3D Charges and Configurations with Sharp Edges

September 27, 2012
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Michael Trott
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In my last blog post, we looked at various examples of electrostatic potentials and magnetostatic fields. We ended with a rectangular current loop. Electrostatic and magnetostatic potentials for squares, cubes, and cuboids typically contain only elementary functions, but the expressions themselves are often quite large compared with simple systems with radial symmetry. In the following, we will discuss some 3D charge configurations that have sharp edges.

Let’s start with a charged 2D rectangle in 3D space. Again, the potential is an elementary function that contains a few logarithms.

WolframAlpha["electric potential charged rectangle"]

Electric potential charged rectangle

We name the computable form for further computational use, and we define the scaled electrical potential for a rectangle.

chargedRectangleData =   WolframAlpha[   "electric potential charged rectangle", {{"Result", 1}, "Input"}]

Equation for the electric potential charged rectangle

Defining the scaled electrical potential for a rectangle

Near the charge itself, the potential decreases linearly with the distance from the potential.

Measuring the potential near the charge itself

The potential near the charge itself

At a corner, we have a slightly more complicated distance dependence.

Measuring the potential near a corner

The potential near a corner

The following graphic of the constant electric-field-magnitude surface reflects the shape of the charged rectangle.

Building a graphic of the constant electric-field-magnitude surface reflecting the shape of the charged sphere

Graphic of the constant electric-field-magnitude surface reflecting the shape of the charged sphere

Having at hand the closed form of the potential of a single rectangle, we can use the superposition principle to calculate the electric field distribution of an arrangement of multiple rectangles. In the absence of a jack, we will have a look at a sphere-in-a-box. The box we model as five squares; we consider an already open box and omit the top square (the lid). We will use the same charge for each face of the box.

Using the superposition principle to calculate the electric field distribution of an arrangement of multiple rectangles

Here are the equipotential surfaces of the resulting open box (we show part of the box cut open).

Equipotential surfaces of the resulting open box

Building a 3D plot of the surfaces

3D plot of the resulting open box

We now add a sphere of total charge Qs at the vertical symmetry line of the box and calculate the resulting potential and electric field strength.

Adding a sphere of total charge Qs at the vertical symmetry line of the box and calculating the resulting potential and electric field strength

To visualize the resulting field distribution, we show field lines originating from the faces of the box and ending on the sphere.

Showing field lines originating from the faces of the box and ending on the sphere

We implement the visualization as a Manipulate. This allows us to change the charge and size of the box and to move it in and out. If the sphere’s charge compensates or exceeds the box’s charge, all field lines end at the sphere. Otherwise some field lines will escape to infinity.

Implementing the visualization as a Manipulate

The natural generalization of a charged square is a charged cube. Fortunately Wolfram|Alpha knows the electrostatic potential of a uniformly charged cube; it is already quite a formidable expression that is neither easy to remember nor easy to derive.

ImageResize[Rasterize[   WolframAlpha["electric potential of a charged cube"]], {Automatic,    600}]

Electric potential of a charged cube

A charged cube is a nice nontrivial model of a charge distribution that isn’t trivial, yet can still be used for exact calculations. We will do some of these in the remaining part of this post.

We specialize to a unit cube (edge length 1) extending from -1/2 to 1/2 along the three coordinate axes. In the chosen units we have ΔV = -4π.

Specializing to a unit cube (edge length 1) extending from -1/2 to 1/2 along the three coordinate axes

Unit cube of the electric potential of a charged cube

Just to convince ourselves that this long result is really correct, let’s quickly compare with the numerical result from explicit integration of Poisson’s equation with the Green’s function.

Poisson's equation with Green's function

Comparing with the numerical result from explicit integration of Poisson's equation with the Green's function

Here are the values of the potential in closed form and shown numerically at geometrically exceptional points. Because the expression for the potential is often undefined at the faces and edges itself (of the form 0/0), we use Limit to calculate the potential values.

Building a grid of the values of the potential in closed form and numerically at geometrically exceptional points

Grid of the values of the potential in closed form and numerically at geometrically exceptional points

The complete agreement between the symbolically and numerically derived values gives us confidence that the model is correct.

It is instructive to plot the potential and its derivatives along a line connecting the center of the cube and the center of a face. The potential itself is a smooth function, the field strength (first derivative of the potential) has a kink at the surface of the cube, and the second derivative of the potential (via Poisson’s equation related to the charge density) is discontinuous at the surface of the cube. (Note that the Laplacian of the potential is proportional to the charge density, and in the following plot we show just the second derivative with respect to z. As a result the yellow line is not piecewise constant.)

Plotting the potential and its derivatives along a line connecting the center of the cube and the center of a face

Plot of the potential and its derivatives along a line connecting the center of the cube and the center of a face

Let us explore the electric field of a charged cube in more detail.

Finding the electric field of a charged cube

It’s a pretty large expression, measuring nearly 700 KB in size.

Making a grid of leaf count and byte count results

Leaf count, 19804; byte count, 679672

Here is a plot of the field strength and direction in the equatorial plane.

Plotting the field strength and direction in the equatorial plane

Field strength and direction in equatorial plane

The maximum field strength occurs at the centers of the faces. This is not unexpected, as the potential of a charged ball exhibits its maximum also on the surface of the ball.

Plotting the potential of a charged ball

Potential of a charged ball

Comparing the field strengths at the face centers, edge centers, and corners shows that the face centers indeed are the points of the largest magnitude of the field strengths.

Finding the field strength at the edge center

Field strength at the edge center

Finding the field strength at the corner

Field strength at the corner

Finding the field strength at the face center

Field strength at the face center

N[Norm /@ {%%%, %%, %}]

{2.19443, 1.67903, 2.5969}

Plotting the constant field-magnitude surface in 3D in one octant gives the following picture. (The surface of the charged unit cube is shown in light yellow.)

Plotting the constant field-magnitude surface in 3D in one octant

Constant field-magnitude surface in 3D in one octant

While most equi-field strength surfaces look like parts of a single deformed sphere-like surface, near the edge of the cube some more interestingly shaped equi-field strength surfaces exist that are topologically not a sphere.

Using Manipulate to show approximately spherical equipotential and equi-field strength surfaces near the center of the cube and far away from the cube

The images above show approximately spherical equipotential and equi-field strength surfaces near the center of the cube and far away from the cube. From far, far away, a charged cube looks in first order as a point charge. To quantify the deviation from the charge of a point charge, we carry out a multipole expansion.

We could just expand the Green’s function for large distances and integrate term-wise over a cube. This shows that far away the potential has the form ((V(r))~(|r|-1) + c5|r|-5 + c7 c5|r|-7 + …. The absence of the lower order terms proportional to r-2, r-3, and r-4 explains why for large distances the equipotential surfaces appear so spherical.

Integrate term-wise over a cube

Simplifying the equation

Result of the multipole expansion

Alternatively, we can use the well-known multipole expansion in spherical coordinates.

Equation showing the multipole expansion in spherical coordinates

This yields the same result, here demonstrated for the l = 4 term.

Using an alternative method to find the same result

Continuing to solve the equation

Simplifying the equation

(7 (x^4 + y^4 - 3 y^2 z^2 + z^4 - 3 x^2 (y^2 + z^2)))/(480 (x^2 + y^2 + z^2)^(9/2))

Before leaving our explorations of an electric field of a charged cube (or the mathematically equivalent problem of a gravitational field of a massive cube), we should have a quick look at the equivalent of the Kepler problem: the motion of a test particle in the field of a charged cube, as pioneered by Liu, Baoyin, and Ma and recently studied in more detail by Chappel, Iqbal, and Abbott. We can easily do this using the numerical differential equation solving capabilities of Mathematica. To quickly solve the equations of motions, we define a compiled version of the electric field strength of our charged unit cube.

Defining a compiled version of the electric field strength of our charged unit cube

To easily see and explore the orbits, we use a Manipulate. Here we assume the cube to be penetrable. The orbits around a point mass are well-known Kepler’s ellipses. We leave it to the reader to investigate if there are closed orbits around a cube, and for which initial conditions such closed orbits exist.

Using Manipulate to see and explore the orbits

Most real-world charged cubes are not penetrable (and a charged plasma in the form of a cube is difficult to keep in this shape), so a more realistic model would model the cube as a solid, and a test charge would be reflected when it hits one of its faces. We can make the test charge bounce at the cube relatively straightforwardly by adding the "EventLocator" method of NDSolve. We initiate a bounce if the trajectory would cross a face, and we assume a totally elastic reflection process.

Using Manipulate to make the test charge bounce at the cube

A bouncing test charge revolving around a charged cube ends our second blog post about charge distributions with edges. Not that there is not more to be calculated. For instance, what is a closed form for the total electrostatic self-energy of a charged cube
Σ = ∫R3ρ(r)V(r)dr? Here is the numerical result.

Closed form for the total electrostatic self-energy of a charged cube

23.6538

Carrying out the integration over the potential over the unit cube is challenging, and the conjectured exact form is:

Integration over the potential over the unit cube
f
Integration over the potential over the unit cube

23.6538

Download this post as a Computable Document Format (CDF) file.

3 Comments

Is there any way that you can make this a bit more digestible for low-techies :)?

Posted by Frederik October 1, 2012 at 6:12 am Reply

Good Work!

One of your references seems to have a good idea for stacking the solutions for cubes together, in some square cases.

For density functions with the appropriate boundaries, integration can then be replaced by a summation.

However, the analysis of density functions with curvy boundaries will not improve much from the knowledge of analytic solutions for cuboids, because in the limit of a -> 0, the potential function approaches spherical symmetry, ie becomes a point charge, as some of your plots show. Also, copying the cuboid solution for some integral could actually present memory problems for the computer if a is taken to be small enough and the volume has very curvy boundaries.

Thanks again,

Brad

Posted by brad klee October 1, 2012 at 10:35 am Reply

What about to calculate the potential of a charged metal triange?

(I am joking. It’s another task).

Posted by Vladislav October 11, 2012 at 6:41 am Reply
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