We establish both upper and lower bounds on the electric field in the case where two circular conductivity inclusions are very close but not touching. We also obtain such bounds when a circular inclusion is very close to the boundary of a circular domain which contains the inclusion. The novelty of these estimates, which improve and make complete our earlier results in [H. Ammari, H. Kang, M. Lim, Gradient estimates for solutions to the conductivity problem, Math. Ann. 332 (2005) 277-286], is that they give an optimal information about the blow-up of the electric field as the conductivities of the inclusions degenerate. (C) 2007 Elsevier Masson SAS. All rights reserved.