Question

For which condition is Ampere's law not valid?

Answer 1

Ampere's Law is a fundamental principle in electromagnetism, but it has limitations in certain conditions. It states that the magnetic field around a closed loop is proportional to the electric current passing through the loop. The law, in its simple form, is valid when the current is steady and the situation involves classical, ideal conditions.

However, Ampere's Law does not hold in the following cases:

1. Time-varying Electric Fields:
   

   Ampere's Law in its simple form assumes that the electric current is constant. However, if the electric field is changing with time (for example, in the case of a time-varying electric field), the law doesn’t account for the magnetic field generated by these changing fields. This situation requires the inclusion of Maxwell's correction to Ampere's Law, which includes a term related to the rate of change of the electric field. The full form of Ampere's Law in this case is:

   \[
   \nabla \times \mathbf{B} = \mu_0 \mathbf{J} + \mu_0 \epsilon_0 \frac{\partial \mathbf{E}}{\partial t}
   \]

   where \( \mu_0 \) is the permeability of free space, \( \mathbf{J} \) is the current density, and \( \frac{\partial \mathbf{E}}{\partial t} \) is the time rate of change of the electric field.

1. Non-static Current Distributions:
   

   When the current is not steady or when it’s in a dynamic or oscillating form (e.g., in alternating currents or when current is varying rapidly), the simple form of Ampere's Law becomes inadequate. For these cases, again, the modified version with Maxwell’s correction must be used.

In summary, Ampere's Law is not valid when the electric field is time-varying. To cover these cases, we need to use the full Maxwell-Ampere equation, which accounts for both the current and the changing electric field.