for a conservative field ….....,,,,,.,.,. .,......

∫E.dl=? .. and why

The vanishing of closed line integrals means that the field is conservative.

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0=→E×→∇, the "physical interpretation" is the the electric field is irrotational, i.e. it has no "vortices". The, more valuable, mathematical implication is that there is a scalar potential whose gradient is the electric field.