The linear transient eddy current electromagnetic field problem for determining solution is characterized by replacing the interface condition of the tangential component of the electric field intensity by the interface condition of the normal component of the magnetic flux density, and ignoring the displacement current in the constraint equations. In is a fundamental theoretical problem whether the solution to the irregular problem for determining solution in unique and stable. In this paper, we introduce a scalar potential function in the nonconducting region,and prove the existence of an auxiliary function which is important to the problem. After deriving the energy estimate inequality, the uniqueness and stability of the solution to the linear transient eddy current electromagnetic field problem for determining solution are proved. The results of the paper can provide theoretical basis for analyzing and calculating the linear transient eddy current electromagnetic field problem. As an application example, we present the analytical solution to the eddy current problem that a single-turn coil driven by solitary pulse current is arranged coaxially with a spherical conductor.