Study the property of $W^{\prime}$ at future $e^-p$ collider

As a strong candidate for new physics beyond the Standard Model, the exotic charged gauge boson $W^{\prime}$ has attracted extensive research interest. In this work we investigate the interactions of the $W^{\prime}$ boson at the electron-proton colliders. The process $e^- u \to \nu_e d$ and $e^- u \to e^\pm jjj$ with $t$-channel $W^{\prime}$ exchange are studied. The polarization of the initial-state electrons has a significant impact on the cross section of the studied process, while the angular distribution of the final-state leptons serves as an important observable for the interactions of the $W^{\prime}$ boson. In some specific regions of the parameter space, the detectable mass range for the $W^{\prime}$ boson can reach around 10 TeV, and the coupling strength can achieve a precision of approximately 1\% relative to the interaction strength of the Standard Model. Especially, $e^- u \to e^+ jjj$ process is forbidden within the Standard Model, which would constitute important evidence in the search for the Left-Right Symmetric Model.