Capacitance 1: Tesla Coil Topload

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A Practical Situation: Spherical Topload Fields and Capacitance for Tesla Coil or Van De Graaf Generator

Using the concepts of charge, dipoles, potentials and images, it is possible to construct a model for the electric fields of a spherical topload above a metallic ground plane for a tesla coil or

Van de Graaf generator (as seen in the figure).
Geometry of simplified topload geometry. The various stages of simplification are shown: image of sphere and point charge/dipole approximation.
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Geometry of simplified topload geometry. The various stages of simplification are shown: image of sphere and point charge/dipole approximation.

This will allow us to estimate the capacitance of the topload; useful for practical designs.

As a brute-force boundary-value problem, this would require fairly sophisticated mathematical techniques. However, by using the idea of images as well as judicious application of point charges and dipoles, we can assemble an equivalent problem that is easy to solve and has much in common with modern computational methods in electromagnetics.

Let us start off by hypothesizing that the fields generated by the charges on the surface of the spherical topload will possess a point-charge as well as a dipole nature. The sphere, whose center is at h and whose radius is a, can be replaced with a charge Q and a dipole moment \textbf{p} = p_z\hat{\textbf{z}} both positioned at h. The field solution represents the sum of the field from Q and the dipole pz. This is a technique called a "multipole expansion" and could contain higher order "quadrupole" and "octopole" moments for higher accuracy.

We can also eliminate the ground plane by placing an image charge -Q and another dipole p_z\, at z=-h\,. At this point, we do not know the values of Q or pz, but by using the the method of images, we have eliminated the need to treat the ground plane explicitly, thereby simplifying the problem.

In order to find Q and pz, we specify the boundary condition V = V0 on the spherical surface centered at h with radius a (where the spherical topload is). The potential of the charge/dipole system is given by

V(x, y, z) = \frac{Q}{4\pi\epsilon_0}\left ( \frac{1}{\sqrt{x^2+y^2+(z-h)^2}} - \frac{1}{\sqrt{x^2+y^2+(z+h)^2}}\right ) +
\frac{p_z}{4\pi\epsilon_0}\left ( \frac{z-h}{\left (x^2+y^2+(z-h)^2\right )^{3/2}} + \frac{z+h}{\left (x^2+y^2+(z+h)^2\right )^{3/2}} \right )

The easiest way to set V = V0 on the spherical surface is to pick two points on the sphere's surface and enforce the condition there. In fact, let us pick the points (0,0,h+a) and (0,0,h-a), i.e. on the top and the bottom of the sphere, we set V = V0. This gives us at (0,0,h+a)

4\pi\epsilon_0 V_0 = Q\left (\frac{1}{a} - \frac{1}{2h+a}\right ) + p_z \left (\frac{1}{a^2} + \frac{1}{(2h+a)^2}\right ),

and at (0,0,h-a)

4\pi\epsilon_0 V_0 = Q\left (\frac{1}{a} - \frac{1}{2h-a}\right ) + p_z \left (-\frac{1}{a^2} + \frac{1}{(2h-a)^2}\right ).

We now have two equations with two unknowns that can be solved by elimination or by matrix methods. The equations can be written in matrix form as

4\pi\epsilon_0V_0 \left \{ \begin{matrix} 1 \\ 1 \end{matrix}\right \} = \left [\begin{matrix} \frac{1}{a}-\frac{1}{2h+a} & \frac{1}{a^2}+\frac{1}{(2h+a)^2} \\ \frac{1}{a}-\frac{1}{2h-a} & -\frac{1}{a^2}+\frac{1}{(2h-a)^2} \end{matrix} \right ] \left \{\begin{matrix} Q \\ p_z \end{matrix} \right \}

Solving this system, we get

Q = 4\pi\epsilon_0 V_0\frac{1}{D} \left ( \frac{2}{a^2} + \frac{1}{(2h+a)^2} - \frac{1}{(2h-a)^2}\right )

and

p_z = -4\pi\epsilon_0 V_0 \frac{1}{D} \left ( \frac{1}{2h-a} - \frac{1}{2h+a}\right ),

where

D =\left (\frac{1}{a}-\frac{1}{2h-a}\right ) \left (\frac{1}{a^2} + \frac{1}{(2h+a)^2}\right ) + \left (\frac{1}{a}-\frac{1}{2h+a}\right ) \left (\frac{1}{a^2}-\frac{1}{(2h-a)^2}\right )..

The illustration shows an example of the equipotential contours of the

system computed with this method.
Equipotential contours for a spherical topload of radius 0.3m and height (from center) of 1.0m.
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Equipotential contours for a spherical topload of radius 0.3m and height (from center) of 1.0m.


Of considerable practical interest is the capacitance, which is defined as

C = \frac{Q}{V_0}\,.

It is clear that from the expression for the charge on the sphere, the capacitance is given in terms of the sphere height h and radius a as

C = 4\pi\epsilon_0\frac{1}{D} \left ( \frac{2}{a^2} + \frac{1}{(2h+a)^2} - \frac{1}{(2h-a)^2}\right )

A plot of topload capacitance versus sphere radius and height from the groundplane. When the sphere radius is less than 10% of the height, the capacitance of the sphere is practically that of the isolated sphere (infinite height from the ground plane), that is

C\rightarrow 4\pi\epsilon_0 a\,

as

h \rightarrow \infty.
Plots of topload capacitance for several heights. The horizontal axis is the radius of the sphere and the left vertical axis is the capacitance. The right vertical axis is the field error estimate for h=0.55. The error plot is marked with an "O" where the field error is near 10%.
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Plots of topload capacitance for several heights. The horizontal axis is the radius of the sphere and the left vertical axis is the capacitance. The right vertical axis is the field error estimate for h=0.55. The error plot is marked with an "O" where the field error is near 10%.
This is because the dipole contribution

to the field decays very quickly with distance (as 1 / r3) and we are left only with the contribution from the charge Q.

The question remains, however: how good is the underlying field solution from which we compute the capacitance? Fortunately, because we have the whole solution for the potential everywhere in space, we can test other points on the sphere and see how well the boundary conditions are satisfied. Let us choose to test the error at the point (0,a,h). Inserting the solution for Q,pz and computing a normalised error Ierror = (V(0,a,h) - V0) / V0, we can check the quality of the solution as a and h are varied (the error plot here is for a height of 0.55m). The illustration shows that the approximation is very good as long as the height is more than about 1.5 times the radius. The capacitance is likely to not suffer more than a few percent error as long as h / a > 1.5 or 2. This is because as the gap between the sphere and ground plane grows narrower, the simple charge/dipole model begins to break down as higher order effects come into play.

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