Field Lines and Potentials – Reality and Imagination Reloaded

I am creating Physics Art very much like I create Found Poetry. I need tough constraints, silly rules, and tools not designed for art. In a world of perfect renderings and deep fakes, I restrict myself to programming parametric curves. Just lines.

There are millions of beautiful parametric functions you could choose from. So, I constrain myself to functions with relevance to physics.  Weird, not well-behaved functions were appealing; so I haven’t followed this rule in the beginning. Or so I thought. I have presented the complex function 1/z two times – without a reference to physics. It was a textbook example of a simple holomorphic (complex-differentiable) function. But I missed a chance to refer to electrostatics. It’s the introduction to the theory of electrostatics where you might get introduced to holomorphic functions as a budding physicist!

Real and imaginary part of 1/z are symmetric, in a sense. You can consider them functions of two coordinates x and y in a two-dimensional world. Real and imaginary part are surfaces spanning the x-y-plane.  There are two sets of lines: The curves of either the real or the imaginary part being constant. At every point in the plane, these lines are perpendicular to each other.

Either real or imaginary part represents field lines (integral curves) of some electrostatic field, and the other part encapsulates the electrostatic potential. Field lines are always perpendicular to equipotential surfaces. Surfaces in a two-dimensional world are lines, hence the symmetry in 2D.

I am depicting a bunch of these lines again, now using Ukraine’s exact national colors, as I did when drawing the temperature waves. (I had ‘enhanced’ the colors, when I applied them to Reality and Imagination the first time). Where the lines of constant real and constant imaginary part meet, I insert red lines.  The ‘towers’ of real and imaginary part meet in the vertical axis, as well as in a hyperbola in the plane defined by x being equal to y.

I am using only a small number of lines this time, and formulae are created with Unicode characters. Expect me to frantically troubleshoot my characters as soon as I will have hit publish and look at the gibberish on a smartphone.

Some math is in the original Reality and Imagination post (from the ‘white background phase’), more math and physics is inserted below.

Two functions U(x,y) and V(x,y).
Lines U=const and V=const

Complex z = x + iy
Complex function f(z) = 1/z = U + iV = (x – iy) / (x² + y²)

Complex differentiable: df/dz must make sense.
Conjugate z̅ = x – iy
df/dz̅  must not exist independently of df/dz
∂U/∂x + i ∂V/∂x = ∂f/∂x = df/dz · ∂z/∂x = df/dz · 1
-i ∂U/∂y + ∂V/∂y = -i∂f/∂y = -idf/dz · ∂z/∂y = -df/dz · i² = df/dz

Thus U and V are connected by Cauchy-Riemann equations.
∂U/∂x = ∂V/∂y
∂U/∂y = –∂V/∂x

What does that mean for two functions U and V?
Their gradients (∂U/∂x, ∂U/∂y) and (∂V/∂x, ∂V/∂y) are intertwined:
Perpendicular:
∂U/∂x · ∂V/∂x + ∂U/∂y · ∂V/∂y = –∂U/∂x · ∂U/∂y + ∂U/∂y · ∂U/∂x = 0
Same magnitude:
(∂U/∂x)² + (∂U/∂y)² = (∂V/∂y)² + (-∂V/∂x)²

The electric field is (some factor times) the gradient of the potential.
Setting the factor to 1 here. Curly characters for vectors / vector fields.
𝒰 = ∂U/∂𝓇
𝒱 = ∂V/∂𝓇

Second derivative – Laplacian is zero for U:
∆U = ∂²U/∂x² + ∂²U/∂x²
= ∂/∂x ∂U/∂x + ∂/∂y ∂U/∂y = ∂/∂x ∂V/∂y – ∂/∂y ∂V/∂x = ∂V/∂x∂y∂V/∂y∂x = 0
∆U = ∂/∂𝓇 ∂U/∂𝓇 = ∂𝒰/∂𝓇 = 0

Same for V (Symmetry).
Divergence of the field is zero in a region without charges (says Maxwell).
Thus both U and V could – potentially – be electric potentials.

Field lines are integral curves.
Follow a line by incrementing parameter t.
Line: (x(t), y(t))
Vector field: A vector 𝒰 or 𝒱 at each point (x,y)
Field lines are the lines ‘connecting’ the vectors.
Tangent vector (dx/dt, dy/dt) is equal to the vector 𝒰 or 𝒱.
These are coupled differential equations.
Solve them to recover the field lines.
One for each integration constant.

Lines are given by U(x(t),y(t)) = c or V(x(t),y(t)) = c.
Apply d/dt
dx/dt · ∂U/∂x + dy/dt · ∂U/∂y = 0

Fulfilled if the tangent vector is proportional to the gradient vector of corresponding V:
dx/dt = ∂V/∂x = -λ ∂U/∂y
dy/dt = ∂V/∂y = λ ∂U/∂x

As U and V are related by Cauchy-Riemann, thus factor λ has to be 1.

What kind of charge distribution would cause the field / potential encapsulated in 1/z?
It’s the field of a very small dipole or the field of a dipole far away from its two charges!
Add a dimension to this a 2D world to make it 3D:
The 2D dipole becomes a dipole line made up of dipoles stacked upon each other,
like two charged wires close to each other.

How can you see it is a dipole?
The charge(s) are located where the functions have their singularities;
where the Cauchy-Riemann equations become of the kind ∞ = ∞.
The electric field lines are closed. How can that happen if this is an electrostatic field (no electromagnetic waves)?
Field lines start at the positive charge and end in the negative charge.
Make the dipole smaller and smaller, until it shrinks to a single point: That’s a loop!
Look at the field lines of a dipole with measurable size, and move the charges closer and closer to each other!

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For other examples of fields ‘originating from’ holomorphic functions see Complex Functions and Electrostatics by Prof. Kirk T. McDonald who has one of those timeless, classic, encyclopedic science websites that put a lot of ‘modern’ and sleek but empty and fluffed up sites to shame (Looking at you, sites who delete all the true content and changing URLs all the time!).

He retired in 2016 and a colleague said this:

To me, Kirk is a scholar of the old school — the kind who viewed it as their duty to tend with great love and rigor to a well-defined intellectual field. Such people make no apologies for the importance of getting things right even when no sexy headline is at issue and without their kind science would never have got off the ground.

_____

I suppose I will be creating images with yellow and blue loops for a while, and with dark backgrounds. I am supporting Cards for Ukraine  – and I can absolutely vouch for them! This is a social project that supports Ukrainian refugees in Austria, filling the alarmingly huge gaps not catered for by broken official ‘systems’. Follow Tanja Maier and Mario Zechner on Twitter for more details – there is dark humor amid all the depressing and kafkaesque stuff.

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