According to Maxwell's equations, static magnetic fields created by ferromagnets should create ever-increasing electric fields in insulators and constant current in conductors.
Equation 1 is Maxwell's full fourth equation, which is an extension of Ampere's law. It states that curl (rotation) in the magnetic field is created by both current density and a time varying electric field.
By replaced current density with equation 2 (sigma is the conductivity) we see that this is simply describing magnetic fields created by conventional current in conductors as well as by displacement current in devices like capacitors.
However, there's nothing stopping us from solving for the time varying electric field. For initial simplicity let's assume we're not in a conductor (conductivity is zero and so is the current density). This gives us equation 3, which clearly states curl in the magnetic field should always be associated with a time varying electric field. If our magnetic field curl is coming from a static magnet then we certainly aren't using a time varying E-field to generate the B-field, so the B-field must then be generating a time varying E-field. Weird.
As a quick aside, if the B-field is nonzero then the curl of the B-field must be nonzero due to the lack of magnetic monopoles (Maxwell's second equation). This is why a static magnet guarantees nonzero magnetic curl around it.
If we allow our scenario to take place inside of a conductor then our conductivity is nonzero, giving us equation 4. This equation implies the strength of the E-field will approach the value necessary to create a current density that would produce our B-field curl in the absence of our static magnet. If our conductivity is zero like before then the E-field can never produce the necessary current density aforementioned, which is why equation 3 implies the E-field will be increasing indefinitely.
Equation 1 is Maxwell's full fourth equation, which is an extension of Ampere's law. It states that curl (rotation) in the magnetic field is created by both current density and a time varying electric field.
By replaced current density with equation 2 (sigma is the conductivity) we see that this is simply describing magnetic fields created by conventional current in conductors as well as by displacement current in devices like capacitors.
However, there's nothing stopping us from solving for the time varying electric field. For initial simplicity let's assume we're not in a conductor (conductivity is zero and so is the current density). This gives us equation 3, which clearly states curl in the magnetic field should always be associated with a time varying electric field. If our magnetic field curl is coming from a static magnet then we certainly aren't using a time varying E-field to generate the B-field, so the B-field must then be generating a time varying E-field. Weird.
As a quick aside, if the B-field is nonzero then the curl of the B-field must be nonzero due to the lack of magnetic monopoles (Maxwell's second equation). This is why a static magnet guarantees nonzero magnetic curl around it.
If we allow our scenario to take place inside of a conductor then our conductivity is nonzero, giving us equation 4. This equation implies the strength of the E-field will approach the value necessary to create a current density that would produce our B-field curl in the absence of our static magnet. If our conductivity is zero like before then the E-field can never produce the necessary current density aforementioned, which is why equation 3 implies the E-field will be increasing indefinitely.
