 We have developed a theoretical model to study the interaction between two particles on a graphing sheet with a gap. This gap can be caused by external forces such as electric fields or thermal gradients. The model takes into account both quantum and thermal fluctuations in the electromagnetic field. We found that the presence of a gap causes the dispersion force to become more attractive, which could be used to create new types of devices. This article was authored by Galena L. Klamchitsky, Konstantin C. Korakoff, Vladimir M. Mostaponenko and others.