The Sombor index for graphs is given by Gutman in 2021. Since Hypergraphs can more accurately describe certain chemical scenarios. Then it has been proposed to generalize the Sombor index from graphs to hypergraphs. Recently, Shetty and Bhat defined the Sombor index \( SO(H) \) of a hypergraph \( H \) as \( SO(H)=\sum\limits_{e_{i} \in E(H)}\sqrt{\sum\limits_{u \in e_{i}}d_{H}(u)^{2}} \), where \( d_{H}(u) \) is the degree of the vertex \( u \) of \( H \). In this paper, we study the Sombor indices of uniform hypergraphs by hypergraph operations. The extremal hypergraph with minimum Sombor index is obtained among uniform hypertrees with maximum degree \( \Delta \geq 3 \), and the corresponding value of minimum Sombor index is also obtained. Furthermore, we consider the Sombor index for uniform unicyclic hypergraphs. The extremal hypergraph with maximum(minimum) Sombor index for uniform unicyclic hypergraphs is given, and the corresponding values for maximum(minimum) Sombor index are also given.