Repelling Conductors with Charge? Part 1

Repelling Conductors with Charge? Part 1

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Note: this post will soon be updated with diagrams, thank you for your patience!

1. Background

Opposites attract, likes repel. Straight off the first page of Electrostatics 101, this forms the ground rule for all basic interactions between charged objects. We can be more precise: the exact force between two objects with charges q_1 and q_2 separated by a distance of r equals

    \begin{equation*} F=\frac{1}{4\pi\epsilon_0} * \frac{q_1q_2}{r^2} \end{equation*}

This is known as Coulomb’s Law. Here, \epsilon_0 is an important physical constant related to how well electric field can permeate through a vacuum. The important part is just that it is constant.

Now, what happens when one of the two objects is neutral, i.e. has net charge equal to 0? Plugging 0 into Coulomb’s law may lead us to believe that there will be no net force, but this is not quite true. A famous demonstration in introductory physics uses a powered Van de Graaff generator, a metal device capable of building up charge, and brings it close to an electrically neutral metal sphere.

Rather than experiencing no force due to its lack of net charge, the sphere is actually pulled towards the generator. In another counterintuitive property, this attraction occurs regardless of whether the Van de Graaff generator had built up positive or negative charge: it does not switch to a repulsive force upon inverting the charge. So why does this happen?

The key word here is metal. Most metals are good electrical conductors, which means they allow charges to flow within them. Ordinarily, these charges would be evenly distributed in a neutral object, so no net charge is measured in any location. However, if the Van de Graaff generator is negatively charged, this provides an impulse for the charges in the neutral sphere to move in response; negative charges are repelled by the generator and move away, leaving the side closer to the generator positively charged.

Now, this is where Coulomb’s law comes in. The positive charges closer to the generator are attracted by the negative generator, while the farther negative charges are repelled. However, because the electric force gets stronger with closer distance, the attraction ends up overpowering the repulsion, leading to a net attractive force. Checking the case of a positively charged generator is left to the reader; the same reasoning should produce another net attractive force.

2. Repulsive Forces?

Now, here is where my post truly starts. When I learned of this property, I felt a little strange that this force should seemingly always be attractive: due to the very symmetric nature of charges, one would expect than any sort of attractive electrical force should have a repulsive counterpart. So I tried to think of a situation where a neutral conductor could be repelled by charged object through the same mechanism of polarization.

To do this, let us revisit the original case, but this time thinking in terms of electric fields. Whenever a neutral conductor is immersed in an electric field, the charges will move in response to it: specifically, positive charges will be pushed in the direction of the field, whereas negative charges will experience a force in the opposite direction. If the field is “more or less” pointing the same way throughout, this will result in a visible polarization of the conductor, with positive charges at one end and negative charges at the other. Now, we should ask ourselves: why did this result in an attractive net force? The key part about the original setup is that if you were to move along in the direction of the electric field, the strength of the field itself changes. Take a positively charged generator that emanates an outward electric field. As you move outward along the field lines, the field gets weaker; hence, the closer attractive force is able to win out.

Now, we clearly see what we need to do to produce repulsion instead: what if there were a way to use positive charges to produce a field that got stronger along its direction, rather than weaker? If this were possible, the farther repulsive force would overpower the closer attractive force, producing a net repulsion.

The first thing that occurred to me is that if we use a bounded/finite charge to generate this field, then it cannot continue to grow indefinitely: to see this, we can bound the volume of charge within a sphere, and consider what happens when you leave that sphere and move away from it. It is intuitively clear that once you get “far enough away,” this charge will just “look like” a spherical dot: all nuances will be smoothed out, and the field will seem to decay according to Coulombs law. In other words, if there is a repulsive force, it cannot last indefinitely: it is limited to a certain range around the charged object. Once that range is exceeded, the force becomes attractive again.

Now, we just have to think of a charge distribution that will permit at least some range of repulsion. I first tried to visualize a strange “crescent moon” shape of charge, where the corners would provide an extra strong field when they are approached. However, the net field at a particular point depends not just on the magnitudes of constituent fields, but also how much they cancel each other through their directions. Through this observation, I realized that there was a much simpler arrangement that exploited this fact.

Consider a simple arrangement of two positive charges, each equalling q, separated by some distance 2d as shown above. We will consider the behavior of the electric field along the line (actually plane in 3D) that symmetrically bisects these charges. Pick a point located a distance of x from the midpoint of the charges. We can calculate the field generated by one of the charges as

    \begin{equation*} E_1(r) = \frac{1}{4\pi\epsilon_0} * \frac{q}{r^2+\left(\frac{2d}{2}\right)^2} = \frac{q}{4\pi\epsilon_0} * \frac{1}{r^2+d^2} \end{equation*}

A similar field E_2(r) is generated by the second charge. However, their fields are not perfectly aligned: the components perpendicular to the bisecting line are exactly cancelled, while the component parallel to the line doubles up. So the net field E(r) is

    \begin{equation*} E(r) = 2\cos(\theta(r))*E_1(r) \end{equation*}

where \theta(r) is the angle each field forms with the horizontal. It is clear that \cos(\theta(r)) = \frac{r}{\sqrt{r^2+d^2}}, so substituting yields

    \begin{equation*} E(r) = \frac{q}{2\pi\epsilon_0} * \frac{r}{\left(r^2+d^2\right)^{\frac{3}{2}}} \end{equation*}

Remember why we are calculating this field: if a conductor is placed between the charges at a distance of r, it will be subject to an electric field of E(r), inducing a polarization that will cause either a repulsive or attractive force. To find out which, we must analyze E(r): if E(r) increases upon moving away from the charges (increasing r), then the conductor will be repelled. Taking the derivative of E(r) with respect to r provides just this information: a positive derivative indicates repulsion, while a negative derivative indicates attraction. For simplicity, we will ignore \frac{q}{2\pi\epsilon_0}: it is simply a positive constant and will not affect the sign of the derivative.

    \begin{equation*} \frac{d}{dr} \frac{r}{\left(r^2+d^2\right)^{\frac{3}{2}}} > 0 \rightarrow \frac{\left(r^2+d^2\right)^{\frac{3}{2}} - 3r^2\left(r^2+d^2\right)^{\frac{1}{2}}}{\left(r^2+d^2\right)^{3}} > 0 \end{equation*}

    \begin{equation*} \rightarrow \left(r^2+d^2\right)^{\frac{3}{2}} - 3r^2\left(r^2+d^2\right)^{\frac{1}{2}} > 0 \end{equation*}

    \begin{equation*} [\rightarrow r^2+d^2-3r^2 = d^2-2r^2>0 \end{equation*}

    \begin{equation*} \rightarrow r<\frac{d}{\sqrt{2}} \end{equation*}

So, it appears that there is a repulsive force! Given two equal charges separated by a distance of 2d, one can place a small conductor between them at a distance of r from the center. If r is within a range of \frac{d}{\sqrt{2}}, the conductor will be repelled from the charges; after this threshold, attractive forces will resume.

3. Caveats

As of now, this model is quite simple, and the second installment of this post will provide more mathematical refinement. To start off, there is an important question regarding the field generated by these charges: we only calculated this field on a mathematical, infinitely thin plane separating the two charges. However, real conductors are 3-dimensional, and will take up some space above and below this plane. These points are no longer symmetric across the charges, so the electric fields there do not point directly horizontally. One can ask: does this skewing of the electric fields affect the conductor’s polarization in any meaningful way, opposing the repulsion?

As it turns out, if you make the conductor very thin along the axis of symmetry of the charges, this “sideways skew” becomes arbitrarily small and is negligible with respect to the dimensions of the conductor. In the following post, I will discuss this result in further detail, along with attempting to actually quantify the net force on the polarized conductor. Stay tuned!

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