Fractional order Legendre functions and electric fields (why are lightning-rods pointy)?

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This is a bit of mathematical amusement which came out of a study of field concentration and wave scattering by sharp edges and points. It is well known that electric fields are strongly concentrated in the vicinity of sharp points. In fact, near perfectly conducting points, the electric field will approach infinity. What is the nature of this singularity?

To answer this question, let us assume that we can model the pointy structure as an infinite circular cone with the point at the origin (0,0,0) with angle alpha.

In this situation, we shall solve the Laplace Equation for the scalar electric potential V

subjected to the boundary conditions

At this point, we shall focus on azimuthally symmetric solutions (where V does not depend on phi, the azimuthal angle). This allows us to write Laplace's equation in spherical coordinates as

This partial differential equation (PDE) is separable by assuming

where R(r) and P() are functions of their respective variables only! This allows us to write the PDE as two ordinary differential equations

where l(l+1) is the so-called separation constant.

It is easy to show that the differential equation in r has the solutions

where Al and Bl are constants yet to be determined from the boundary conditions. We also need to know the values for l. These will come from the second differential equation (for P) and its boundary conditions. In order for the potential to be regular as r approaches the point of the cone, we will focus on solutions in terms of .

We can simplify the appearance of the differential equation for P by making the substitution

The second DE now looks like

This is the famous Legendre differential equation and it is an eigenvalue equation in , where l is the eigenvalue. If there are no boundary conditions, but we constrain the solution to regular (non-singular) solutions in , we get the familiar Legendre functions of integer order (the Legendre polynomials), viz.

The eigenvalues l in this case are 0, 1, 2, 3, 4, .....

In the case of the pointy circular cone, however, things are not quite so simple. The eigenvalues do not necessarily take integer values and the polynomial series is not necessarily finite. Perhaps the most straightforward way to see what these functions look like is to use some type of numerical method to generate P given the boundary condition

We constructed a simple solution method based on a 1-dimensional finite element method to generate the solutions to the eigenvalue problem. By assuming the cone angle alpha to be 180 degrees, we get the flat ground plane problem (in effect, the cone becomes a perfect electric conducting plane). We expect the solutions to be all the odd Legendre polynomials of integer order. In fact, using 100 cubic Hermite polynomials to approximate the Legendre function we get the eigenvalues

Order

Computed Eigenvalue

1

1.0000000

2

3.0000000

3

5.0000000

4

7.0000000

5

9.0000000

6

11.000000

7

13.000000

8

15.000000

9

17.000001

10

19.000002

We see that the expected eigenvalues are computed to 7 decimal places quite easily. The first five functions look like

By comparing with the analytic expressions with the computed solutions, we are satisfied the numerical method is working properly. We see that there are no singular fields in this case, because the eigenvalues l are all greater than 1.

Now, let's move on to something more interesting. Let us assume that we have a cone whose angle is 150 degrees (i.e. the cone point spans an angle of 60 degrees). The first 10 eigenvalues l are given by

Eigenvalue Order

Computed Eigenvalue

1

0.3461839

2

1.568297

3

2.777734

4

3.982932

5

5.186204

6

6.388442

7

7.590066

8

8.791295

9

9.992257

10

11.19303

If we consider the first eigenvalue, we see that the solution for the potential near the tip of the cone can be written as

where refers to the fractional order Legendre function which describes the angular behaviour of the potential near the cone tip. It looks like

By recognising that the electric field is or

Numerically speaking, we have

We can see that the electric field goes to infinity as we get closer to the cone tip (r=0) .

Now, we'll try a very pointy cone, where alpha is 350 degrees. The computed eigenvalue that gives rise to singular fields is l = 0.1581469. The fractional-order Legendre function looks like P_0.158 in the plot below.

It is interesting to see that for the sharper cone point, the slope of the Legendre function drops off much more rapidly. This makes sense physically, because the this cone is thinner at any given r than the previous one. The fields will vary much more rapidly. In fact, we have

The singularity is significantly stronger in this case. The exponent is -0.8418531 vesus -0.6538161 in the previous “less pointy” cone case... Hence, if we want the enhance the possibility of causing a discharge (as in the case of the lightning rod), we want a high electric field. This means, the pointier we make the lightningrod, the higher the field and the more likely we can cause charge to leak away, avoiding a possibly damaging lightning strike!

References:

1. S. Ramo, J. R. Whinnery and T. D. Van Duzer, Fields and Waves in Communication Electronics, Wiley, 1984.

2. J. Van Bladel, Singular Electromagnetic Fields and Sources, IEEE Press, 1995.