Casimir surface force on a dilute dielectric ball

The Casimir surface force density F on a dielectric dilute spherical ball of radius a, surrounded by a vacuum, is calculated at zero temperature. We treat (n-1)(n being the refractive index) as a small parameter. The dispersive properties of the material are taken into account by adopting a simple model for the dispersion relation, involving a sharp high frequency cutoff at ω=ω₀. For a nondispersive medium there appears (after regularization) a finite, physical force Fⁿᵒⁿᵈⁱˢᵖ which is repulsive. By means of a uniform asymptotic expansion of the Riccati-Bessel functions we calculate Fⁿᵒⁿᵈⁱˢᵖ up to the fourth order in 1/ν. For a dispersive medium the main part of the force Fᵈⁱˢᵖ is also repulsive. The dominant term in Fᵈⁱˢᵖ is proportional to (n-1)²ω₀³/a, and will under usual physical conditions outweigh Fⁿᵒⁿᵈⁱˢᵖ by several orders of magnitude.