1. **Problem Statement:** Determine the current $I_y$ in the given circuit using mesh analysis. 2. **Understanding Mesh Analysis:** Mesh analysis involves writing Kirchhoff's Voltage Law (KVL) equations for independent loops (meshes) in the circuit. The sum of voltage drops and rises around each mesh equals zero. 3. **Identify Meshes and Assign Currents:** - Assign mesh currents $I_1$, $I_2$, etc., clockwise in each loop. - Note the current source of 2 A sets a mesh current directly or relates mesh currents. 4. **Write KVL Equations:** - For each mesh, sum voltage drops ($IR$) and voltage sources. - Use Ohm's law: voltage drop across resistor $R$ is $IR$ where $I$ is mesh current through $R$. 5. **Apply Current Source Constraint:** - The 2 A current source fixes the current in its branch, so $I_y = 2$ A if $I_y$ is the current through that branch. 6. **Analyze the Circuit:** - Since $I_y$ is the current through the 2 $ m \, \Omega$ resistor near the 2 V source and the current source is 2 A in that branch, by the definition of current source, $I_y = 2$ A. 7. **Final Answer:** $$I_y = 2$$ This is because the current source enforces the current in that branch directly.