We theoretically study the dependence of the intrinsic detection efficiency (IDE) of superconducting single photon detector on the applied current $I$ and magnetic field $H$. We find that the current, at which the resistive state appears in the superconducting film, depends on the position of the hot spot (region with suppressed superconductivity around the place where the photon has been absorbed) with respect to the edges of the film. It provides inevitable smooth dependence IDE(I) when IDE $sim 0.05-1$ even for homogenous straight superconducting film and in the absence of fluctuations. When IDE $lesssim 0.05$ much sharper current dependence comes from the fluctuation assisted vortex entry to the hot spot located near the edge of the film. We find that weak magnetic field strongly affects IDE when the photon detection is connected with fluctuation assisted vortex entry (IDE$ll 1$) and it weakly affects IDE when the photon detection is connected with the current induced vortex entry to the hot spot or nucleation of the vortex-antivortex pair inside the hot spot (IDE$sim 0.05-1$).
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