The question asks about a fundamental law in physics that describes the relationship between electric current and potential difference across a metallic conductor when the temperature is constant. We are provided with four options to identify this law:

1. Ampere's law: This law relates the integrated magnetic field around a closed loop to the electric current passing through it. It is a fundamental law of magnetism and does not pertain to the direct relationship between current and potential difference in a conductor.
2. Ohm's law: This is the correct answer, as Ohm’s law states that the current (\(I\)) flowing through a conductor between two points is directly proportional to the voltage (\(V\)) across the two points, provided the temperature remains constant. Mathematically, it is expressed as: \(V = IR\), where \(R\) is the resistance of the conductor.
3. Coulomb's law: This law describes the electrostatic interaction between electrically charged particles. It is not related to the direct relationship between current and potential difference.
4. Joule's law: This law concerns the heat produced by an electric current passing through a conductor. It is not directly describing the relationship between current and potential difference.

Hence, the correct answer is Ohm's law.

To summarize: Ohm’s law is a pivotal law in the field of electricity, fundamental to understanding how electrical circuits operate. It establishes a direct proportionality between voltage and current in a metallic conductor, controlling for resistance, when the conditions are maintained at a constant temperature.