A charged pendulum suspended in mechanical equilibrium by an insulated string and deflected by a uniform electric field was investigated theoretically and experimentally. When the uniform electric field is generated by a pair of large parallel conducting plates, there is an additional force of attraction between the pendulum and its corresponding image charges, which can dramatically affect the pendulum's equilibrium deflection. Consequently, the deflection depends non-linearly on the pendulum's net charge and the plate potentials, which results in additional unstable-equilibrium solutions, saddle-point solutions, or conditions for which no equilibrium occurs. Inclusion of this experiment within introductory and/or advanced undergraduate physics courses is discussed.

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According to Earnshaw's Theorem (Ref. 19), a purely electrostatic force field cannot create a stable mechanical equilibrium in all three dimensions, levitating a charged body against gravity. However, the string's tension force on the bob is not electrostatic, allowing for the stable equilibrium to occur.
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Poisson's equation is still satisfied when a voltage difference is applied to the plates. The physical surface charge density associated with the image charge σI yields an equipotential surface on each plate when combined with the potential associated with the physical pendulum charge. Separately, the surface charge density associated with the applied voltage difference between the plates σ0 can be added to σI, uniformly offsetting each plate's total potential.
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It takes a finite amount of time for the electromagnetic field to propagate back and forth, continuously updating each plate's charge distribution and producing an extra image charge every trip. Therefore, the charge distribution should be represented by large but finite number of image charges. However, the infinite sum of image-charge forces converges quickly and the partial sum of image-charge forces on the stationary pendulum is essentially indistinguishable (Ref. 15) from that of the infinite sum.
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We used the software simion (Ref. 20) to numerically solve Laplace's equation using the actual finite-plate dimensions under the most extreme experimental conditions considered in our results, 0.1 m deflection with a plate separation of 0.4 m. We found that the difference between the numerically determined field magnitude and the prediction of Eq. (3) to be 0.01%.
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Supplementary Material

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