electric field and electric potential for *this* problem, in the region $x>0$. - Finally, make a plot of the *induced surface charge density* on the plate. You may restrict your plot to the $z=0$ cross section, and plot $\sigma(y, z=0)$ as a function of $y$. 2. Now, consider a spherical conductor, of radius 2m, centered at the origin. The spherical conductor is held at a fixed potential $V=0$. A charge $+1$nC is placed at $(4.0, 0.0, 0.0)$. What are the electric field and the electric potential everywhere outside of the spherical conductor? - Review the plots you made in Exercise 3. *Do any of the plots satisfy this current problem's specifications*? - Now, use the method of images to plot the electric field and electric potential for *this* problem, outside of the spherical conductor. 3. Now, consider a plane conductor that is "bent" into a 90$^\circ$ angle (so that one of the two flat pieces line up with $x=0$, and the other with $y=0$). A charge $1$nC is placed at $(1.0, 1.0, 0.0)$. What are the electric field and the electric potential in the region $x>0, y>0$? - Review the plots you made in Exercise 3. *Do any of the plots satisfy this current problem's specifications*? - Now, use the method of images to plot the electric field and electric potential for *this* problem, in the region $x>0, y>0$.