  We consider a U(1) gauge field theory with fermion fields (or with scalar
fields) that live in a space with $\delta$ extra compact dimensions, and we
compute the fermion-induced quantum energy in the presence of a constant
magnetic field, which is directed towards the x_3 axis. Our motivation is to
study the effect of extra dimensions on the asymptotic behavior of the quantum
energy in the strong field limit (eB>>M^{2}), where M=1/R. We see that the weak
logarithmic growth of the quantum energy for four dimensions, is modified by a
rapid power growth in the case of the extra dimensions.
