Final answer:
In classical electromagnetism, Maxwell's equations describe the relationship between the changing electric and magnetic fields. The condition of a zero time derivative of the electric field ( ∂E/∂t=0 ) does not imply that the magnetic field’s time derivative (∂B/∂t) is also zero, as they can be independently varied.
Step-by-step explanation:
The relationship between electric field (E) and magnetic field (B) changes in time is governed by Maxwell's equations, which represent one of the cornerstones of classical electromagnetism. Specifically, Faraday's law of induction implies that a changing magnetic field induces an electric field. Conversely, Maxwell added the displacement current term to Ampere's law, which implies that a changing electric field can produce a magnetic field.
If the partial derivative of the electric field with respect to time (∂E/∂t) is zero, that does not necessarily mean that the partial derivative of the magnetic field with respect to time (∂B/∂t) is also zero. The absence of a time-varying electric field does not imply the absence of a time-varying magnetic field and vice versa, as these are independent phenomena even though they are related through Maxwell's equations.
For example, in the case of electromagnetic waves, a changing electric field produces a changing magnetic field and vice versa, allowing the wave to propagate through free space. However, in a static case where (∂E/∂t=0), we can still have scenarios where magnetic fields change in time due to other causes aside from the electric field changes, such as the presence of time-varying currents.