  In this work, we investigate the Josephson effect between a two-band
superconductor either with the $s++$ (two energy gaps have the same sign and
are fully gapped) pairing symmetry or $s\pm$ (two energy gaps have $\pi$ phase
difference and are fully gapped) pairing symmetry and a conventional s-wave
superconductors. The ground state, critical current, plasma modes, flux flow
dynamics, and response to external ac electric field, possible soliton
solutions are investigated. For junctions with the charge neutrality breaking,
we find a new plasma mode for junctions, which gives rise to new resonance
peaks in the Josephson flux flow region. Because of the frustrated interaction
in junctions with $s\pm$ pairing symmetry, time-reversal-symmetry (TRS) can be
broken if the frustration is optimized. In the TRS broken (TRSB) state, there
is a non-trivial phase difference between the two Josephson tunnelling
channels, which results in a non-trivial interference. Furthermore, we find a
novel massless plasma mode at the TRSB transition for junctions with the charge
neutrality breaking. In the TRSB state, a spontaneous magnetic flux appears
where there is a spatial inhomogeneity in the Josephson coupling, thus provides
a possible smoking-gun evidence for the underlying pairing symmetry.
