Anti-gravity effect from charged capacitor? (correction) A Reddit user called GratefulForGodGift has hypothesized that an electric field could warp spacetime.
Consider the electrostatic field energy $\epsilon$ stored in a parallel plate capacitor given by:
$$\epsilon = \frac{1}{2}k \epsilon_0E^2Ax,$$
where $k$ is the relative permittivity of the dielectric, $E$ is the applied electric field strength and $x$ is the separation distance of the plates.
The energy is stored in the tension between the displaced positive nuclei and the displaced negative electrons in the dielectric material. Thus there is negative pressure inside the dielectric parallel to the applied electric field.
The pressure inside the capacitor, $P_x$, is given by
\begin{eqnarray}
P_x &=& \frac{F_x}{A},\\
&=& -\frac{1}{A}\frac{d\epsilon}{dx},\\
&=& -\frac{1}{2}k\epsilon_0E^2.
\end{eqnarray}
John Baez gives a simplified version of Einstein's equations in terms of the shrinking of a volume $V$ of test particles around a region of space containing mass density and pressure:
$$\left.\frac{\ddot{V}}{V}\right\vert_{t=0}=-4\pi G(\rho_m+(\frac{1}{c^2}(P_x+P_y+P_z)).$$
In order to cancel the effect of the positive mass density $\rho_m$ we need a sufficiently negative pressure $P_x$ so that:
\begin{eqnarray}
\frac{P_x}{c^2}&=&-\rho_m,\\
E^2 &=& \frac{2\rho_m c^2}{k\epsilon_0}.
\end{eqnarray}
Let us assume that:
\begin{eqnarray}
c &\approx& 10^8 \rm{\ m}/\rm{s},\\
\epsilon_0 &\approx& 10^{-11} \rm{\ F}/\rm{m},\\
k &\approx& 10^4,\\
\rho_m &\approx& 10^3 \rm{\ kg}/\rm{m}^3.\\
\end{eqnarray}
We find that we can nullify the effect of gravity with an electric field E given by
$$E \approx 10^{13} \rm{\ V}/\rm{m}.$$
This is a large electric field but significantly less than the Schwinger limit $E_c\approx 10^{18}$ V/m where the electromagnetic vacuum breaks down.
Is there any positive pressure in the system which would counteract the anti-gravity effect of the negative pressure in the dielectric?
Correction
I should have included the electric field mass density $\rho^\epsilon_m$ given by
$$\rho^\epsilon_m = \frac{1}{2}\frac{k\epsilon_0E^2}{c^2}$$
in Einstein's Equations. So that we have
\begin{eqnarray}
\left.\frac{\ddot{V}}{V}\right\vert_{t=0}&=&-4\pi G(\rho_m+\rho^\epsilon_m+\frac{1}{c^2}P_x),\\
&=& -4\pi G(\rho_m+\frac{1}{2}\frac{k\epsilon_0E^2}{c^2}-\frac{1}{2}\frac{k\epsilon_0E^2}{c^2}),\\
&=& -4\pi G\rho_m.
\end{eqnarray}
Therefore the mass density of the electrostatic energy in the capacitor cancels out the anti-gravity effect of the negative pressure in the dielectric.
 A: 
Can the negative pressure in a capacitor cancel its mass?

No.
First, if there is “dielectric material” inside the capacitor, one has to compare characteristic value of electric field not with the Schwinger limit but with values at which dielectric breakdown occurs and those are much lower than values at which the energy density of electrostatic field becomes comparable to rest energy density of the dielectric. And while there could be interesting electroelastic effects in materials, the absolute magnitude of stresses appearing would be tiny in comparison with the rest energy density of the material. So only for vacuum capacitors the electric fields can produce stresses with gravitational effects comparable to the effects of energy densities.
Second, the stress–energy tensor of (vacuum) electromagnetic field is traceless:
$$ T^{\mu}{}_\mu=0.$$
This means that in a purely electrostatic configuration stress tensor eigenvalue corresponding to negative pressure would be accompanied by two positive eigenvalues of the same absolute value so that
$$
P_x+P_y+P_z=W,
$$
where $W$ is the energy density of electromagnetic field. In other words, the net effect of all pressure components of electromagnetic field does not cancel but doubles gravitational effects of field energy density.
Third point, is that the stresses of the electromagnetic fields would lead to the elastic stresses in the mechanical structure of capacitor (in  plates, walls, etc.) surrounding the field and gravitational effects of these stresses would largely compensate for the gravitational effects of the field stresses themselves. A relevant concept here is Tolman's paradox: suppose that part of the mass inside some container udergoes conversion into radiation, as a result the stress-energy tensor of matter changes so that now there are large pressure components. But because the walls of the container must now develop negative pressure (tension) to contain this radiation the net gravitational field outside the container remains unchanged.
All these points do not mean that the pressures (positive or negative) of electromagnetic field have no gravitational effects at all but just that there cannot be a simple “mass negator” type device. Note that the single component equation could not contain all the information of Einstein field equations but only one most directly relevant in the Newtonian-like setting. The effects of EM field pressure would thus be more subtle, relevant for e.g. higher multipoles or for coupling (generation/absorption/amplification) with gravitational waves.
For more discussion of the gravitational effects of stresses/pressures aimed at students of general relativity see:

*

*Ehlers, J., Ozsváth, I., & Schucking, E. L. (2006). Active mass under pressure. American journal of physics, 74(7), 607-613, doi:10.1119/1.2198881, arXiv:gr-qc/0505040.

