The magnetic field is a non-conservative vector field. In classical electromagnetism, the magnetic field is described by Maxwell's equations, which include Faraday's law of electromagnetic induction and Ampère's law with Maxwell's addition.
Faraday's law states that a changing magnetic field induces an electric field, while Ampère's law describes how a current or changing electric field generates a magnetic field. These laws imply that the magnetic field does not possess a scalar potential function in the same way that the electric field does.
In contrast, conservative vector fields, such as the gravitational field, can be expressed as the gradient of a scalar potential. This property allows conservative fields to have certain important properties, such as the independence of the path taken in line integrals (e.g., work done against gravity only depends on the initial and final positions). Non-conservative fields, like the magnetic field, do not have this property.
It is worth noting that the magnetic vector potential, which is related to the magnetic field through mathematical equations, can be thought of as having some similarities to a scalar potential. However, it is important to distinguish between the magnetic vector potential and the magnetic field itself. The magnetic field, as a vector quantity, does not possess a scalar potential and is considered non-conservative.