  We consider the generalization to String and M-theory of the Melvin solution.
These are flux p-branes which have (p+1)-dimensional Poincare invariance and
are associated to an electric (p+1)-form field strength along their
worldvolume. When a stack of Dp-branes is placed along the worldvolume of a
flux (p+3)-brane it will expand to a spherical D(p+2)-brane due to the
dielectric effect. This provides a new setup to consider the gauge
theory/gravity duality. Compactifying M-theory on a circle we find the exact
gravity solution of the type IIA theory describing the dielectric expansion of
N D4-branes into a spherical bound state of D4-D6-branes, due to the presence
of a flux 7-brane. In the decoupling limit, the deformation of the dual field
theory associated with the presence of the flux brane is irrelevant in the UV.
We calculate the gravitational radius and energy of the dielectric brane which
give, respectively, a prediction for the VEV of scalars and vacuum energy of
the dual field theory. Consideration of a spherical D6-brane probe with n units
of D4-brane charge in the dielectric brane geometry suggests that the dual
theory arises as the Scherk-Schwarz reduction of the M5-branes (2,0) conformal
field theory. The probe potential has one minimum placed at the locus of the
bulk dielectric brane and another associated to an inner dielectric brane
shell.
