To solve this problem, we need to understand the behavior of current, drift velocity, electric field, and current density in a metallic conductor with a non-uniform cross-section.$\\$Given: A constant potential difference is applied across a metallic conductor of non-uniform cross-section.$\\$Let's analyze each option:$\\$• Current:$\\$In a conductor, the current $(I)$ is defined as the rate of flow of charge. According to the principle of conservation of charge, the current remains constant throughout the conductor, even if the cross-section is non-uniform.$\\$• Drift Velocity:$\\$Drift velocity $(v_d)$ is the average velocity of charge carriers in a conductor. It is given by $v_d = \frac{I}{nAe},$ where $n$ is the number density of charge carriers, $A$ is the cross-sectional area, and $e$ is the charge of an electron. Since $A$ varies, $v_d$ will also vary.$\\$• Electric Field:$\\$The electric field $(E)$ in a conductor is related to the potential difference and the length of the conductor. In a non-uniform conductor, the electric field is not constant because it depends on the local potential gradient, which can vary with the cross-section.$\\$• Current Density:$\\$Current density $(J)$ is defined as the current per unit area, $J = \frac{I}{A}.$ Since the area $(A)$ changes along the conductor, the current density will also change.$\\$Conclusion:$\\$The quantity that remains constant along the conductor is the current. Therefore, the correct option is Option 1: current.